Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 954098, 12 pages http://dx.doi.org/10.1155/2013/954098 Research Article Generalized Analytic Fourier-Feynman Transform of Functionals in a Banach Algebra F , 1 , 2 Jae Gil Choi, 1 David Skoug, 2 and Seung Jun Chang 1 1 Department of Mathematics, Dankook University, Cheonan 330-714, Republic of Korea 2 Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, USA Correspondence should be addressed to Seung Jun Chang; [email protected] Received 18 July 2013; Accepted 26 September 2013 Academic Editor: Kari Ylinen Copyright © 2013 Jae Gil Choi et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce the Fresnel type class F , 1 , 2 . We also establish the existence of the generalized analytic Fourier-Feynman transform for functionals in the Banach algebra F , 1 , 2 . 1. Introduction Let be a separable Hilbert space and let M() be the space of all complex-valued Borel measures on . e Fourier transform of in M() is defined by () (ℎ )≡ (ℎ )=∫ exp { ⟨ℎ, ℎ ⟩} (ℎ) , ℎ ∈ . (1) e set of all functions of the form (1) is denoted by F() and is called the Fresnel class of . Let (, , ]) be an abstract Wiener space. It is known [1, 2] that each functional of the form (1) can be extended to uniquely by () = ∫ exp {(ℎ, ) ∼ } (ℎ) , ∈ , (2) where (⋅, ⋅) ∼ is a stochastic inner product between and . e Fresnel class F() of is the space of (equivalence classes of) all functionals of the form (2). ere has been a tremendous amount of papers and books in the literature on the Fresnel integral theory and Fresnel classes F() and F() on abstract Wiener and Hilbert spaces. For an elementary introduction see [3, Chapter 20]. Furthermore, in [1], Kallianpur and Bromley introduced a larger class F 1 , 2 than the Fresnel class F() and showed the existence of the analytic Feynman integral of functionals in F 1 , 2 for a successful treatment of certain physical problems by means of a Feynman integral. e Fresnel class F 1 , 2 of 2 is the space of (equivalence classes of) all functionals on 2 of the following form: ( 1 , 2 )=∫ exp { { { 2 ∑ =1 ( 1/2 ℎ, ) ∼} } } (ℎ) , (3) where 1 and 2 are bounded, nonnegative, and self-adjoint operators on and ∈ M(). In this paper we study the functionals of the form (3) with ( 1 , 2 ) in a very general function space 2 , [0, ] ≡ , [0, ] × , [0, ]. e function space , [0, ], induced by generalized Brownian motion process, was introduced by Yeh [4, 5] and was used extensively in [6–13]. In this paper, we also construct a concrete theory of the generalized analytic Fourier-Feynman transform (GFFT) of functionals in a generalized Fresnel type class defined on 2 , [0, ]. Other work involving GFFT theories on , [0, ] include [6, 7, 9, 12, 13]. e Wiener process used in [1, 2, 14–17] is stationary in time and is free of driſt while the stochastic process used in this paper, as well as in [4, 6–13, 18], is nonstationary in time and is subject to a driſt (). It turns out, as noted in Remark 7 below, that including a driſt term () makes establishing the existence of the GFFT
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Hindawi Publishing CorporationJournal of Function Spaces and ApplicationsVolume 2013 Article ID 954098 12 pageshttpdxdoiorg1011552013954098
Research ArticleGeneralized Analytic Fourier-Feynman Transform ofFunctionals in a Banach Algebra F119886119887
11986011198602
Jae Gil Choi1 David Skoug2 and Seung Jun Chang1
1 Department of Mathematics Dankook University Cheonan 330-714 Republic of Korea2Department of Mathematics University of Nebraska-Lincoln Lincoln NE 68588-0130 USA
Correspondence should be addressed to Seung Jun Chang sejchangdankookackr
Received 18 July 2013 Accepted 26 September 2013
Academic Editor Kari Ylinen
Copyright copy 2013 Jae Gil Choi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We introduce the Fresnel type classF1198861198871198601 1198602
We also establish the existence of the generalized analytic Fourier-Feynman transformfor functionals in the Banach algebraF119886119887
1198601 1198602
1 Introduction
Let119867 be a separable Hilbert space and letM(119867) be the spaceof all complex-valued Borel measures on 119867 The Fouriertransform of 120590 inM(119867) is defined by
119891 (120590) (ℎ1015840
) equiv (ℎ1015840
) = int
119867
exp 119894 ⟨ℎ ℎ1015840⟩ 119889120590 (ℎ) ℎ1015840 isin 119867
(1)
The set of all functions of the form (1) is denoted byF(119867) andis called the Fresnel class of 119867 Let (119867 119861 ]) be an abstractWiener space It is known [1 2] that each functional of theform (1) can be extended to 119861 uniquely by
(119909) = int
119867
exp 119894(ℎ 119909)sim 119889120590 (ℎ) 119909 isin 119861 (2)
where (sdot sdot)sim is a stochastic inner product between 119867 and119861 The Fresnel class F(119861) of 119861 is the space of (equivalenceclasses of) all functionals of the form (2) There has beena tremendous amount of papers and books in the literatureon the Fresnel integral theory and Fresnel classes F(119861)and F(119867) on abstract Wiener and Hilbert spaces For anelementary introduction see [3 Chapter 20]
Furthermore in [1] Kallianpur and Bromley introduceda larger classF
11986011198602
than the Fresnel classF(119861) and showedthe existence of the analytic Feynman integral of functionals
in F11986011198602
for a successful treatment of certain physicalproblems by means of a Feynman integral The Fresnel classF11986011198602
of 1198612 is the space of (equivalence classes of) allfunctionals on 1198612 of the following form
119865 (1199091 1199092) = int
119867
exp
2
sum
119895=1
119894(11986012
119895ℎ 119909119895)
sim
119889120590 (ℎ) (3)
where1198601and119860
2are bounded nonnegative and self-adjoint
operators on119867 and 120590 isinM(119867)In this paper we study the functionals 119865 of the form (3)
with (1199091 1199092) in a very general function space 1198622
119886119887[0 119879] equiv
119862119886119887[0 119879] times119862
119886119887[0 119879] The function space 119862
119886119887[0 119879] induced
by generalized Brownian motion process was introducedby Yeh [4 5] and was used extensively in [6ndash13] In thispaper we also construct a concrete theory of the generalizedanalytic Fourier-Feynman transform (GFFT) of functionalsin a generalized Fresnel type class defined on1198622
119886119887[0 119879] Other
work involving GFFT theories on 119862119886119887[0 119879] include [6 7 9
12 13]The Wiener process used in [1 2 14ndash17] is stationary in
time and is free of drift while the stochastic process used inthis paper as well as in [4 6ndash13 18] is nonstationary in timeand is subject to a drift 119886(119905)
It turns out as noted in Remark 7 below that including adrift term 119886(119905) makes establishing the existence of the GFFT
2 Journal of Function Spaces and Applications
of functionals on 1198622119886119887[0 119879] very difficult However when
119886(119905) equiv 0 and 119887(119905) = 119905 on [0 119879] the general function space119862119886119887[0 119879] reduces to the Wiener space 119862
0[0 119879]
2 Definitions and Preliminaries
Let 119886(119905) be an absolutely continuous real-valued function on[0 119879] with 119886(0) = 0 1198861015840(119905) isin 1198712[0 119879] and let 119887(119905) be a strictlyincreasing continuously differentiable real-valued functionwith 119887(0) = 0 and 1198871015840(119905) gt 0 for each 119905 isin [0 119879]The generalizedBrownian motion process 119884 determined by 119886(119905) and 119887(119905) isa Gaussian process with mean function 119886(119905) and covariancefunction 119903(119904 119905) = min119887(119904) 119887(119905) For more details see[6 10 12] By Theorem 142 in [5] the probability measure120583 induced by 119884 taking a separable version is supportedby 119862119886119887[0 119879] (which is equivalent to the Banach space of
continuous functions 119909 on [0 119879]with 119909(0) = 0 under the supnorm) Hence (119862
119886119887[0 119879] B(119862
119886119887[0 119879]) 120583) is the function
space induced by119884whereB(119862119886119887[0 119879]) is the Borel120590-algebra
of 119862119886119887[0 119879] We then complete this function space to obtain
(119862119886119887[0 119879] W(119862
119886119887[0 119879]) 120583) whereW(119862
119886119887[0 119879]) is the set
of all Wiener measurable subsets of 119862119886119887[0 119879]
A subset 119861 of 119862119886119887[0 119879] is said to be scale-invariant
measurable provided 120588119861 is W(119862119886119887[0 119879])-measurable for all
120588 gt 0 and a scale-invariant measurable set 119873 is said to be ascale-invariant null set provided 120583(120588119873) = 0 for all 120588 gt 0A property that holds except on a scale-invariant null setis said to hold scale-invariant almost everywhere (s-ae) Afunctional119865 is said to be scale-invariantmeasurable provided119865 is defined on a scale-invariant measurable set and 119865(120588sdot) isW(119862119886119887[0 119879])-measurable for every 120588 gt 0 If two functionals
119865 and119866 defined on119862119886119887[0 119879] are equal s-ae we write 119865 asymp 119866
Let 11987112119886119887[0 119879] be the space of Lebesgue measurable func-
tions on [0 119879] given by
11987112
119886119887[0 119879] = V int
119879
0
|V (119904)|2119889119887 (119904) lt infin
int
119879
0
|V (119904)| 119889 |119886| (119904) lt infin
(4)
where |119886|(sdot) is the total variation function of 119886(sdot) Then11987112
119886119887[0 119879] is a separable Hilbert space with inner product
defined by
(119906 V)11987112
119886119887
= int
119879
0
119906 (119905) V (119905) 119889119887 (119905)
+ (int
119879
0
119906 (119905) 119889119886 (119905)) (int
119879
0
V (119905) 119889119886 (119905))
(5)
In particular note that 11990611987112
119886119887
equiv [(119906 119906)11987112
119886119887
]12
= 0 if and onlyif 119906(119905) = 0 ae on [0 119879]
Let 120601119895infin
119895=1be a complete orthonormal set in 11987112
119886119887[0 119879]
each of whose elements is of bounded variation on [0 119879] suchthat
int
119879
0
120601119895(119905) 120601119896(119905) 119889119887 (119905) =
0 119895 = 119896
1 119895 = 119896
(6)
Then for each V isin 11987112119886119887[0 119879] the Paley-Wiener-Zygmund
(PWZ) stochastic integral ⟨V 119909⟩ is defined by the followingformula
⟨V 119909⟩ = lim119899rarrinfin
int
119879
0
119899
sum
119895=1
(V 120601119895)119886119887
120601119895(119905) 119889119909 (119905) (7)
for all 119909 isin 119862119886119887[0 119879] for which the limit exists one can show
that for each V isin 11987112119886119887[0 119879] the PWZ stochastic integral
⟨V 119909⟩ exists for 120583-ae 119909 isin 119862119886119887[0 119879] and if V is of bounded
variation on [0 119879] then the PWZ stochastic integral ⟨V 119909⟩equals the Riemann-Stieltjes integral int119879
0
V(119905)119889119909(119905) for s-ae119909 isin 119862
119886119887[0 119879]
Remark 1 (1) For each V isin 11987112119886119887[0 119879] the PWZ stochastic
integral ⟨V 119909⟩ is a Gaussian random variable on 119862119886119887[0 119879]
with mean int1198790
V(119904)119889119886(119904) and variance int1198790
V2(119904)119889119887(119904)(2) For all 119906 V isin 11987112
119886119887[0 119879]
int
119862119886119887[0119879]
⟨119906 119909⟩⟨V 119909⟩ 119889120583 (119909)
= int
119879
0
119906 (119904) V (119904) 119889119887 (119904)
+ (int
119879
0
119906 (119904) 119889119886 (119904)) (int
119879
0
V (119904) 119889119886 (119904))
(8)
Hence we see that for all 119906 V isin 11987112119886119887[0 119879] int119879
0
119906(119904)V(119904)119889119887(119904) =0 if and only if ⟨119906 119909⟩ and ⟨V 119909⟩ are independent randomvariables
The following Cameron-Martin subspace of 119862119886119887[0 119879]
plays an important role throughout this paperLet
1198621015840
119886119887[0 119879] = 119908 isin 119862
119886119887[0 119879] 119908 (119905) = int
119905
0
119911 (119904) 119889119887 (119904)
for some 119911 isin 11987112119886119887[0 119879]
(9)
For 119908 isin 1198621015840119886119887[0 119879] let119863 1198621015840
119886119887[0 119879] rarr 119871
12
119886119887[0 119879] be defined
by the following formula
119863119908 (119905) =
119889120582119908
119889120582119887
(119905) (10)
where 119889120582119908119889120582119887denotes the Radon-Nikodym derivative of
the signed measure 120582119908
induced by 119908 with respect tothe Borel-Stieltjes measure 120582
119887induced by 119887 Then 1198621015840
119886119887equiv
1198621015840
119886119887[0 119879] with inner product
(1199081 1199082)1198621015840
119886119887
= int
119862119886119887[0119879]
⟨1198631199081 119909⟩⟨119863119908
2 119909⟩ 119889120583 (119909) (11)
is a separable Hilbert space
Journal of Function Spaces and Applications 3
Using (8) we observe that the linear operator given by(10) is an isometry In fact the inverse operator 119863minus1 11987112
119886119887[0 119879] rarr 119862
1015840
119886119887[0 119879] is given by
(119863minus1
119911) (119905) = int
119905
0
119911 (119904) 119889119887 (119904) (12)
Moreover the triple (1198621015840119886119887[0 119879] 119862
119886119887[0 119879] 120583) becomes an
abstract Wiener spaceThroughout this paper for 119908 isin 1198621015840
119886119887[0 119879] we will use the
notation (119908 119909)sim instead of ⟨119863119908 119909⟩We also use the followingnotations for 119908
1 1199082 119908 isin 1198621015840
119886119887[0 119879]
(1199081 1199082)119887= int
119879
0
1198631199081(119905) 119863119908
2(119905) 119889119887 (119905)
119908119887= radic(119908119908)
119887
(13)
Then 1198621015840119886119887[0 119879] with the inner product given by (13) is also a
separable Hilbert space It is easy to see that the two norms sdot 1198621015840
119886119887
and sdot 119887are equivalent Furthermore we have the
following assertions
(i) 119886(sdot) is an element of 1198621015840119886119887[0 119879]
(ii) For each 119908 isin 1198621015840119886119887[0 119879] the random variable 119909 997891rarr
(119908 119909)sim is Gaussian with mean (119908 119886)
119887and variance
1199082
119887
(iii) (119908 120572119909)sim = (120572119908 119909)sim = 120572(119908 119909)sim for any real number120572 119908 isin 1198621015840
119886119887[0 119879] and 119909 isin 119862
119886119887[0 119879]
(iv) Let 1199081 119908
119899 be a subset of 1198621015840
119886119887[0 119879] such that
int
119879
0
119863119908119894(119905)119863119908
119895(119905)119889119887(119905) = 120575
119894119895 where 120575
119894119895is the Kro-
necker delta Then the random variables (119908119894 119909)simrsquos are
independent
In this paper we adopt asmuch as possible the definitionsand notations used in [7 9 12 13] for the definitions ofthe generalized analytic Feynman integral and the GFFT offunctionals on 119862
119886119887[0 119879]
The following integration formula is used several times inthis paper
int
R
exp minus1205721199062 + 120573119906 119889119906 = radic120587120572
exp1205732
4120572
(14)
for complex numbers 120572 and 120573 with Re(120572) gt 0
3 The GFFT of Functionals in a BanachAlgebra F119886119887
11986011198602
LetM(1198621015840119886119887[0 119879]) be the space of complex-valued countably
additive (and hence finite) Borel measures on 1198621015840119886119887[0 119879]
M(1198621015840119886119887[0 119879]) is a Banach algebra under the total variation
norm and with convolution as multiplicationWe define the Fresnel type class F(119862
119886119887[0 119879]) of
functionals on 119862119886119887[0 119879] as the space of all stochastic
Fourier transforms of elements of M(1198621015840119886119887[0 119879]) that is
119865 isin F(119862119886119887[0 119879]) if and only if there exists a measure 119891 in
M(1198621015840119886119887[0 119879]) such that
119865 (119909) = int
1198621015840
119886119887[0119879]
exp 119894(119908 119909)sim 119889119891 (119908) (15)
for s-ae 119909 isin 119862119886119887[0 119879] More precisely since we will identify
functionals which coincide s-ae on 119862119886119887[0 119879]F(119862
119886119887[0 119879])
can be regarded as the space of all 119904-equivalence classes offunctionals of the form (15)
The Fresnel type class F(119862119886119887[0 119879]) is a Banach algebra
with norm
119865 =10038171003817100381710038171198911003817100381710038171003817= int
1198621015840
119886119887[0119879]
11988910038161003816100381610038161198911003816100381610038161003816(119908) (16)
In fact the correspondence 119891 997891rarr 119865 is injective carriesconvolution into pointwise multiplication and is a Banachalgebra isomorphism where 119891 and 119865 are related by (15)
Remark 2 The Banach algebra F(119862119886119887[0 119879]) contains sev-
eral interesting functions which arise naturally in quantummechanics Let M(R) be the class of C-valued countablyadditive measures on B(R) the Borel class of R For ] isinM(R) the Fourier transform ] of ] is a complex-valuedfunction defined on R by the following formula
] (119906) = intR
exp 119894119906V 119889] (V) (17)
Let G be the set of all complex-valued functions on[0 119879]timesR of the form 120579(119904 119906) =
119904(119906) where 120590
119904 0 le 119904 le 119879 is
a family fromM(R) satisfying the following two conditions
(i) for every 119864 isinB(R) 120590119904(119864) is Borel measurable in 119904
(ii) int1198790
120590119904119889119887(119904) lt +infin
Let 120579 isin G and let119867 be given by
119867(119909) = expint119879
0
120579 (119905 119909 (119905)) 119889119905 (18)
for s-ae 119909 isin 119862119886119887[0 119879] Then using the methods similar
to those used in [18] we can show that the function 120579(119905 119906)is Borel-measurable and that 120579(119905 119909(119905)) int119879
0
120579(119905 119909(119905))119889119905 and119867(119909) are elements of F(119862
119886119887[0 119879]) These facts are relevant
to quantum mechanics where exponential functions play aprominent role
Let119860 be a nonnegative self-adjoint operator on1198621015840119886119887[0 119879]
and 119891 any complex measure on 1198621015840119886119887[0 119879] Then the func-
tional
119865 (119909) = int
1198621015840
119886119887[0119879]
exp 119894(11986012119908 119909)sim
119889119891 (119908) (19)
belongs to F(119862119886119887[0 119879]) because it can be rewritten as
int1198621015840
119886119887[0119879]
exp119894(119908 119909)sim119889119891119860(119908) for 119891
119860= 119891 ∘ (119860
12
)minus1 Let 119860 be
self-adjoint but not nonnegative Then 119860 has the form
119860 = 119860+
minus 119860minus
(20)
4 Journal of Function Spaces and Applications
where both 119860+ and 119860minus are bounded nonnegative and self-adjoint operators
In this section we will extend the ideas of [1] to obtainexpressions of the generalized analytic Feynman integral andthe GFFT of functionals of the form (19) when119860 is no longerrequired to be nonnegative To do this we will introducedefinitions and notations analogous to those in [7 12 13]
Let W(1198622119886119887[0 119879]) denote the class of all Wiener mea-
surable subsets of the product function space 1198622119886119887[0 119879] A
subset 119861 of 1198622119886119887[0 119879] is said to be scale-invariant measurable
provided (12058811199091 12058821199092) (119909
1 1199092) isin 119861 is W(1198622
119886119887[0 119879])-
measurable for every 1205881gt 0 and 120588
2gt 0 and a scale-
invariantmeasurable subset119873 of1198622119886119887[0 119879] is said to be scale-
invariant null provided (120583 times 120583)((12058811199091 12058821199092) (119909
1 1199092) isin
119873) = 0 for every 1205881gt 0 and 120588
2gt 0 A property that
holds except on a scale-invariant null set is said to hold s-ae on 1198622
119886119887[0 119879] A functional 119865 on 1198622
119886119887[0 119879] is said to be
scale-invariant measurable provided 119865 is defined on a scale-invariant measurable set and 119865(120588
1sdot 1205882sdot) is W(1198622
119886119887[0 119879])-
measurable for every 1205881gt 0 and 120588
2gt 0 If two functionals
119865 and 119866 defined on 1198622119886119887[0 119879] are equal s-ae then we write
119865 asymp 119866We denote the product function space integral of a
W(1198622119886119887[0 119879])-measurable functional 119865 by
119864 [119865] equiv 119864119909[119865 (1199091 1199092)]
= int
1198622
119886119887[0119879]
119865 (1199091 1199092) 119889 (120583 times 120583) (119909
1 1199092)
(21)
whenever the integral existsThroughout this paper letCC
+and C
+denote the set of
complex numbers complex numbers with positive real partand nonzero complex numbers with nonnegative real partrespectively Furthermore for all 120582 isin C
+ 120582minus12 (or 12058212) is
always chosen to have positive real part We also assume thatevery functional 119865 on 1198622
119886119887[0 119879] we consider is s-ae defined
and is scale-invariant measurable
Definition 3 Let C2+equiv C+times C+and let C2
+equivC+timesC+ Let
119865 1198622
119886119887[0 119879] rarr C be such that for each 120582
1gt 0 and 120582
2gt 0
the function space integral
119869 (1205821 1205822)
= int
1198622
119886119887[0119879]
119865 (120582minus12
11199091 120582minus12
21199092) 119889 (120583 times 120583) (119909
1 1199092)
(22)
exists If there exists a function 119869lowast(1205821 1205822) analytic inC2
+such
that 119869lowast(1205821 1205822) = 119869(120582
1 1205822) for all 120582
1gt 0 and 120582
2gt 0 then
119869lowast
(1205821 1205822) is defined to be the analytic function space integral
of 119865 over 1198622119886119887[0 119879] with parameter 120582 = (120582
1 1205822) and for 120582 isin
C2+we write
119864an[119865] equiv 119864
an
119909[119865 (1199091 1199092)]
equiv 119864
an(12058211205822)
11990911199092[119865 (1199091 1199092)] = 119869
lowast
(1205821 1205822)
(23)
Let 1199021and 1199022be nonzero real numbers Let 119865 be a functional
such that 119864an[119865] exists for all 120582 isin C2+ If the following limit
exists we call it the generalized analytic Feynman integral of119865 with parameter 119902 = (119902
1 1199022) and we write
119864anf 119902[119865] equiv 119864
anf 119902119909[119865 (1199091 1199092)]
equiv 119864
anf(11990211199022)
11990911199092[119865 (1199091 1199092)] = lim
120582rarrminus119894 119902
119864an[119865]
(24)
where 120582 = (1205821 1205822) rarr minus119894 119902 = (minus119894119902
1 minus1198941199022) through values in
C2+
Definition 4 Let 1199021and 1199022be nonzero real numbers For 120582 =
(1205821 1205822) isin C2+and (119910
1 1199102) isin 1198622
119886119887[0 119879] let
119879 120582(119865) (119910
1 1199102) equiv 119879(12058211205822)(119865) (119910
1 1199102)
= 119864
an
119909[119865 (1199101+ 1199091 1199102+ 1199092)]
(25)
For 119901 isin (1 2] we define the 119871119901analytic GFFT 119879(119901)
119902(119865) of 119865
by the formula ( 120582 isin C2+)
119879(119901)
119902(119865) (119910
1 1199102) equiv 119879(119901)
(1199021 1199022)(119865) (119910
1 1199102)
= lim120582rarrminus119894 119902
119879 120582(119865) (119910
1 1199102)
(26)
if it exists that is for each 1205881gt 0 and 120588
2gt 0
lim120582rarrminus119894 119902
int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816
119879 120582(119865) (120588
11199101 12058821199102)
minus119879(119901)
119902(119865) (120588
11199101 12058821199102)
100381610038161003816100381610038161003816
1199011015840
119889 (120583 times 120583) (1199101 1199102)
= 0
(27)
where 1119901+11199011015840 = 1We define the 1198711analytic GFFT119879(1)
119902(119865)
of 119865 by the formula ( 120582 isin C2+)
119879(1)
119902(119865) (119910
1 1199102) = lim120582rarrminus119894 119902
119879 120582(119865) (119910
1 1199102) (28)
if it exists
We note that for 1 le 119901 le 2 119879(119901)119902(119865) is defined only s-ae
We also note that if 119879(119901)119902(119865) exists and if 119865 asymp 119866 then 119879(119901)
119902(119866)
exists and 119879(119901)119902(119866) asymp 119879
(119901)
119902(119865) Moreover from Definition 4
we see that for 1199021 1199022isin R minus 0
119864
anf 119902119909[119865 (1199091 1199092)] = 119879
(1)
119902(119865) (0 0) (29)
Next we give the definition of the generalized Fresnel typeclassF119886119887
11986011198602
Journal of Function Spaces and Applications 5
Definition 5 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] The generalized Fresnel
type classF11988611988711986011198602
of functionals on1198622119886119887[0 119879] is defined as the
space of all functionals 119865 on 1198622119886119887[0 119879] of the following form
119865 (1199091 1199092) = int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119909119895)
sim
119889119891 (119908) (30)
for some119891 isinM(1198621015840119886119887[0 119879]) More precisely since we identify
functionals which coincide s-ae on 1198622119886119887[0 119879] F119886119887
11986011198602
canbe regarded as the space of all 119904-equivalence classes offunctionals of the form (30)
Remark 6 (1) In Definition 5 let 1198601be the identity operator
on 1198621015840119886119887[0 119879] and 119860
2equiv 0 Then F 119886119887
11986011198602
is essentially theFresnel type class F(119862
119886119887[0 119879]) and for 119901 isin [1 2] and
nonzero real numbers 1199021and 1199022
119879(119901)
(1199021 1199022)(119865) (119910
1 1199102) = 119879(119901)
1199021
(1198650) (1199101) (31)
if it exists where 1198650(1199091) = 119865(119909
1 1199092) for all (119909
1 1199092) isin
1198622
119886119887[0 119879] and 119879(119901)
1199021
(1198650) means the 119871
119901analytic GFFT on
119862119886119887[0 119879] see [6 12](2) The map 119891 997891rarr 119865 defined by (30) sets up an algebra
isomorphism betweenM(1198621015840119886119887[0 119879]) andF119886119887
11986011198602
if Ran(1198601+
1198602) is dense in1198621015840
119886119887[0 119879] where Ran indicates the range of an
operator In this caseF 11988611988711986011198602
becomes a Banach algebra underthe norm 119865 = 119891 For more details see [1]
Remark 7 Let 119865 be given by (30) In evaluating119864119909[119865(120582minus12
11199091 120582minus12
21199092)] and 119879
(12058211205822)(119865)(1199101 1199102) = 119864
119909[119865(1199101+
120582minus12
11199091 1199102+ 120582minus12
21199092)] for 120582
1gt 0 and 120582
2gt 0 the expression
120595 (120582 119908)
equiv 120595 (1205821 1205822 1198601 1198602 119908)
= exp
2
sum
119895=1
[
[
minus
(119860119895119908119908)
119887
2120582119895
+ 119894120582minus12
119895(11986012
119895119908 119886)119887
]
]
(32)
occurs Clearly for 120582119895gt 0 119895 isin 1 2 |120595( 120582 119908)| le 1 for all
119908 isin 1198621015840
119886119887[0 119879] But for 120582 isin C2
+ |120595( 120582 119908)| is not necessarily
bounded by 1Note that for each 120582
119895isinC+with 120582
119895= 120572119895+ 119894120573119895 119895 isin 1 2
12058212
119895=
radicradic1205722
119895+ 1205732
119895+ 120572119895
2
+ 119894
120573119895
10038161003816100381610038161003816120573119895
10038161003816100381610038161003816
radicradic1205722
119895+ 1205732
119895minus 120572119895
2
120582minus12
119895=radic
radic1205722
119895+ 1205732
119895+ 120572119895
2 (1205722
119895+ 1205732
119895)
minus 119894
120573119895
10038161003816100381610038161003816120573119895
10038161003816100381610038161003816
radic
radic1205722
119895+ 1205732
119895minus 120572119895
2 (1205722
119895+ 1205732
119895)
(33)
Hence for 120582119895isinC+with 120582
119895= 120572119895+ 119894120573119895 119895 isin 1 2
10038161003816100381610038161003816120595 (120582 119908)
10038161003816100381610038161003816
= exp
2
sum
119895=1
[
[
[
[
minus
120572119895
2 (1205722
119895+ 1205732
119895)
(119860119895119908119908)
119887
+
120573119895
10038161003816100381610038161003816120573119895
10038161003816100381610038161003816
radic
radic1205722
119895+ 1205732
119895minus 120572119895
2 (1205722
119895+ 1205732
119895)
(11986012
119895119908 119886)119887
]
]
]
]
(34)
The right hand side of (34) is an unbounded functionof 119908 for 119908 isin 1198621015840
119886119887[0 119879] Thus 119864an[119865] 119864anf 119902[119865] 119879
120582(119865) and
119879(119901)
119902(119865) might not exist Thus throughout this paper we will
need to put additional restrictions on the complex measure119891 corresponding to 119865 in order to obtain our results for theGFFT and the generalized analytic Feynman integral of 119865
In view of Remark 7 we clearly need to impose additionalrestrictions on the functionals 119865 inF119886119887
11986011198602
For a positive real number 119902
0 let
Γ1199020
=
120582 = (120582
1 1205822) isinC2
+| 120582119895= 120572119895+ 119894120573119895
10038161003816100381610038161003816Im (120582minus12
119895)
10038161003816100381610038161003816=radic
radic1205722
119895+ 1205732
119895minus 120572119895
2 (1205722
119895+ 1205732
119895)
lt
1
radic21199020
119895 = 1 2
(35)
and let119896 (1199020 119908) equiv 119896 (119902
0 1198601 1198602 119908)
= exp
2
sum
119895=1
(21199020)minus121003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817119887119886119887
(36)
Then for all 120582 = (1205821 1205822) isin Γ1199020
10038161003816100381610038161003816120595 (120582 119908)
10038161003816100381610038161003816le exp
2
sum
119895=1
radic
radic1205722
119895+ 1205732
119895minus 120572119895
2 (1205722
119895+ 1205732
119895)
100381610038161003816100381610038161003816
(11986012
119895119908 119886)119887
100381610038161003816100381610038161003816
le exp
2
sum
119895=1
radic
radic1205722
119895+ 1205732
119895minus 120572119895
2 (1205722
119895+ 1205732
119895)
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817119887119886119887
lt 119896 (1199020 119908)
(37)
6 Journal of Function Spaces and Applications
We note that for all real 119902119895with |119902
119895| gt 1199020 119895 isin 1 2
(minus119894119902119895)
minus12
=
1
radic
100381610038161003816100381610038162119902119895
10038161003816100381610038161003816
+ sign (119902119895)
119894
radic
100381610038161003816100381610038162119902119895
10038161003816100381610038161003816
(38)
and (minus1198941199021 minus1198941199022) isin Γ1199020
For the existence of the GFFT of 119865 we define a subclass
F1199020
11986011198602
ofF 11988611988711986011198602
by 119865 isin F 119902011986011198602
if and only if
int
1198621015840
119886119887[0119879]
119896 (1199020 119908) 119889
10038161003816100381610038161198911003816100381610038161003816(119908) lt +infin (39)
where 119891 and 119865 are related by (30) and 119896 is given by (36)
Remark 8 Note that in case 119886(119905) equiv 0 and 119887(119905) = 119905 on [0 119879]the function space 119862
119886119887[0 119879] reduces to the classical Wiener
space 1198620[0 119879] and (119908 119886)
119887= 0 for all 119908 isin 1198621015840
119886119887[0 119879] =
1198621015840
0[0 119879] Hence for all 120582 isin C2
+ |120595( 120582 119908)| le 1 and for any
positive real number 1199020F 119902011986011198602
= F11986011198602
theKallianpur andBromleyrsquos class introduced in Section 1
Theorem 9 Let 1199020be a positive real number and let 119865 be an
element ofF 119902011986011198602
Then for any nonzero real numbers 1199021and
1199022with |119902
119895| gt 1199020 119895 isin 1 2 the 119871
1analytic GFFT of 119865 119879(1)
119902(119865)
exists and is given by the following formula
119879(1)
119902(119865) (119910
1 1199102)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
120595(minus119894 119902 119908) 119889119891 (119908)
(40)
for s-ae (1199101 1199102) isin 1198622
119886119887[0 119879] where 120595 is given by (32)
Proof We first note that for 119895 isin 1 2 the PWZ stochasticintegral (11986012
119895119908 119909)sim is a Gaussian random variable withmean
(11986012
119895119908 119886)119887and variance 11986012
119895119908
2
119887
= (119860119895119908119908)119887
Hence using(30) the Fubini theorem the change of variables theorem and(14) we have that for all 120582
1gt 0 and 120582
2gt 0
119869 (1199101 1199102 1205821 1205822)
equiv 119864119909[119865 (1199101+ 120582minus12
11199091 1199102+ 120582minus12
21199092)]
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
times(
2
prod
119895=1
119864119909119895
[exp 119894120582minus12119895(11986012
119895119908 119909119895)
sim
])119889119891 (119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
times[
[
[
2
prod
119895=1
(2120587(119860119895119908119908)
119887
)
minus12
times int
R
exp
119894120582minus12
119895119906119895
minus
[119906119895minus (11986012
119895119908 119886)119887
]
2
2(119860119895119908119908)
119887
119889119906119895
]
]
]
119889119891(119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
120595(120582 119908) 119889119891 (119908)
(41)
Let
119879 120582(119865) (119910
1 1199102)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
120595(120582 119908) 119889119891 (119908)
(42)
for each 120582 isin C2+ Clearly
119879 120582(119865) (119910
1 1199102) = 119869 (119910
1 1199102 1205821 1205822) (43)
for all 1205821gt 0 and 120582
2gt 0 Let Γ
1199020
be given by (35)Then for all120582 isin Int(Γ
1199020
)
1003816100381610038161003816119879 120582(119865) (119910
1 1199102)1003816100381610038161003816lt int
1198621015840
119886119887[0119879]
119896 (1199020 119908) 119889
10038161003816100381610038161198911003816100381610038161003816(119908) lt +infin
(44)
Using this fact and the dominated convergence theoremwe see that 119879
120582(119865)(1199101 1199102) is a continuous function of 120582 =
(1205821 1205822) on Int(Γ
1199020
) For each 119908 isin 1198621015840119886119887[0 119879] 120595( 120582 119908) is an
analytic function of 120582 throughout the domain Int(Γ1199020
) so thatintΔ
120595(120582 119908)119889
120582 = 0 for every rectifiable simple closed curve
Δ in Int(Γ1199020
) By (42) the Fubini theorem and the Moreratheorem we see that 119879
120582(119865)(1199101 1199102) is an analytic function of
120582 throughout the domain Int(Γ
1199020
) Finally using (28) withthe dominated convergence theorem we obtain the desiredresult
Theorem 10 Let 1199020and 119865 be as inTheorem 9Then for all 119901 isin
(1 2] and all nonzero real numbers 1199021and 119902
2with |119902
119895| gt 1199020
119895 isin 1 2 the119871119901analytic GFFT of119865119879(119901)
119902(119865) exists and is given
by the right hand side of (40) for s-ae (1199101 1199102) isin 1198622
119886119887[0 119879]
Journal of Function Spaces and Applications 7
Proof Let Γ1199020
be given by (35) It was shown in the proofof Theorem 9 that 119879
120582(119865)(1199101 1199102) is an analytic function of 120582
throughout the domain Int(Γ1199020
) In viewofDefinition 4 it willsuffice to show that for each 120588
1gt 0 and 120588
2gt 0
lim120582rarrminus119894 119902
int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816
119879 120582(119865) (120588
11199101 12058821199102)
minus119879(119901)
119902(119865) (120588
11199101 12058821199102)
100381610038161003816100381610038161003816
1199011015840
times 119889 (120583 times 120583) (1199101 1199102) = 0
(45)
Fixing 119901 isin (1 2] and using the inequalities (37) and (39)we obtain that for all 120588
119895gt 0 119895 isin 1 2 and all 120582 isin Γ
1199020
100381610038161003816100381610038161003816
119879 120582(119865) (120588
11199101 12058821199102) minus 119879(119901)
119902(119865) (120588
11199101 12058821199102)
100381610038161003816100381610038161003816
1199011015840
le
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894120588119895(11986012
119895119908 119910119895)
sim
times[120595 (120582 119908) minus 120595 (minus119894 119902 119908)] 119889119891 (119908)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199011015840
le (int
1198621015840
119886119887[0119879]
[
10038161003816100381610038161003816120595 (120582 119908)
10038161003816100381610038161003816+
10038161003816100381610038161003816120595 (minus119894 119902 119908)
10038161003816100381610038161003816] 11988910038161003816100381610038161198911003816100381610038161003816(119908))
1199011015840
le (2int
1198621015840
119886119887[0119879]
119896 (1199020 119908) 119889
10038161003816100381610038161198911003816100381610038161003816(119908))
1199011015840
lt +infin
(46)
Hence by the dominated convergence theorem we see thatfor each 119901 isin (1 2] and each 120588
1gt 0 and 120588
2gt 0
lim120582rarrminus119894 119902
int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816
119879 120582(119865) (120588
11199101 12058821199102)
minus119879(119901)
119902(119865) (120588
11199101 12058821199102)
100381610038161003816100381610038161003816
1199011015840
119889 (120583 times 120583) (1199101 1199102)
= lim120582rarrminus119894 119902
int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119910119895)
sim
times 120595(120582 119908) 119889119891 (119908)
minus int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119910119895)
sim
times120595(minus119894 119902 119908) 119889119891 (119908)
10038161003816100381610038161003816100381610038161003816100381610038161003816
1199011015840
times 119889 (120583 times 120583) (1199101 1199102)
= int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119910119895)
sim
times lim120582rarrminus119894 119902
[120595 (120582 119908)
minus120595 (minus119894 119902 119908)] 119889119891 (119908)
10038161003816100381610038161003816100381610038161003816100381610038161003816
1199011015840
times 119889 (120583 times 120583) (1199101 1199102)
= 0
(47)
which concludes the proof of Theorem 10
Remark 11 (1) In view of Theorems 9 and 10 we see thatfor each 119901 isin [1 2] the 119871
119901analytic GFFT of 119865 119879(119901)
119902(119865) is
given by the right hand side of (40) for 1199020 1199021 1199022 and 119865 as
in Theorem 9 and for s-ae (1199101 1199102) isin 1198622
119886119887[0 119879]
119879(119901)
119902(119865) (119910
1 1199102) = 119864
anf 119902119909[119865 (1199101+ 1199091 1199102+ 1199092)] 119901 isin [1 2]
(48)
In particular using this fact and (29) we have that for all 119901 isin[1 2]
119879(119901)
119902(119865) (0 0) = 119864
anf 119902119909[119865 (1199091 1199092)] (49)
(2) For nonzero real numbers 1199021and 119902
2with |119902
119895| gt 119902
0
119895 isin 1 2 define a set function 119891 119860119902B(1198621015840
119886119887[0 119879]) rarr C by
119891
119860
119902(119861) = int
119861
120595 (minus119894 119902 119908) 119889119891 (119908) 119861 isinB (1198621015840
119886119887[0 119879])
(50)
where 119891 and 119865 are related by (30) and B(1198621015840119886119887[0 119879]) is the
Borel 120590-algebra of 1198621015840119886119887[0 119879] Then it is obvious that 119891 119860
119902
belongs to M(1198621015840119886119887[0 119879]) and for all 119901 isin [1 2] 119879(119901)
119902(119865) can
be expressed as
119879(119901)
119902(119865) (119910
1 1199102) = int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
119889119891
119860
119902(119908)
(51)
for s-ae (1199101 1199102) isin 119862
2
119886119887[0 119879] Hence 119879(119901)
119902(119865) belongs to
F 11988611988711986011198602
for all 119901 isin [1 2]
4 Relationships between the GFFT and theFunction Space Integral of Functionals inF11988611988711986011198602
In this section we establish a relationship between the GFFTand the function space integral of functionals in the Fresneltype classF119886119887
11986011198602
8 Journal of Function Spaces and Applications
Throughout this section for convenience we use thefollowing notation for given 120582 isin C
+and 119899 = 1 2 let
119866119899(120582 119909)
= exp[1 minus 1205822
]
119899
sum
119896=1
[(119890119896 119909)sim
]
2
+ (12058212
minus 1)
119899
sum
119896=1
(119890119896 119886)1198621015840
119886119887
(119890119896 119909)sim
(52)
where 119890119899infin
119899=1is a complete orthonormal set in (1198621015840
119886119887[0 119879]
sdot 119887)To obtain our main results Theorems 14 and 17 below
we state a fundamental integration formula for the functionspace 119862
119886119887[0 119879]
Let 1198901 119890
119899 be an orthonormal set in (1198621015840
119886119887[0 119879] sdot
119887)
let 119896 R119899 rarr C be a Lebesgue measurable function and let119870 119862119886119887[0 119879] rarr C be given by
119870 (119909) = 119896 ((1198901 119909)sim
(119890119899 119909)sim
) (53)
Then
119864 [119870] = int
119862119886119887[0119879]
119896 ((1198901 119909)sim
(119890119899 119909)sim
) 119889120583 (119909)
= (2120587)minus1198992
int
R119899119896 (1199061 119906
119899)
times exp
minus
119899
sum
119895=1
[119906119895minus (119890119895 119886)119887
]
2
2
1198891199061 119889119906
119899
(54)
in the sense that if either side of (54) exists both sides existand equality holds
We also need the following lemma to obtain our maintheorem in this section
Lemma 12 Let 1198901 119890
119899 be an orthonormal subset of
(1198621015840
119886119887[0 119879] sdot
119887) and let 119908 be an element of 1198621015840
119886119887[0 119879] Then
for each 120582 isin C+ the function space integral
119864119909[119866119899(120582 119909) exp 119894(119908 119909)sim] (55)
exists and is given by the formula
119864119909[119866119899(120582 119909) exp 119894(119908 119909)sim]
= 120582minus1198992 exp
[
120582 minus 1
2120582
]
119899
sum
119896=1
(119890119896 119908)2
119887minus
1
2
1199082
119887
+ 119894120582minus12
119899
sum
119896=1
(119890119896 119886)119887(119890119896 119908)119887
+ 119894(119890119899+1 119886)119887[1199082
119887minus
119899
sum
119896=1
(119890119896 119908)2
119887]
12
(56)
where 119866119899is given by (52) above and
119890119899+1=[
[
1199082
119887minus
119899
sum
119895=1
(119890119895 119908)
2
119887
]
]
minus12
119908 minus
119899
sum
119895=1
(119890119895 119908)119887
119890119895
(57)
Proof (Outline) Using the Gram-Schmidt process for any119908 isin 119862
1015840
119886119887[0 119879] we can write 119908 = sum
119899+1
119896=1119888119896119890119896 where
1198901 119890
119899 119890119899+1 is an orthonormal set in (1198621015840
119886119887[0 119879] sdot
119887)
and
119888119896=
(119890119896 119908)119887 119896 = 1 119899
[1199082
119887minus
119899
sum
119895=1
(119890119895 119908)
2
119887
]
12
119896 = 119899 + 1
(58)
Then using (52) (54) the Fubini theorem and (14) it followsthat (56) holds for all 120582 isin C
+
The following remark will be very useful in the proof ofour main theorem in this section
Remark 13 Let 1199020be a positive real number and let Γ
1199020
begiven by (35) For real numbers 119902
1and 1199022with |119902
119895| gt 1199020 119895 isin
1 2 let 120582119899infin
119899=1= (1205821119899 1205822119899)infin
119899=1be a sequence in C2
+such
that
120582119899= (1205821119899 1205822119899) 997888rarr minus119894 119902 = (minus119894119902
1 minus1198941199022) (59)
Let 120582119895119899= 120572119895119899+ 119894120573119895119899
for 119895 isin 1 2 and 119899 isin N Then for119895 isin 1 2 Re(120582
119895119899) = 120572119895119899gt 0 and
120582minus1
119895119899= (120572119895119899+ 119894120573119895119899)
minus1
=
120572119895119899minus 119894120573119895119899
1205722
119895119899+ 1205732
119895119899
(60)
for each 119899 isin N Since |Im ((minus119894119902119895)minus12
)| = 1radic2|119902119895| lt 1radic2119902
0
for 119895 isin 1 2 there exists a sufficiently large 119871 isin N such thatfor any 119899 ge 119871 120582
1119899and 120582
2119899are in Int(Γ
1199020
) and
120575 (1199021 1199022) equiv sup ( 1003816100381610038161003816
1003816Im (120582minus12
1119899)
10038161003816100381610038161003816 119899 ge 119871
cup
10038161003816100381610038161003816Im (120582minus12
2119899)
10038161003816100381610038161003816 119899 ge 119871
cup
100381610038161003816100381610038161003816
Im ((minus1198941199021)minus12
)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
Im ((minus1198941199022)minus12
)
100381610038161003816100381610038161003816
)
lt
1
radic21199020
(61)
Thus there exists a positive real number 120576 gt 1 such that120575(1199021 1199022) lt 1(120576radic2119902
0)
Let 119890119899infin
119899=1be a complete orthonormal set in (1198621015840
119886119887[0 119879]
sdot 119887) Using Parsevalrsquos identity it follows that
(1198921 1198922)119887=
infin
sum
119899=1
(119890119899 1198921)119887(119890119899 1198922)119887
(62)
Journal of Function Spaces and Applications 9
for all 1198921 1198922isin 1198621015840
119886119887[0 119879] In addition for each 119892 isin 1198621015840
119886119887[0 119879]
10038171003817100381710038171198921003817100381710038171003817
2
119887minus
119899
sum
119896=1
(119890119896 119892)2
119887=
infin
sum
119896=119899+1
(119890119896 119892)2
119887ge 0 (63)
for every 119899 isin NSince
(119892 119886)119887=
infin
sum
119899=1
(119890119899 119892)119887(119890119899 119886)119887
(64)
and for 120576 gt 1
minus12057610038171003817100381710038171198921003817100381710038171003817119887119886119887lt minus10038171003817100381710038171198921003817100381710038171003817119887119886119887le (119892 119886)
119887
le10038171003817100381710038171198921003817100381710038171003817119887119886119887lt 12057610038171003817100381710038171198921003817100381710038171003817119887119886119887
(65)
there exists a sufficiently large119870119895isin N such that for any 119899 ge 119870
119895
1003816100381610038161003816100381610038161003816100381610038161003816
119899
sum
119896=1
(119890119896 11986012
119895119908)119887
(119890119896 119886)119887
1003816100381610038161003816100381610038161003816100381610038161003816
lt 120576
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817119887119886119887
(66)
for 119895 isin 1 2Using these and a long and tedious calculation we can
show that for every 119899 ge max119871 1198701 1198702
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
exp
2
sum
119895=1
([
120582119895119899minus 1
2120582119895119899
]
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
minus
1
2
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
+ 119894120582minus12
119895119899
119899
sum
119896=1
(119890119896 11986012
119895119908)119887
(119890119896 119886)119887+ 119894(119890119899+1 119886)119887
times [
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
minus
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
]
12
)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119896 (1199020 119908)
(67)
where 119896(1199020 119908) is given by (36)
In our next theorem for119865 isin F11988611988711986011198602
we express theGFFTof 119865 as the limit of a sequence of function space integrals on1198622
119886119887[0 119879]
Theorem 14 Let 1199020and 119865 be as in Theorem 10 Let 119890
119899infin
119899=1
be a complete orthonormal set in (1198621015840119886119887[0 119879] sdot
119887) and let
(1205821119899 1205822119899)infin
119899=1be a sequence in C2
+such that 120582
119895119899rarr minus119894119902
119895
where 119902119895is a real number with |119902
119895| gt 1199020 119895 isin 1 2 Then for
119901 isin [1 2] and for s-ae (1199101 1199102) isin 1198622
119886119887[0 119879]
119879(119901)
119902(119865) (119910
1 1199102)
= lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865 (119910
1+ 1199091 1199102+ 1199092)]
(68)
where 119866119899is given by (52)
Proof ByTheorems9 and 10we know that for each119901 isin [1 2]the 119871119901analytic GFFT of 119865 119879(119901)
119902(119865) exists and is given by the
right hand side of (40) Thus it suffices to show that
119879(1)
119902(119865) (119910
1 1199102) = 119864
anf 119902119909[119865 (1199101+ 1199091 1199102+ 1199092)]
= lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times 119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092)
times 119865 (1199101+ 1199091 1199102+ 1199092)] sdot
(69)
Using (30) the Fubini theorem and (56) with 120582 and 119908replaced with 120582
119895119899and 11986012
119895119908 119895 isin 1 2 respectively we see
that
1205821198992
11198991205821198992
2119899119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865 (119910
1+ 1199091 1199102+ 1199092)]
= 1205821198992
11198991205821198992
2119899int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
times(
2
prod
119895=1
119864119909119895
[119866119899(120582minus12
119895119899 119909119895)
times exp 119894(11986012119895119908 119909119895)
sim
])119889119891 (119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
(119894(11986012
119895119908 119910119895)
sim
+ [
120582119895119899minus 1
2120582119895119899
]
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
minus
1
2
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
+ 119894120582minus12
119895119899
119899
sum
119896=1
(119890119896 119886)119887(119890119896 11986012
119895119908)119887
+ 119894(119890119899+1 119886)119887[
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
minus
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
]
12
)
119889119891 (119908)
(70)
But by Remark 13 we see that the last expression of (70) isdominated by (39) on the region Γ
1199020
given by (35) for allbut a finite number of values of 119899 Next using the dominatedconvergence theorem Parsevalrsquos relation and (40) we obtainthe desired result
Corollary 15 Let 1199020 119865 119890
119899infin
119899=1 (1205821119899 1205822119899)infin
119899=1and (119902
1 1199022)
be as in Theorem 14 Then
119864
anf119902
119909[119865 (1199091 1199092)]
= lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times 119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865 (119909
1 1199092)]
(71)
where 119866119899is given by (52)
10 Journal of Function Spaces and Applications
Corollary 16 Let 1199020 119865 and 119890
119899infin
119899=1be as in Theorem 14 and
let Γ1199020
be given by (35) Let 120582 = (1205821 1205822) isin Int (Γ
1199020
) and(1205821119899 1205822119899)infin
119899=1be a sequence in C2
+such that 120582
119895119899rarr 120582
119895
119895 isin 1 2 Then
119864
an
119909[119865 (1199091 1199092)] = lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865(1199091 1199092)]
(72)
where 119866119899is given by (52)
Our another result namely a change of scale formula forfunction space integrals now follows fromCorollary 16 above
Theorem 17 Let 119865 isin F11988611988711986011198602
and let 119890119899infin
119899=1be a complete
orthonormal set in (1198621015840119886119887[0 119879] sdot
119887) Then for any 120588
1gt 0 and
1205882gt 0
119864119909[119865 (12058811199091 12058821199092)]
= lim119899rarrinfin
120588minus119899
1120588minus119899
2
times 119864119909[119866119899(120588minus2
1 1199091)119866119899(120588minus2
2 1199092) 119865 (119909
1 1199092)]
(73)
where 119866119899is given by (52)
Proof Simply choose 120582119895= 120588minus2
119895for 119895 isin 1 2 and 120582
119895119899= 120588minus2
119895
for 119895 isin 1 2 and 119899 isin N in (72)
Remark 18 Of course if we choose 119886(119905) equiv 0 119887(119905) = 1199051198601= 119868 (identity operator) and 119860
2= 0 (zero operator) then
the function space 119862119886119887[0 119879] reduces to the classical Wiener
space 1198620[0 119879] and the generalized Fresnel type class F 119886119887
11986011198602
reduces to the Fresnel class F(1198620[0 119879]) It is known that
F(1198620[0 119879]) forms a Banach algebra over the complex field
In this case we have the relationships between the analyticFeynman integral and theWiener integral on classicalWienerspace as discussed in [14 15]
In recent paper [19] Yoo et al have studied a change ofscale formula for function space integral of the functionalsin the Banach algebra S(1198712
119886119887[0 119879]) the Banach algebra
S(1198712119886119887[0 119879]) is introduced in [12]
5 Functionals in F11988611988711986011198602
In this section we prove a theorem ensuring that variousfunctionals are inF119886119887
11986011198602
Theorem 19 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] Let (119884Y 120574) be a 120590-finite
measure space and let 120593119897 119884 rarr 119862
1015840
119886119887[0 119879] beYndashB(1198621015840
119886119887[0 119879])
measurable for 119897 isin 1 119889 Let 120579 119884 times R119889 rarr C be given by120579(120578 sdot) = ]
120578(sdot) where ]
120578isin M(R119889) for every 120578 isin 119884 and where
the family ]120578 120578 isin 119884 satisfies
(i) ]120578(119864) is a Y-measurable function of 120578 for every 119864 isin
B(R119889)(ii) ]
120578 isin 1198711
(119884Y 120574)
Under these hypothesis the functional 119865 1198622119886119887[0 119879] rarr C
given by
119865 (1199091 1199092) = int
119884
120579(120578
2
sum
119895=1
(11986012
1198951205931(120578) 119909
119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 119909
119895)
sim
)119889120574 (120578)
(74)
belongs toF11988611988711986011198602
and satisfies the inequality
119865 le int
119884
10038171003817100381710038171003817]120578
10038171003817100381710038171003817119889120574 (120578) (75)
Proof Using the techniques similar to those used in [20] wecan show that ]
120578 is measurable as a function of 120578 that 120579 is
Y-measurable and that the integrand in (74) is a measurablefunction of 120578 for every (119909
1 1199092) isin 1198622
119886119887[0 119879]
We define a measure 120591 onY timesB(R119889) by
120591 (119864) = int
119884
]120578(119864(120578)
) 119889120574 (120578) for 119864 isin Y timesB (R119889) (76)
Then by the first assertion of Theorem 31 in [17] 120591 satisfies120591 le int
119884
]120578119889120574(120578) Now let Φ 119884 times R119889 rarr 119862
1015840
119886119887[0 119879] be
defined by Φ(120578 V1 V
119889) = sum
119889
119897=1V119897120593119897(120578) Then Φ is Y times
B(R119889) ndashB(1198621015840119886119887[0 119879])-measurable on the hypothesis for 120593
119897
119897 isin 1 119889 Let 120590 = 120591 ∘Φminus1 Then clearly 120590 isinM(1198621015840119886119887[0 119879])
and satisfies 120590 le 120591From the change of variables theorem and the second
assertion of Theorem 31 in [17] it follows that for ae(1199091 1199092) isin 1198622
119886119887[0 119879] and for every 120588
1gt 0 and 120588
2gt 0
119865 (12058811199091 12058821199092)
= int
119884
]120578(
2
sum
119895=1
(11986012
1198951205931(120578) 120588
119895119909119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 120588
119895119909119895)
sim
)119889120574 (120578)
= int
119884
[
[
int
R119889exp
119894
119889
sum
119897=1
V119897
[
[
2
sum
119895=1
(11986012
119895120593119897(120578)
120588119895119909119895)
sim
]
]
119889]120578
times (V1 V
119889)]
]
119889120574 (120578)
= int
119884timesR119889exp
119894
119889
sum
119897=1
V119897
[
[
2
sum
119895=1
(11986012
119895120593119897(120578) 120588
119895119909119895)
sim
]
]
119889120591
times (120578 V1 V
119889)
Journal of Function Spaces and Applications 11
= int
119884timesR119889exp
2
sum
119895=1
119894(11986012
119895Φ(120578 V
1 V
119889) 120588119895119909119895)
sim
119889120591
times (120578 V1 V
119889)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119909119895)
sim
119889120591 ∘ Φminus1
(119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119909119895)
sim
119889120590 (119908)
(77)
Thus the functional 119865 given by (74) belongs to F11988611988711986011198602
andsatisfies the inequality
119865 = 120590 le 120591 le int
119884
10038171003817100381710038171003817]120578
10038171003817100381710038171003817119889120574 (120578) (78)
As mentioned in (2) of Remark 6 F 11988611988711986011198602
is a Banachalgebra if Ran(119860
1+ 1198602) is dense in 1198621015840
119886119887[0 119879] In this case
many analytic functionals of 119865 can be formed The followingcorollary is relevant to Feynman integration theories andquantum mechanics where exponential functions play animportant role
Corollary 20 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] such that Ran(119860
1+ 1198602)
is dense in 1198621015840119886119887[0 119879] Let 119865 be given by (74) with 120579 as in
Theorem 19 and let 120573 C rarr C be an entire function Then(120573 ∘ 119865)(119909
1 1199092) is in F119886119887
11986011198602
In particular exp119865(1199091 1199092) isin
F11988611988711986011198602
Corollary 21 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] and let 119892
1 119892
119889 be a
finite subset of 1198621015840119886119887[0 119879] Given 120573 = ] where ] isin M(R119889)
define 119865 1198622119886119887[0 119879] rarr C by
119865 (1199091 1199092) = 120573(
2
sum
119895=1
(11986012
1198951198921 119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 119909119895)
sim
)
(79)
Then 119865 is an element ofF11988611988711986011198602
Proof Let (119884Y 120574) be a probability space and for 119897 isin1 119889 let 120593
119897(120578) equiv 119892
119897 Take 120579(120578 sdot) = 120573(sdot) = ](sdot) Then for
all 1205881gt 0 and 120588
2gt 0 and for ae (119909
1 1199092) isin 1198622
119886119887[0 119879]
int
119884
120579(120578
2
sum
119895=1
(11986012
1198951205931(120578) 120588
119895119909119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 120588
119895119909119895)
sim
)119889120574 (120578)
= int
119884
120573(
2
sum
119895=1
(11986012
1198951198921 120588119895119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 120588119895119909119895)
sim
)119889120574 (120578)
= 120573(
2
sum
119895=1
(11986012
1198951198921 120588119895119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 120588119895119909119895)
sim
)
= 119865 (12058811199091 12058821199092)
(80)
Hence 119865 isin F11988611988711986011198602
Remark 22 Let 119889 = 1 and let (119884Y 120574) = ([0 119879]B([0 119879])119898119871) in Theorem 19 where 119898
119871denotes the Lebesgue measure
on [0 119879] ThenTheorems 46 47 and 49 in [18] follow fromthe results in this section by letting119860
1be the identity operator
and letting 1198602equiv 0 on 1198621015840
119886119887[0 119879] The function 120579 studied in
[18] (and mentioned in Remark 2 above) is interpreted as thepotential energy in quantum mechanics
Acknowledgments
The authors would like to express their gratitude to the refer-ees for their valuable comments and suggestions which haveimproved the original paper This research was supported bythe Basic Science Research Program through the NationalResearch Foundation of Korea (NRF) funded by theMinistryof Education (2011-0014552)
References
[1] G Kallianpur and C Bromley ldquoGeneralized Feynman integralsusing analytic continuation in several complex variablesrdquo inStochastic Analysis and Applications M A Pinsky Ed vol 7pp 217ndash267 Marcel Dekker New York NY USA 1984
[2] G Kallianpur D Kannan and R L Karandikar ldquoAnalytic andsequential Feynman integrals on abstract Wiener and Hilbertspaces and a Cameron-Martin formulardquo Annales de lrsquoInstitutHenri Poincare vol 21 no 4 pp 323ndash361 1985
[3] G W Johnson and M L Lapidus The Feynman Integral andFeynmanrsquos Operational Calculus Clarendon Press Oxford UK2000
[4] J Yeh ldquoSingularity of Gaussian measures on function spacesinduced by Brownian motion processes with non-stationaryincrementsrdquo Illinois Journal of Mathematics vol 15 pp 37ndash461971
[5] J Yeh Stochastic Processes and the Wiener Integral MarcelDekker New York NY USA 1973
[6] S J Chang J G Choi and D Skoug ldquoIntegration by partsformulas involving generalized Fourier-Feynman transformson function spacerdquo Transactions of the American MathematicalSociety vol 355 no 7 pp 2925ndash2948 2003
[7] S J Chang J G Choi and D Skoug ldquoParts formulas involvingconditional generalized Feynman integrals and conditionalgeneralized Fourier-Feynman transforms on function spacerdquo
12 Journal of Function Spaces and Applications
Integral Transforms and Special Functions vol 15 no 6 pp 491ndash512 2004
[8] S J Chang J G Choi and D Skoug ldquoEvaluation formulas forconditional function space integralsmdashIrdquo Stochastic Analysis andApplications vol 25 no 1 pp 141ndash168 2007
[9] S J Chang J G Choi and D Skoug ldquoGeneralized Fourier-Feynman transforms convolution products and first variationson function spacerdquoTheRockyMountain Journal ofMathematicsvol 40 no 3 pp 761ndash788 2010
[10] S J Chang and D M Chung ldquoConditional function spaceintegrals with applicationsrdquo The Rocky Mountain Journal ofMathematics vol 26 no 1 pp 37ndash62 1996
[11] S J Chang H S Chung and D Skoug ldquoIntegral transformsof functionals in 119871
2(119862119886119887[0 119879])rdquoThe Journal of Fourier Analysis
and Applications vol 15 no 4 pp 441ndash462 2009[12] S J Chang and D Skoug ldquoGeneralized Fourier-Feynman
transforms and a first variation on function spacerdquo IntegralTransforms and Special Functions vol 14 pp 375ndash393 2003
[13] J G Choi and S J Chang ldquoGeneralized Fourier-Feynmantransform and sequential transforms on function spacerdquo Jour-nal of the Korean Mathematical Society vol 49 pp 1065ndash10822012
[14] R H Cameron and D A Storvick ldquoRelationships between theWiener integral and the analytic Feynman integralrdquo Rendicontidel Circolo Matematico di Palermo no 17 supplement pp 117ndash133 1987
[15] RHCameron andDA Storvick ldquoChange of scale formulas forWiener integralrdquo Rendiconti del Circolo Matematico di Palermono 17 pp 105ndash115 1987
[16] G W Johnson ldquoThe equivalence of two approaches to theFeynman integralrdquo Journal of Mathematical Physics vol 23 pp2090ndash2096 1982
[17] G W Johnson and D L Skoug ldquoNotes on the FeynmanintegralmdashIII the Schroedinger equationrdquo Pacific Journal ofMathematics vol 105 pp 321ndash358 1983
[18] S J Chang J G Choi and S D Lee ldquoA Fresnel type class onfunction spacerdquo Journal of the Korean Society of MathematicalEducation Series B vol 16 no 1 pp 107ndash119 2009
[19] I Yoo B J Kim and B S Kim ldquoA change of scale formulafor a function space integral on 119862
119886119887[0 119879]rdquo Proceedings of the
American Mathematical Society vol 141 no 8 pp 2729ndash27392013
[20] K S Chang G W Johnson and D L Skoug ldquoFunctions inthe Fresnel classrdquo Proceedings of the American MathematicalSociety vol 100 no 2 pp 309ndash318 1987
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Function Spaces and Applications
of functionals on 1198622119886119887[0 119879] very difficult However when
119886(119905) equiv 0 and 119887(119905) = 119905 on [0 119879] the general function space119862119886119887[0 119879] reduces to the Wiener space 119862
0[0 119879]
2 Definitions and Preliminaries
Let 119886(119905) be an absolutely continuous real-valued function on[0 119879] with 119886(0) = 0 1198861015840(119905) isin 1198712[0 119879] and let 119887(119905) be a strictlyincreasing continuously differentiable real-valued functionwith 119887(0) = 0 and 1198871015840(119905) gt 0 for each 119905 isin [0 119879]The generalizedBrownian motion process 119884 determined by 119886(119905) and 119887(119905) isa Gaussian process with mean function 119886(119905) and covariancefunction 119903(119904 119905) = min119887(119904) 119887(119905) For more details see[6 10 12] By Theorem 142 in [5] the probability measure120583 induced by 119884 taking a separable version is supportedby 119862119886119887[0 119879] (which is equivalent to the Banach space of
continuous functions 119909 on [0 119879]with 119909(0) = 0 under the supnorm) Hence (119862
119886119887[0 119879] B(119862
119886119887[0 119879]) 120583) is the function
space induced by119884whereB(119862119886119887[0 119879]) is the Borel120590-algebra
of 119862119886119887[0 119879] We then complete this function space to obtain
(119862119886119887[0 119879] W(119862
119886119887[0 119879]) 120583) whereW(119862
119886119887[0 119879]) is the set
of all Wiener measurable subsets of 119862119886119887[0 119879]
A subset 119861 of 119862119886119887[0 119879] is said to be scale-invariant
measurable provided 120588119861 is W(119862119886119887[0 119879])-measurable for all
120588 gt 0 and a scale-invariant measurable set 119873 is said to be ascale-invariant null set provided 120583(120588119873) = 0 for all 120588 gt 0A property that holds except on a scale-invariant null setis said to hold scale-invariant almost everywhere (s-ae) Afunctional119865 is said to be scale-invariantmeasurable provided119865 is defined on a scale-invariant measurable set and 119865(120588sdot) isW(119862119886119887[0 119879])-measurable for every 120588 gt 0 If two functionals
119865 and119866 defined on119862119886119887[0 119879] are equal s-ae we write 119865 asymp 119866
Let 11987112119886119887[0 119879] be the space of Lebesgue measurable func-
tions on [0 119879] given by
11987112
119886119887[0 119879] = V int
119879
0
|V (119904)|2119889119887 (119904) lt infin
int
119879
0
|V (119904)| 119889 |119886| (119904) lt infin
(4)
where |119886|(sdot) is the total variation function of 119886(sdot) Then11987112
119886119887[0 119879] is a separable Hilbert space with inner product
defined by
(119906 V)11987112
119886119887
= int
119879
0
119906 (119905) V (119905) 119889119887 (119905)
+ (int
119879
0
119906 (119905) 119889119886 (119905)) (int
119879
0
V (119905) 119889119886 (119905))
(5)
In particular note that 11990611987112
119886119887
equiv [(119906 119906)11987112
119886119887
]12
= 0 if and onlyif 119906(119905) = 0 ae on [0 119879]
Let 120601119895infin
119895=1be a complete orthonormal set in 11987112
119886119887[0 119879]
each of whose elements is of bounded variation on [0 119879] suchthat
int
119879
0
120601119895(119905) 120601119896(119905) 119889119887 (119905) =
0 119895 = 119896
1 119895 = 119896
(6)
Then for each V isin 11987112119886119887[0 119879] the Paley-Wiener-Zygmund
(PWZ) stochastic integral ⟨V 119909⟩ is defined by the followingformula
⟨V 119909⟩ = lim119899rarrinfin
int
119879
0
119899
sum
119895=1
(V 120601119895)119886119887
120601119895(119905) 119889119909 (119905) (7)
for all 119909 isin 119862119886119887[0 119879] for which the limit exists one can show
that for each V isin 11987112119886119887[0 119879] the PWZ stochastic integral
⟨V 119909⟩ exists for 120583-ae 119909 isin 119862119886119887[0 119879] and if V is of bounded
variation on [0 119879] then the PWZ stochastic integral ⟨V 119909⟩equals the Riemann-Stieltjes integral int119879
0
V(119905)119889119909(119905) for s-ae119909 isin 119862
119886119887[0 119879]
Remark 1 (1) For each V isin 11987112119886119887[0 119879] the PWZ stochastic
integral ⟨V 119909⟩ is a Gaussian random variable on 119862119886119887[0 119879]
with mean int1198790
V(119904)119889119886(119904) and variance int1198790
V2(119904)119889119887(119904)(2) For all 119906 V isin 11987112
119886119887[0 119879]
int
119862119886119887[0119879]
⟨119906 119909⟩⟨V 119909⟩ 119889120583 (119909)
= int
119879
0
119906 (119904) V (119904) 119889119887 (119904)
+ (int
119879
0
119906 (119904) 119889119886 (119904)) (int
119879
0
V (119904) 119889119886 (119904))
(8)
Hence we see that for all 119906 V isin 11987112119886119887[0 119879] int119879
0
119906(119904)V(119904)119889119887(119904) =0 if and only if ⟨119906 119909⟩ and ⟨V 119909⟩ are independent randomvariables
The following Cameron-Martin subspace of 119862119886119887[0 119879]
plays an important role throughout this paperLet
1198621015840
119886119887[0 119879] = 119908 isin 119862
119886119887[0 119879] 119908 (119905) = int
119905
0
119911 (119904) 119889119887 (119904)
for some 119911 isin 11987112119886119887[0 119879]
(9)
For 119908 isin 1198621015840119886119887[0 119879] let119863 1198621015840
119886119887[0 119879] rarr 119871
12
119886119887[0 119879] be defined
by the following formula
119863119908 (119905) =
119889120582119908
119889120582119887
(119905) (10)
where 119889120582119908119889120582119887denotes the Radon-Nikodym derivative of
the signed measure 120582119908
induced by 119908 with respect tothe Borel-Stieltjes measure 120582
119887induced by 119887 Then 1198621015840
119886119887equiv
1198621015840
119886119887[0 119879] with inner product
(1199081 1199082)1198621015840
119886119887
= int
119862119886119887[0119879]
⟨1198631199081 119909⟩⟨119863119908
2 119909⟩ 119889120583 (119909) (11)
is a separable Hilbert space
Journal of Function Spaces and Applications 3
Using (8) we observe that the linear operator given by(10) is an isometry In fact the inverse operator 119863minus1 11987112
119886119887[0 119879] rarr 119862
1015840
119886119887[0 119879] is given by
(119863minus1
119911) (119905) = int
119905
0
119911 (119904) 119889119887 (119904) (12)
Moreover the triple (1198621015840119886119887[0 119879] 119862
119886119887[0 119879] 120583) becomes an
abstract Wiener spaceThroughout this paper for 119908 isin 1198621015840
119886119887[0 119879] we will use the
notation (119908 119909)sim instead of ⟨119863119908 119909⟩We also use the followingnotations for 119908
1 1199082 119908 isin 1198621015840
119886119887[0 119879]
(1199081 1199082)119887= int
119879
0
1198631199081(119905) 119863119908
2(119905) 119889119887 (119905)
119908119887= radic(119908119908)
119887
(13)
Then 1198621015840119886119887[0 119879] with the inner product given by (13) is also a
separable Hilbert space It is easy to see that the two norms sdot 1198621015840
119886119887
and sdot 119887are equivalent Furthermore we have the
following assertions
(i) 119886(sdot) is an element of 1198621015840119886119887[0 119879]
(ii) For each 119908 isin 1198621015840119886119887[0 119879] the random variable 119909 997891rarr
(119908 119909)sim is Gaussian with mean (119908 119886)
119887and variance
1199082
119887
(iii) (119908 120572119909)sim = (120572119908 119909)sim = 120572(119908 119909)sim for any real number120572 119908 isin 1198621015840
119886119887[0 119879] and 119909 isin 119862
119886119887[0 119879]
(iv) Let 1199081 119908
119899 be a subset of 1198621015840
119886119887[0 119879] such that
int
119879
0
119863119908119894(119905)119863119908
119895(119905)119889119887(119905) = 120575
119894119895 where 120575
119894119895is the Kro-
necker delta Then the random variables (119908119894 119909)simrsquos are
independent
In this paper we adopt asmuch as possible the definitionsand notations used in [7 9 12 13] for the definitions ofthe generalized analytic Feynman integral and the GFFT offunctionals on 119862
119886119887[0 119879]
The following integration formula is used several times inthis paper
int
R
exp minus1205721199062 + 120573119906 119889119906 = radic120587120572
exp1205732
4120572
(14)
for complex numbers 120572 and 120573 with Re(120572) gt 0
3 The GFFT of Functionals in a BanachAlgebra F119886119887
11986011198602
LetM(1198621015840119886119887[0 119879]) be the space of complex-valued countably
additive (and hence finite) Borel measures on 1198621015840119886119887[0 119879]
M(1198621015840119886119887[0 119879]) is a Banach algebra under the total variation
norm and with convolution as multiplicationWe define the Fresnel type class F(119862
119886119887[0 119879]) of
functionals on 119862119886119887[0 119879] as the space of all stochastic
Fourier transforms of elements of M(1198621015840119886119887[0 119879]) that is
119865 isin F(119862119886119887[0 119879]) if and only if there exists a measure 119891 in
M(1198621015840119886119887[0 119879]) such that
119865 (119909) = int
1198621015840
119886119887[0119879]
exp 119894(119908 119909)sim 119889119891 (119908) (15)
for s-ae 119909 isin 119862119886119887[0 119879] More precisely since we will identify
functionals which coincide s-ae on 119862119886119887[0 119879]F(119862
119886119887[0 119879])
can be regarded as the space of all 119904-equivalence classes offunctionals of the form (15)
The Fresnel type class F(119862119886119887[0 119879]) is a Banach algebra
with norm
119865 =10038171003817100381710038171198911003817100381710038171003817= int
1198621015840
119886119887[0119879]
11988910038161003816100381610038161198911003816100381610038161003816(119908) (16)
In fact the correspondence 119891 997891rarr 119865 is injective carriesconvolution into pointwise multiplication and is a Banachalgebra isomorphism where 119891 and 119865 are related by (15)
Remark 2 The Banach algebra F(119862119886119887[0 119879]) contains sev-
eral interesting functions which arise naturally in quantummechanics Let M(R) be the class of C-valued countablyadditive measures on B(R) the Borel class of R For ] isinM(R) the Fourier transform ] of ] is a complex-valuedfunction defined on R by the following formula
] (119906) = intR
exp 119894119906V 119889] (V) (17)
Let G be the set of all complex-valued functions on[0 119879]timesR of the form 120579(119904 119906) =
119904(119906) where 120590
119904 0 le 119904 le 119879 is
a family fromM(R) satisfying the following two conditions
(i) for every 119864 isinB(R) 120590119904(119864) is Borel measurable in 119904
(ii) int1198790
120590119904119889119887(119904) lt +infin
Let 120579 isin G and let119867 be given by
119867(119909) = expint119879
0
120579 (119905 119909 (119905)) 119889119905 (18)
for s-ae 119909 isin 119862119886119887[0 119879] Then using the methods similar
to those used in [18] we can show that the function 120579(119905 119906)is Borel-measurable and that 120579(119905 119909(119905)) int119879
0
120579(119905 119909(119905))119889119905 and119867(119909) are elements of F(119862
119886119887[0 119879]) These facts are relevant
to quantum mechanics where exponential functions play aprominent role
Let119860 be a nonnegative self-adjoint operator on1198621015840119886119887[0 119879]
and 119891 any complex measure on 1198621015840119886119887[0 119879] Then the func-
tional
119865 (119909) = int
1198621015840
119886119887[0119879]
exp 119894(11986012119908 119909)sim
119889119891 (119908) (19)
belongs to F(119862119886119887[0 119879]) because it can be rewritten as
int1198621015840
119886119887[0119879]
exp119894(119908 119909)sim119889119891119860(119908) for 119891
119860= 119891 ∘ (119860
12
)minus1 Let 119860 be
self-adjoint but not nonnegative Then 119860 has the form
119860 = 119860+
minus 119860minus
(20)
4 Journal of Function Spaces and Applications
where both 119860+ and 119860minus are bounded nonnegative and self-adjoint operators
In this section we will extend the ideas of [1] to obtainexpressions of the generalized analytic Feynman integral andthe GFFT of functionals of the form (19) when119860 is no longerrequired to be nonnegative To do this we will introducedefinitions and notations analogous to those in [7 12 13]
Let W(1198622119886119887[0 119879]) denote the class of all Wiener mea-
surable subsets of the product function space 1198622119886119887[0 119879] A
subset 119861 of 1198622119886119887[0 119879] is said to be scale-invariant measurable
provided (12058811199091 12058821199092) (119909
1 1199092) isin 119861 is W(1198622
119886119887[0 119879])-
measurable for every 1205881gt 0 and 120588
2gt 0 and a scale-
invariantmeasurable subset119873 of1198622119886119887[0 119879] is said to be scale-
invariant null provided (120583 times 120583)((12058811199091 12058821199092) (119909
1 1199092) isin
119873) = 0 for every 1205881gt 0 and 120588
2gt 0 A property that
holds except on a scale-invariant null set is said to hold s-ae on 1198622
119886119887[0 119879] A functional 119865 on 1198622
119886119887[0 119879] is said to be
scale-invariant measurable provided 119865 is defined on a scale-invariant measurable set and 119865(120588
1sdot 1205882sdot) is W(1198622
119886119887[0 119879])-
measurable for every 1205881gt 0 and 120588
2gt 0 If two functionals
119865 and 119866 defined on 1198622119886119887[0 119879] are equal s-ae then we write
119865 asymp 119866We denote the product function space integral of a
W(1198622119886119887[0 119879])-measurable functional 119865 by
119864 [119865] equiv 119864119909[119865 (1199091 1199092)]
= int
1198622
119886119887[0119879]
119865 (1199091 1199092) 119889 (120583 times 120583) (119909
1 1199092)
(21)
whenever the integral existsThroughout this paper letCC
+and C
+denote the set of
complex numbers complex numbers with positive real partand nonzero complex numbers with nonnegative real partrespectively Furthermore for all 120582 isin C
+ 120582minus12 (or 12058212) is
always chosen to have positive real part We also assume thatevery functional 119865 on 1198622
119886119887[0 119879] we consider is s-ae defined
and is scale-invariant measurable
Definition 3 Let C2+equiv C+times C+and let C2
+equivC+timesC+ Let
119865 1198622
119886119887[0 119879] rarr C be such that for each 120582
1gt 0 and 120582
2gt 0
the function space integral
119869 (1205821 1205822)
= int
1198622
119886119887[0119879]
119865 (120582minus12
11199091 120582minus12
21199092) 119889 (120583 times 120583) (119909
1 1199092)
(22)
exists If there exists a function 119869lowast(1205821 1205822) analytic inC2
+such
that 119869lowast(1205821 1205822) = 119869(120582
1 1205822) for all 120582
1gt 0 and 120582
2gt 0 then
119869lowast
(1205821 1205822) is defined to be the analytic function space integral
of 119865 over 1198622119886119887[0 119879] with parameter 120582 = (120582
1 1205822) and for 120582 isin
C2+we write
119864an[119865] equiv 119864
an
119909[119865 (1199091 1199092)]
equiv 119864
an(12058211205822)
11990911199092[119865 (1199091 1199092)] = 119869
lowast
(1205821 1205822)
(23)
Let 1199021and 1199022be nonzero real numbers Let 119865 be a functional
such that 119864an[119865] exists for all 120582 isin C2+ If the following limit
exists we call it the generalized analytic Feynman integral of119865 with parameter 119902 = (119902
1 1199022) and we write
119864anf 119902[119865] equiv 119864
anf 119902119909[119865 (1199091 1199092)]
equiv 119864
anf(11990211199022)
11990911199092[119865 (1199091 1199092)] = lim
120582rarrminus119894 119902
119864an[119865]
(24)
where 120582 = (1205821 1205822) rarr minus119894 119902 = (minus119894119902
1 minus1198941199022) through values in
C2+
Definition 4 Let 1199021and 1199022be nonzero real numbers For 120582 =
(1205821 1205822) isin C2+and (119910
1 1199102) isin 1198622
119886119887[0 119879] let
119879 120582(119865) (119910
1 1199102) equiv 119879(12058211205822)(119865) (119910
1 1199102)
= 119864
an
119909[119865 (1199101+ 1199091 1199102+ 1199092)]
(25)
For 119901 isin (1 2] we define the 119871119901analytic GFFT 119879(119901)
119902(119865) of 119865
by the formula ( 120582 isin C2+)
119879(119901)
119902(119865) (119910
1 1199102) equiv 119879(119901)
(1199021 1199022)(119865) (119910
1 1199102)
= lim120582rarrminus119894 119902
119879 120582(119865) (119910
1 1199102)
(26)
if it exists that is for each 1205881gt 0 and 120588
2gt 0
lim120582rarrminus119894 119902
int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816
119879 120582(119865) (120588
11199101 12058821199102)
minus119879(119901)
119902(119865) (120588
11199101 12058821199102)
100381610038161003816100381610038161003816
1199011015840
119889 (120583 times 120583) (1199101 1199102)
= 0
(27)
where 1119901+11199011015840 = 1We define the 1198711analytic GFFT119879(1)
119902(119865)
of 119865 by the formula ( 120582 isin C2+)
119879(1)
119902(119865) (119910
1 1199102) = lim120582rarrminus119894 119902
119879 120582(119865) (119910
1 1199102) (28)
if it exists
We note that for 1 le 119901 le 2 119879(119901)119902(119865) is defined only s-ae
We also note that if 119879(119901)119902(119865) exists and if 119865 asymp 119866 then 119879(119901)
119902(119866)
exists and 119879(119901)119902(119866) asymp 119879
(119901)
119902(119865) Moreover from Definition 4
we see that for 1199021 1199022isin R minus 0
119864
anf 119902119909[119865 (1199091 1199092)] = 119879
(1)
119902(119865) (0 0) (29)
Next we give the definition of the generalized Fresnel typeclassF119886119887
11986011198602
Journal of Function Spaces and Applications 5
Definition 5 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] The generalized Fresnel
type classF11988611988711986011198602
of functionals on1198622119886119887[0 119879] is defined as the
space of all functionals 119865 on 1198622119886119887[0 119879] of the following form
119865 (1199091 1199092) = int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119909119895)
sim
119889119891 (119908) (30)
for some119891 isinM(1198621015840119886119887[0 119879]) More precisely since we identify
functionals which coincide s-ae on 1198622119886119887[0 119879] F119886119887
11986011198602
canbe regarded as the space of all 119904-equivalence classes offunctionals of the form (30)
Remark 6 (1) In Definition 5 let 1198601be the identity operator
on 1198621015840119886119887[0 119879] and 119860
2equiv 0 Then F 119886119887
11986011198602
is essentially theFresnel type class F(119862
119886119887[0 119879]) and for 119901 isin [1 2] and
nonzero real numbers 1199021and 1199022
119879(119901)
(1199021 1199022)(119865) (119910
1 1199102) = 119879(119901)
1199021
(1198650) (1199101) (31)
if it exists where 1198650(1199091) = 119865(119909
1 1199092) for all (119909
1 1199092) isin
1198622
119886119887[0 119879] and 119879(119901)
1199021
(1198650) means the 119871
119901analytic GFFT on
119862119886119887[0 119879] see [6 12](2) The map 119891 997891rarr 119865 defined by (30) sets up an algebra
isomorphism betweenM(1198621015840119886119887[0 119879]) andF119886119887
11986011198602
if Ran(1198601+
1198602) is dense in1198621015840
119886119887[0 119879] where Ran indicates the range of an
operator In this caseF 11988611988711986011198602
becomes a Banach algebra underthe norm 119865 = 119891 For more details see [1]
Remark 7 Let 119865 be given by (30) In evaluating119864119909[119865(120582minus12
11199091 120582minus12
21199092)] and 119879
(12058211205822)(119865)(1199101 1199102) = 119864
119909[119865(1199101+
120582minus12
11199091 1199102+ 120582minus12
21199092)] for 120582
1gt 0 and 120582
2gt 0 the expression
120595 (120582 119908)
equiv 120595 (1205821 1205822 1198601 1198602 119908)
= exp
2
sum
119895=1
[
[
minus
(119860119895119908119908)
119887
2120582119895
+ 119894120582minus12
119895(11986012
119895119908 119886)119887
]
]
(32)
occurs Clearly for 120582119895gt 0 119895 isin 1 2 |120595( 120582 119908)| le 1 for all
119908 isin 1198621015840
119886119887[0 119879] But for 120582 isin C2
+ |120595( 120582 119908)| is not necessarily
bounded by 1Note that for each 120582
119895isinC+with 120582
119895= 120572119895+ 119894120573119895 119895 isin 1 2
12058212
119895=
radicradic1205722
119895+ 1205732
119895+ 120572119895
2
+ 119894
120573119895
10038161003816100381610038161003816120573119895
10038161003816100381610038161003816
radicradic1205722
119895+ 1205732
119895minus 120572119895
2
120582minus12
119895=radic
radic1205722
119895+ 1205732
119895+ 120572119895
2 (1205722
119895+ 1205732
119895)
minus 119894
120573119895
10038161003816100381610038161003816120573119895
10038161003816100381610038161003816
radic
radic1205722
119895+ 1205732
119895minus 120572119895
2 (1205722
119895+ 1205732
119895)
(33)
Hence for 120582119895isinC+with 120582
119895= 120572119895+ 119894120573119895 119895 isin 1 2
10038161003816100381610038161003816120595 (120582 119908)
10038161003816100381610038161003816
= exp
2
sum
119895=1
[
[
[
[
minus
120572119895
2 (1205722
119895+ 1205732
119895)
(119860119895119908119908)
119887
+
120573119895
10038161003816100381610038161003816120573119895
10038161003816100381610038161003816
radic
radic1205722
119895+ 1205732
119895minus 120572119895
2 (1205722
119895+ 1205732
119895)
(11986012
119895119908 119886)119887
]
]
]
]
(34)
The right hand side of (34) is an unbounded functionof 119908 for 119908 isin 1198621015840
119886119887[0 119879] Thus 119864an[119865] 119864anf 119902[119865] 119879
120582(119865) and
119879(119901)
119902(119865) might not exist Thus throughout this paper we will
need to put additional restrictions on the complex measure119891 corresponding to 119865 in order to obtain our results for theGFFT and the generalized analytic Feynman integral of 119865
In view of Remark 7 we clearly need to impose additionalrestrictions on the functionals 119865 inF119886119887
11986011198602
For a positive real number 119902
0 let
Γ1199020
=
120582 = (120582
1 1205822) isinC2
+| 120582119895= 120572119895+ 119894120573119895
10038161003816100381610038161003816Im (120582minus12
119895)
10038161003816100381610038161003816=radic
radic1205722
119895+ 1205732
119895minus 120572119895
2 (1205722
119895+ 1205732
119895)
lt
1
radic21199020
119895 = 1 2
(35)
and let119896 (1199020 119908) equiv 119896 (119902
0 1198601 1198602 119908)
= exp
2
sum
119895=1
(21199020)minus121003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817119887119886119887
(36)
Then for all 120582 = (1205821 1205822) isin Γ1199020
10038161003816100381610038161003816120595 (120582 119908)
10038161003816100381610038161003816le exp
2
sum
119895=1
radic
radic1205722
119895+ 1205732
119895minus 120572119895
2 (1205722
119895+ 1205732
119895)
100381610038161003816100381610038161003816
(11986012
119895119908 119886)119887
100381610038161003816100381610038161003816
le exp
2
sum
119895=1
radic
radic1205722
119895+ 1205732
119895minus 120572119895
2 (1205722
119895+ 1205732
119895)
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817119887119886119887
lt 119896 (1199020 119908)
(37)
6 Journal of Function Spaces and Applications
We note that for all real 119902119895with |119902
119895| gt 1199020 119895 isin 1 2
(minus119894119902119895)
minus12
=
1
radic
100381610038161003816100381610038162119902119895
10038161003816100381610038161003816
+ sign (119902119895)
119894
radic
100381610038161003816100381610038162119902119895
10038161003816100381610038161003816
(38)
and (minus1198941199021 minus1198941199022) isin Γ1199020
For the existence of the GFFT of 119865 we define a subclass
F1199020
11986011198602
ofF 11988611988711986011198602
by 119865 isin F 119902011986011198602
if and only if
int
1198621015840
119886119887[0119879]
119896 (1199020 119908) 119889
10038161003816100381610038161198911003816100381610038161003816(119908) lt +infin (39)
where 119891 and 119865 are related by (30) and 119896 is given by (36)
Remark 8 Note that in case 119886(119905) equiv 0 and 119887(119905) = 119905 on [0 119879]the function space 119862
119886119887[0 119879] reduces to the classical Wiener
space 1198620[0 119879] and (119908 119886)
119887= 0 for all 119908 isin 1198621015840
119886119887[0 119879] =
1198621015840
0[0 119879] Hence for all 120582 isin C2
+ |120595( 120582 119908)| le 1 and for any
positive real number 1199020F 119902011986011198602
= F11986011198602
theKallianpur andBromleyrsquos class introduced in Section 1
Theorem 9 Let 1199020be a positive real number and let 119865 be an
element ofF 119902011986011198602
Then for any nonzero real numbers 1199021and
1199022with |119902
119895| gt 1199020 119895 isin 1 2 the 119871
1analytic GFFT of 119865 119879(1)
119902(119865)
exists and is given by the following formula
119879(1)
119902(119865) (119910
1 1199102)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
120595(minus119894 119902 119908) 119889119891 (119908)
(40)
for s-ae (1199101 1199102) isin 1198622
119886119887[0 119879] where 120595 is given by (32)
Proof We first note that for 119895 isin 1 2 the PWZ stochasticintegral (11986012
119895119908 119909)sim is a Gaussian random variable withmean
(11986012
119895119908 119886)119887and variance 11986012
119895119908
2
119887
= (119860119895119908119908)119887
Hence using(30) the Fubini theorem the change of variables theorem and(14) we have that for all 120582
1gt 0 and 120582
2gt 0
119869 (1199101 1199102 1205821 1205822)
equiv 119864119909[119865 (1199101+ 120582minus12
11199091 1199102+ 120582minus12
21199092)]
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
times(
2
prod
119895=1
119864119909119895
[exp 119894120582minus12119895(11986012
119895119908 119909119895)
sim
])119889119891 (119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
times[
[
[
2
prod
119895=1
(2120587(119860119895119908119908)
119887
)
minus12
times int
R
exp
119894120582minus12
119895119906119895
minus
[119906119895minus (11986012
119895119908 119886)119887
]
2
2(119860119895119908119908)
119887
119889119906119895
]
]
]
119889119891(119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
120595(120582 119908) 119889119891 (119908)
(41)
Let
119879 120582(119865) (119910
1 1199102)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
120595(120582 119908) 119889119891 (119908)
(42)
for each 120582 isin C2+ Clearly
119879 120582(119865) (119910
1 1199102) = 119869 (119910
1 1199102 1205821 1205822) (43)
for all 1205821gt 0 and 120582
2gt 0 Let Γ
1199020
be given by (35)Then for all120582 isin Int(Γ
1199020
)
1003816100381610038161003816119879 120582(119865) (119910
1 1199102)1003816100381610038161003816lt int
1198621015840
119886119887[0119879]
119896 (1199020 119908) 119889
10038161003816100381610038161198911003816100381610038161003816(119908) lt +infin
(44)
Using this fact and the dominated convergence theoremwe see that 119879
120582(119865)(1199101 1199102) is a continuous function of 120582 =
(1205821 1205822) on Int(Γ
1199020
) For each 119908 isin 1198621015840119886119887[0 119879] 120595( 120582 119908) is an
analytic function of 120582 throughout the domain Int(Γ1199020
) so thatintΔ
120595(120582 119908)119889
120582 = 0 for every rectifiable simple closed curve
Δ in Int(Γ1199020
) By (42) the Fubini theorem and the Moreratheorem we see that 119879
120582(119865)(1199101 1199102) is an analytic function of
120582 throughout the domain Int(Γ
1199020
) Finally using (28) withthe dominated convergence theorem we obtain the desiredresult
Theorem 10 Let 1199020and 119865 be as inTheorem 9Then for all 119901 isin
(1 2] and all nonzero real numbers 1199021and 119902
2with |119902
119895| gt 1199020
119895 isin 1 2 the119871119901analytic GFFT of119865119879(119901)
119902(119865) exists and is given
by the right hand side of (40) for s-ae (1199101 1199102) isin 1198622
119886119887[0 119879]
Journal of Function Spaces and Applications 7
Proof Let Γ1199020
be given by (35) It was shown in the proofof Theorem 9 that 119879
120582(119865)(1199101 1199102) is an analytic function of 120582
throughout the domain Int(Γ1199020
) In viewofDefinition 4 it willsuffice to show that for each 120588
1gt 0 and 120588
2gt 0
lim120582rarrminus119894 119902
int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816
119879 120582(119865) (120588
11199101 12058821199102)
minus119879(119901)
119902(119865) (120588
11199101 12058821199102)
100381610038161003816100381610038161003816
1199011015840
times 119889 (120583 times 120583) (1199101 1199102) = 0
(45)
Fixing 119901 isin (1 2] and using the inequalities (37) and (39)we obtain that for all 120588
119895gt 0 119895 isin 1 2 and all 120582 isin Γ
1199020
100381610038161003816100381610038161003816
119879 120582(119865) (120588
11199101 12058821199102) minus 119879(119901)
119902(119865) (120588
11199101 12058821199102)
100381610038161003816100381610038161003816
1199011015840
le
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894120588119895(11986012
119895119908 119910119895)
sim
times[120595 (120582 119908) minus 120595 (minus119894 119902 119908)] 119889119891 (119908)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199011015840
le (int
1198621015840
119886119887[0119879]
[
10038161003816100381610038161003816120595 (120582 119908)
10038161003816100381610038161003816+
10038161003816100381610038161003816120595 (minus119894 119902 119908)
10038161003816100381610038161003816] 11988910038161003816100381610038161198911003816100381610038161003816(119908))
1199011015840
le (2int
1198621015840
119886119887[0119879]
119896 (1199020 119908) 119889
10038161003816100381610038161198911003816100381610038161003816(119908))
1199011015840
lt +infin
(46)
Hence by the dominated convergence theorem we see thatfor each 119901 isin (1 2] and each 120588
1gt 0 and 120588
2gt 0
lim120582rarrminus119894 119902
int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816
119879 120582(119865) (120588
11199101 12058821199102)
minus119879(119901)
119902(119865) (120588
11199101 12058821199102)
100381610038161003816100381610038161003816
1199011015840
119889 (120583 times 120583) (1199101 1199102)
= lim120582rarrminus119894 119902
int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119910119895)
sim
times 120595(120582 119908) 119889119891 (119908)
minus int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119910119895)
sim
times120595(minus119894 119902 119908) 119889119891 (119908)
10038161003816100381610038161003816100381610038161003816100381610038161003816
1199011015840
times 119889 (120583 times 120583) (1199101 1199102)
= int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119910119895)
sim
times lim120582rarrminus119894 119902
[120595 (120582 119908)
minus120595 (minus119894 119902 119908)] 119889119891 (119908)
10038161003816100381610038161003816100381610038161003816100381610038161003816
1199011015840
times 119889 (120583 times 120583) (1199101 1199102)
= 0
(47)
which concludes the proof of Theorem 10
Remark 11 (1) In view of Theorems 9 and 10 we see thatfor each 119901 isin [1 2] the 119871
119901analytic GFFT of 119865 119879(119901)
119902(119865) is
given by the right hand side of (40) for 1199020 1199021 1199022 and 119865 as
in Theorem 9 and for s-ae (1199101 1199102) isin 1198622
119886119887[0 119879]
119879(119901)
119902(119865) (119910
1 1199102) = 119864
anf 119902119909[119865 (1199101+ 1199091 1199102+ 1199092)] 119901 isin [1 2]
(48)
In particular using this fact and (29) we have that for all 119901 isin[1 2]
119879(119901)
119902(119865) (0 0) = 119864
anf 119902119909[119865 (1199091 1199092)] (49)
(2) For nonzero real numbers 1199021and 119902
2with |119902
119895| gt 119902
0
119895 isin 1 2 define a set function 119891 119860119902B(1198621015840
119886119887[0 119879]) rarr C by
119891
119860
119902(119861) = int
119861
120595 (minus119894 119902 119908) 119889119891 (119908) 119861 isinB (1198621015840
119886119887[0 119879])
(50)
where 119891 and 119865 are related by (30) and B(1198621015840119886119887[0 119879]) is the
Borel 120590-algebra of 1198621015840119886119887[0 119879] Then it is obvious that 119891 119860
119902
belongs to M(1198621015840119886119887[0 119879]) and for all 119901 isin [1 2] 119879(119901)
119902(119865) can
be expressed as
119879(119901)
119902(119865) (119910
1 1199102) = int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
119889119891
119860
119902(119908)
(51)
for s-ae (1199101 1199102) isin 119862
2
119886119887[0 119879] Hence 119879(119901)
119902(119865) belongs to
F 11988611988711986011198602
for all 119901 isin [1 2]
4 Relationships between the GFFT and theFunction Space Integral of Functionals inF11988611988711986011198602
In this section we establish a relationship between the GFFTand the function space integral of functionals in the Fresneltype classF119886119887
11986011198602
8 Journal of Function Spaces and Applications
Throughout this section for convenience we use thefollowing notation for given 120582 isin C
+and 119899 = 1 2 let
119866119899(120582 119909)
= exp[1 minus 1205822
]
119899
sum
119896=1
[(119890119896 119909)sim
]
2
+ (12058212
minus 1)
119899
sum
119896=1
(119890119896 119886)1198621015840
119886119887
(119890119896 119909)sim
(52)
where 119890119899infin
119899=1is a complete orthonormal set in (1198621015840
119886119887[0 119879]
sdot 119887)To obtain our main results Theorems 14 and 17 below
we state a fundamental integration formula for the functionspace 119862
119886119887[0 119879]
Let 1198901 119890
119899 be an orthonormal set in (1198621015840
119886119887[0 119879] sdot
119887)
let 119896 R119899 rarr C be a Lebesgue measurable function and let119870 119862119886119887[0 119879] rarr C be given by
119870 (119909) = 119896 ((1198901 119909)sim
(119890119899 119909)sim
) (53)
Then
119864 [119870] = int
119862119886119887[0119879]
119896 ((1198901 119909)sim
(119890119899 119909)sim
) 119889120583 (119909)
= (2120587)minus1198992
int
R119899119896 (1199061 119906
119899)
times exp
minus
119899
sum
119895=1
[119906119895minus (119890119895 119886)119887
]
2
2
1198891199061 119889119906
119899
(54)
in the sense that if either side of (54) exists both sides existand equality holds
We also need the following lemma to obtain our maintheorem in this section
Lemma 12 Let 1198901 119890
119899 be an orthonormal subset of
(1198621015840
119886119887[0 119879] sdot
119887) and let 119908 be an element of 1198621015840
119886119887[0 119879] Then
for each 120582 isin C+ the function space integral
119864119909[119866119899(120582 119909) exp 119894(119908 119909)sim] (55)
exists and is given by the formula
119864119909[119866119899(120582 119909) exp 119894(119908 119909)sim]
= 120582minus1198992 exp
[
120582 minus 1
2120582
]
119899
sum
119896=1
(119890119896 119908)2
119887minus
1
2
1199082
119887
+ 119894120582minus12
119899
sum
119896=1
(119890119896 119886)119887(119890119896 119908)119887
+ 119894(119890119899+1 119886)119887[1199082
119887minus
119899
sum
119896=1
(119890119896 119908)2
119887]
12
(56)
where 119866119899is given by (52) above and
119890119899+1=[
[
1199082
119887minus
119899
sum
119895=1
(119890119895 119908)
2
119887
]
]
minus12
119908 minus
119899
sum
119895=1
(119890119895 119908)119887
119890119895
(57)
Proof (Outline) Using the Gram-Schmidt process for any119908 isin 119862
1015840
119886119887[0 119879] we can write 119908 = sum
119899+1
119896=1119888119896119890119896 where
1198901 119890
119899 119890119899+1 is an orthonormal set in (1198621015840
119886119887[0 119879] sdot
119887)
and
119888119896=
(119890119896 119908)119887 119896 = 1 119899
[1199082
119887minus
119899
sum
119895=1
(119890119895 119908)
2
119887
]
12
119896 = 119899 + 1
(58)
Then using (52) (54) the Fubini theorem and (14) it followsthat (56) holds for all 120582 isin C
+
The following remark will be very useful in the proof ofour main theorem in this section
Remark 13 Let 1199020be a positive real number and let Γ
1199020
begiven by (35) For real numbers 119902
1and 1199022with |119902
119895| gt 1199020 119895 isin
1 2 let 120582119899infin
119899=1= (1205821119899 1205822119899)infin
119899=1be a sequence in C2
+such
that
120582119899= (1205821119899 1205822119899) 997888rarr minus119894 119902 = (minus119894119902
1 minus1198941199022) (59)
Let 120582119895119899= 120572119895119899+ 119894120573119895119899
for 119895 isin 1 2 and 119899 isin N Then for119895 isin 1 2 Re(120582
119895119899) = 120572119895119899gt 0 and
120582minus1
119895119899= (120572119895119899+ 119894120573119895119899)
minus1
=
120572119895119899minus 119894120573119895119899
1205722
119895119899+ 1205732
119895119899
(60)
for each 119899 isin N Since |Im ((minus119894119902119895)minus12
)| = 1radic2|119902119895| lt 1radic2119902
0
for 119895 isin 1 2 there exists a sufficiently large 119871 isin N such thatfor any 119899 ge 119871 120582
1119899and 120582
2119899are in Int(Γ
1199020
) and
120575 (1199021 1199022) equiv sup ( 1003816100381610038161003816
1003816Im (120582minus12
1119899)
10038161003816100381610038161003816 119899 ge 119871
cup
10038161003816100381610038161003816Im (120582minus12
2119899)
10038161003816100381610038161003816 119899 ge 119871
cup
100381610038161003816100381610038161003816
Im ((minus1198941199021)minus12
)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
Im ((minus1198941199022)minus12
)
100381610038161003816100381610038161003816
)
lt
1
radic21199020
(61)
Thus there exists a positive real number 120576 gt 1 such that120575(1199021 1199022) lt 1(120576radic2119902
0)
Let 119890119899infin
119899=1be a complete orthonormal set in (1198621015840
119886119887[0 119879]
sdot 119887) Using Parsevalrsquos identity it follows that
(1198921 1198922)119887=
infin
sum
119899=1
(119890119899 1198921)119887(119890119899 1198922)119887
(62)
Journal of Function Spaces and Applications 9
for all 1198921 1198922isin 1198621015840
119886119887[0 119879] In addition for each 119892 isin 1198621015840
119886119887[0 119879]
10038171003817100381710038171198921003817100381710038171003817
2
119887minus
119899
sum
119896=1
(119890119896 119892)2
119887=
infin
sum
119896=119899+1
(119890119896 119892)2
119887ge 0 (63)
for every 119899 isin NSince
(119892 119886)119887=
infin
sum
119899=1
(119890119899 119892)119887(119890119899 119886)119887
(64)
and for 120576 gt 1
minus12057610038171003817100381710038171198921003817100381710038171003817119887119886119887lt minus10038171003817100381710038171198921003817100381710038171003817119887119886119887le (119892 119886)
119887
le10038171003817100381710038171198921003817100381710038171003817119887119886119887lt 12057610038171003817100381710038171198921003817100381710038171003817119887119886119887
(65)
there exists a sufficiently large119870119895isin N such that for any 119899 ge 119870
119895
1003816100381610038161003816100381610038161003816100381610038161003816
119899
sum
119896=1
(119890119896 11986012
119895119908)119887
(119890119896 119886)119887
1003816100381610038161003816100381610038161003816100381610038161003816
lt 120576
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817119887119886119887
(66)
for 119895 isin 1 2Using these and a long and tedious calculation we can
show that for every 119899 ge max119871 1198701 1198702
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
exp
2
sum
119895=1
([
120582119895119899minus 1
2120582119895119899
]
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
minus
1
2
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
+ 119894120582minus12
119895119899
119899
sum
119896=1
(119890119896 11986012
119895119908)119887
(119890119896 119886)119887+ 119894(119890119899+1 119886)119887
times [
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
minus
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
]
12
)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119896 (1199020 119908)
(67)
where 119896(1199020 119908) is given by (36)
In our next theorem for119865 isin F11988611988711986011198602
we express theGFFTof 119865 as the limit of a sequence of function space integrals on1198622
119886119887[0 119879]
Theorem 14 Let 1199020and 119865 be as in Theorem 10 Let 119890
119899infin
119899=1
be a complete orthonormal set in (1198621015840119886119887[0 119879] sdot
119887) and let
(1205821119899 1205822119899)infin
119899=1be a sequence in C2
+such that 120582
119895119899rarr minus119894119902
119895
where 119902119895is a real number with |119902
119895| gt 1199020 119895 isin 1 2 Then for
119901 isin [1 2] and for s-ae (1199101 1199102) isin 1198622
119886119887[0 119879]
119879(119901)
119902(119865) (119910
1 1199102)
= lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865 (119910
1+ 1199091 1199102+ 1199092)]
(68)
where 119866119899is given by (52)
Proof ByTheorems9 and 10we know that for each119901 isin [1 2]the 119871119901analytic GFFT of 119865 119879(119901)
119902(119865) exists and is given by the
right hand side of (40) Thus it suffices to show that
119879(1)
119902(119865) (119910
1 1199102) = 119864
anf 119902119909[119865 (1199101+ 1199091 1199102+ 1199092)]
= lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times 119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092)
times 119865 (1199101+ 1199091 1199102+ 1199092)] sdot
(69)
Using (30) the Fubini theorem and (56) with 120582 and 119908replaced with 120582
119895119899and 11986012
119895119908 119895 isin 1 2 respectively we see
that
1205821198992
11198991205821198992
2119899119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865 (119910
1+ 1199091 1199102+ 1199092)]
= 1205821198992
11198991205821198992
2119899int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
times(
2
prod
119895=1
119864119909119895
[119866119899(120582minus12
119895119899 119909119895)
times exp 119894(11986012119895119908 119909119895)
sim
])119889119891 (119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
(119894(11986012
119895119908 119910119895)
sim
+ [
120582119895119899minus 1
2120582119895119899
]
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
minus
1
2
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
+ 119894120582minus12
119895119899
119899
sum
119896=1
(119890119896 119886)119887(119890119896 11986012
119895119908)119887
+ 119894(119890119899+1 119886)119887[
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
minus
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
]
12
)
119889119891 (119908)
(70)
But by Remark 13 we see that the last expression of (70) isdominated by (39) on the region Γ
1199020
given by (35) for allbut a finite number of values of 119899 Next using the dominatedconvergence theorem Parsevalrsquos relation and (40) we obtainthe desired result
Corollary 15 Let 1199020 119865 119890
119899infin
119899=1 (1205821119899 1205822119899)infin
119899=1and (119902
1 1199022)
be as in Theorem 14 Then
119864
anf119902
119909[119865 (1199091 1199092)]
= lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times 119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865 (119909
1 1199092)]
(71)
where 119866119899is given by (52)
10 Journal of Function Spaces and Applications
Corollary 16 Let 1199020 119865 and 119890
119899infin
119899=1be as in Theorem 14 and
let Γ1199020
be given by (35) Let 120582 = (1205821 1205822) isin Int (Γ
1199020
) and(1205821119899 1205822119899)infin
119899=1be a sequence in C2
+such that 120582
119895119899rarr 120582
119895
119895 isin 1 2 Then
119864
an
119909[119865 (1199091 1199092)] = lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865(1199091 1199092)]
(72)
where 119866119899is given by (52)
Our another result namely a change of scale formula forfunction space integrals now follows fromCorollary 16 above
Theorem 17 Let 119865 isin F11988611988711986011198602
and let 119890119899infin
119899=1be a complete
orthonormal set in (1198621015840119886119887[0 119879] sdot
119887) Then for any 120588
1gt 0 and
1205882gt 0
119864119909[119865 (12058811199091 12058821199092)]
= lim119899rarrinfin
120588minus119899
1120588minus119899
2
times 119864119909[119866119899(120588minus2
1 1199091)119866119899(120588minus2
2 1199092) 119865 (119909
1 1199092)]
(73)
where 119866119899is given by (52)
Proof Simply choose 120582119895= 120588minus2
119895for 119895 isin 1 2 and 120582
119895119899= 120588minus2
119895
for 119895 isin 1 2 and 119899 isin N in (72)
Remark 18 Of course if we choose 119886(119905) equiv 0 119887(119905) = 1199051198601= 119868 (identity operator) and 119860
2= 0 (zero operator) then
the function space 119862119886119887[0 119879] reduces to the classical Wiener
space 1198620[0 119879] and the generalized Fresnel type class F 119886119887
11986011198602
reduces to the Fresnel class F(1198620[0 119879]) It is known that
F(1198620[0 119879]) forms a Banach algebra over the complex field
In this case we have the relationships between the analyticFeynman integral and theWiener integral on classicalWienerspace as discussed in [14 15]
In recent paper [19] Yoo et al have studied a change ofscale formula for function space integral of the functionalsin the Banach algebra S(1198712
119886119887[0 119879]) the Banach algebra
S(1198712119886119887[0 119879]) is introduced in [12]
5 Functionals in F11988611988711986011198602
In this section we prove a theorem ensuring that variousfunctionals are inF119886119887
11986011198602
Theorem 19 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] Let (119884Y 120574) be a 120590-finite
measure space and let 120593119897 119884 rarr 119862
1015840
119886119887[0 119879] beYndashB(1198621015840
119886119887[0 119879])
measurable for 119897 isin 1 119889 Let 120579 119884 times R119889 rarr C be given by120579(120578 sdot) = ]
120578(sdot) where ]
120578isin M(R119889) for every 120578 isin 119884 and where
the family ]120578 120578 isin 119884 satisfies
(i) ]120578(119864) is a Y-measurable function of 120578 for every 119864 isin
B(R119889)(ii) ]
120578 isin 1198711
(119884Y 120574)
Under these hypothesis the functional 119865 1198622119886119887[0 119879] rarr C
given by
119865 (1199091 1199092) = int
119884
120579(120578
2
sum
119895=1
(11986012
1198951205931(120578) 119909
119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 119909
119895)
sim
)119889120574 (120578)
(74)
belongs toF11988611988711986011198602
and satisfies the inequality
119865 le int
119884
10038171003817100381710038171003817]120578
10038171003817100381710038171003817119889120574 (120578) (75)
Proof Using the techniques similar to those used in [20] wecan show that ]
120578 is measurable as a function of 120578 that 120579 is
Y-measurable and that the integrand in (74) is a measurablefunction of 120578 for every (119909
1 1199092) isin 1198622
119886119887[0 119879]
We define a measure 120591 onY timesB(R119889) by
120591 (119864) = int
119884
]120578(119864(120578)
) 119889120574 (120578) for 119864 isin Y timesB (R119889) (76)
Then by the first assertion of Theorem 31 in [17] 120591 satisfies120591 le int
119884
]120578119889120574(120578) Now let Φ 119884 times R119889 rarr 119862
1015840
119886119887[0 119879] be
defined by Φ(120578 V1 V
119889) = sum
119889
119897=1V119897120593119897(120578) Then Φ is Y times
B(R119889) ndashB(1198621015840119886119887[0 119879])-measurable on the hypothesis for 120593
119897
119897 isin 1 119889 Let 120590 = 120591 ∘Φminus1 Then clearly 120590 isinM(1198621015840119886119887[0 119879])
and satisfies 120590 le 120591From the change of variables theorem and the second
assertion of Theorem 31 in [17] it follows that for ae(1199091 1199092) isin 1198622
119886119887[0 119879] and for every 120588
1gt 0 and 120588
2gt 0
119865 (12058811199091 12058821199092)
= int
119884
]120578(
2
sum
119895=1
(11986012
1198951205931(120578) 120588
119895119909119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 120588
119895119909119895)
sim
)119889120574 (120578)
= int
119884
[
[
int
R119889exp
119894
119889
sum
119897=1
V119897
[
[
2
sum
119895=1
(11986012
119895120593119897(120578)
120588119895119909119895)
sim
]
]
119889]120578
times (V1 V
119889)]
]
119889120574 (120578)
= int
119884timesR119889exp
119894
119889
sum
119897=1
V119897
[
[
2
sum
119895=1
(11986012
119895120593119897(120578) 120588
119895119909119895)
sim
]
]
119889120591
times (120578 V1 V
119889)
Journal of Function Spaces and Applications 11
= int
119884timesR119889exp
2
sum
119895=1
119894(11986012
119895Φ(120578 V
1 V
119889) 120588119895119909119895)
sim
119889120591
times (120578 V1 V
119889)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119909119895)
sim
119889120591 ∘ Φminus1
(119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119909119895)
sim
119889120590 (119908)
(77)
Thus the functional 119865 given by (74) belongs to F11988611988711986011198602
andsatisfies the inequality
119865 = 120590 le 120591 le int
119884
10038171003817100381710038171003817]120578
10038171003817100381710038171003817119889120574 (120578) (78)
As mentioned in (2) of Remark 6 F 11988611988711986011198602
is a Banachalgebra if Ran(119860
1+ 1198602) is dense in 1198621015840
119886119887[0 119879] In this case
many analytic functionals of 119865 can be formed The followingcorollary is relevant to Feynman integration theories andquantum mechanics where exponential functions play animportant role
Corollary 20 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] such that Ran(119860
1+ 1198602)
is dense in 1198621015840119886119887[0 119879] Let 119865 be given by (74) with 120579 as in
Theorem 19 and let 120573 C rarr C be an entire function Then(120573 ∘ 119865)(119909
1 1199092) is in F119886119887
11986011198602
In particular exp119865(1199091 1199092) isin
F11988611988711986011198602
Corollary 21 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] and let 119892
1 119892
119889 be a
finite subset of 1198621015840119886119887[0 119879] Given 120573 = ] where ] isin M(R119889)
define 119865 1198622119886119887[0 119879] rarr C by
119865 (1199091 1199092) = 120573(
2
sum
119895=1
(11986012
1198951198921 119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 119909119895)
sim
)
(79)
Then 119865 is an element ofF11988611988711986011198602
Proof Let (119884Y 120574) be a probability space and for 119897 isin1 119889 let 120593
119897(120578) equiv 119892
119897 Take 120579(120578 sdot) = 120573(sdot) = ](sdot) Then for
all 1205881gt 0 and 120588
2gt 0 and for ae (119909
1 1199092) isin 1198622
119886119887[0 119879]
int
119884
120579(120578
2
sum
119895=1
(11986012
1198951205931(120578) 120588
119895119909119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 120588
119895119909119895)
sim
)119889120574 (120578)
= int
119884
120573(
2
sum
119895=1
(11986012
1198951198921 120588119895119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 120588119895119909119895)
sim
)119889120574 (120578)
= 120573(
2
sum
119895=1
(11986012
1198951198921 120588119895119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 120588119895119909119895)
sim
)
= 119865 (12058811199091 12058821199092)
(80)
Hence 119865 isin F11988611988711986011198602
Remark 22 Let 119889 = 1 and let (119884Y 120574) = ([0 119879]B([0 119879])119898119871) in Theorem 19 where 119898
119871denotes the Lebesgue measure
on [0 119879] ThenTheorems 46 47 and 49 in [18] follow fromthe results in this section by letting119860
1be the identity operator
and letting 1198602equiv 0 on 1198621015840
119886119887[0 119879] The function 120579 studied in
[18] (and mentioned in Remark 2 above) is interpreted as thepotential energy in quantum mechanics
Acknowledgments
The authors would like to express their gratitude to the refer-ees for their valuable comments and suggestions which haveimproved the original paper This research was supported bythe Basic Science Research Program through the NationalResearch Foundation of Korea (NRF) funded by theMinistryof Education (2011-0014552)
References
[1] G Kallianpur and C Bromley ldquoGeneralized Feynman integralsusing analytic continuation in several complex variablesrdquo inStochastic Analysis and Applications M A Pinsky Ed vol 7pp 217ndash267 Marcel Dekker New York NY USA 1984
[2] G Kallianpur D Kannan and R L Karandikar ldquoAnalytic andsequential Feynman integrals on abstract Wiener and Hilbertspaces and a Cameron-Martin formulardquo Annales de lrsquoInstitutHenri Poincare vol 21 no 4 pp 323ndash361 1985
[3] G W Johnson and M L Lapidus The Feynman Integral andFeynmanrsquos Operational Calculus Clarendon Press Oxford UK2000
[4] J Yeh ldquoSingularity of Gaussian measures on function spacesinduced by Brownian motion processes with non-stationaryincrementsrdquo Illinois Journal of Mathematics vol 15 pp 37ndash461971
[5] J Yeh Stochastic Processes and the Wiener Integral MarcelDekker New York NY USA 1973
[6] S J Chang J G Choi and D Skoug ldquoIntegration by partsformulas involving generalized Fourier-Feynman transformson function spacerdquo Transactions of the American MathematicalSociety vol 355 no 7 pp 2925ndash2948 2003
[7] S J Chang J G Choi and D Skoug ldquoParts formulas involvingconditional generalized Feynman integrals and conditionalgeneralized Fourier-Feynman transforms on function spacerdquo
12 Journal of Function Spaces and Applications
Integral Transforms and Special Functions vol 15 no 6 pp 491ndash512 2004
[8] S J Chang J G Choi and D Skoug ldquoEvaluation formulas forconditional function space integralsmdashIrdquo Stochastic Analysis andApplications vol 25 no 1 pp 141ndash168 2007
[9] S J Chang J G Choi and D Skoug ldquoGeneralized Fourier-Feynman transforms convolution products and first variationson function spacerdquoTheRockyMountain Journal ofMathematicsvol 40 no 3 pp 761ndash788 2010
[10] S J Chang and D M Chung ldquoConditional function spaceintegrals with applicationsrdquo The Rocky Mountain Journal ofMathematics vol 26 no 1 pp 37ndash62 1996
[11] S J Chang H S Chung and D Skoug ldquoIntegral transformsof functionals in 119871
2(119862119886119887[0 119879])rdquoThe Journal of Fourier Analysis
and Applications vol 15 no 4 pp 441ndash462 2009[12] S J Chang and D Skoug ldquoGeneralized Fourier-Feynman
transforms and a first variation on function spacerdquo IntegralTransforms and Special Functions vol 14 pp 375ndash393 2003
[13] J G Choi and S J Chang ldquoGeneralized Fourier-Feynmantransform and sequential transforms on function spacerdquo Jour-nal of the Korean Mathematical Society vol 49 pp 1065ndash10822012
[14] R H Cameron and D A Storvick ldquoRelationships between theWiener integral and the analytic Feynman integralrdquo Rendicontidel Circolo Matematico di Palermo no 17 supplement pp 117ndash133 1987
[15] RHCameron andDA Storvick ldquoChange of scale formulas forWiener integralrdquo Rendiconti del Circolo Matematico di Palermono 17 pp 105ndash115 1987
[16] G W Johnson ldquoThe equivalence of two approaches to theFeynman integralrdquo Journal of Mathematical Physics vol 23 pp2090ndash2096 1982
[17] G W Johnson and D L Skoug ldquoNotes on the FeynmanintegralmdashIII the Schroedinger equationrdquo Pacific Journal ofMathematics vol 105 pp 321ndash358 1983
[18] S J Chang J G Choi and S D Lee ldquoA Fresnel type class onfunction spacerdquo Journal of the Korean Society of MathematicalEducation Series B vol 16 no 1 pp 107ndash119 2009
[19] I Yoo B J Kim and B S Kim ldquoA change of scale formulafor a function space integral on 119862
119886119887[0 119879]rdquo Proceedings of the
American Mathematical Society vol 141 no 8 pp 2729ndash27392013
[20] K S Chang G W Johnson and D L Skoug ldquoFunctions inthe Fresnel classrdquo Proceedings of the American MathematicalSociety vol 100 no 2 pp 309ndash318 1987
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces and Applications 3
Using (8) we observe that the linear operator given by(10) is an isometry In fact the inverse operator 119863minus1 11987112
119886119887[0 119879] rarr 119862
1015840
119886119887[0 119879] is given by
(119863minus1
119911) (119905) = int
119905
0
119911 (119904) 119889119887 (119904) (12)
Moreover the triple (1198621015840119886119887[0 119879] 119862
119886119887[0 119879] 120583) becomes an
abstract Wiener spaceThroughout this paper for 119908 isin 1198621015840
119886119887[0 119879] we will use the
notation (119908 119909)sim instead of ⟨119863119908 119909⟩We also use the followingnotations for 119908
1 1199082 119908 isin 1198621015840
119886119887[0 119879]
(1199081 1199082)119887= int
119879
0
1198631199081(119905) 119863119908
2(119905) 119889119887 (119905)
119908119887= radic(119908119908)
119887
(13)
Then 1198621015840119886119887[0 119879] with the inner product given by (13) is also a
separable Hilbert space It is easy to see that the two norms sdot 1198621015840
119886119887
and sdot 119887are equivalent Furthermore we have the
following assertions
(i) 119886(sdot) is an element of 1198621015840119886119887[0 119879]
(ii) For each 119908 isin 1198621015840119886119887[0 119879] the random variable 119909 997891rarr
(119908 119909)sim is Gaussian with mean (119908 119886)
119887and variance
1199082
119887
(iii) (119908 120572119909)sim = (120572119908 119909)sim = 120572(119908 119909)sim for any real number120572 119908 isin 1198621015840
119886119887[0 119879] and 119909 isin 119862
119886119887[0 119879]
(iv) Let 1199081 119908
119899 be a subset of 1198621015840
119886119887[0 119879] such that
int
119879
0
119863119908119894(119905)119863119908
119895(119905)119889119887(119905) = 120575
119894119895 where 120575
119894119895is the Kro-
necker delta Then the random variables (119908119894 119909)simrsquos are
independent
In this paper we adopt asmuch as possible the definitionsand notations used in [7 9 12 13] for the definitions ofthe generalized analytic Feynman integral and the GFFT offunctionals on 119862
119886119887[0 119879]
The following integration formula is used several times inthis paper
int
R
exp minus1205721199062 + 120573119906 119889119906 = radic120587120572
exp1205732
4120572
(14)
for complex numbers 120572 and 120573 with Re(120572) gt 0
3 The GFFT of Functionals in a BanachAlgebra F119886119887
11986011198602
LetM(1198621015840119886119887[0 119879]) be the space of complex-valued countably
additive (and hence finite) Borel measures on 1198621015840119886119887[0 119879]
M(1198621015840119886119887[0 119879]) is a Banach algebra under the total variation
norm and with convolution as multiplicationWe define the Fresnel type class F(119862
119886119887[0 119879]) of
functionals on 119862119886119887[0 119879] as the space of all stochastic
Fourier transforms of elements of M(1198621015840119886119887[0 119879]) that is
119865 isin F(119862119886119887[0 119879]) if and only if there exists a measure 119891 in
M(1198621015840119886119887[0 119879]) such that
119865 (119909) = int
1198621015840
119886119887[0119879]
exp 119894(119908 119909)sim 119889119891 (119908) (15)
for s-ae 119909 isin 119862119886119887[0 119879] More precisely since we will identify
functionals which coincide s-ae on 119862119886119887[0 119879]F(119862
119886119887[0 119879])
can be regarded as the space of all 119904-equivalence classes offunctionals of the form (15)
The Fresnel type class F(119862119886119887[0 119879]) is a Banach algebra
with norm
119865 =10038171003817100381710038171198911003817100381710038171003817= int
1198621015840
119886119887[0119879]
11988910038161003816100381610038161198911003816100381610038161003816(119908) (16)
In fact the correspondence 119891 997891rarr 119865 is injective carriesconvolution into pointwise multiplication and is a Banachalgebra isomorphism where 119891 and 119865 are related by (15)
Remark 2 The Banach algebra F(119862119886119887[0 119879]) contains sev-
eral interesting functions which arise naturally in quantummechanics Let M(R) be the class of C-valued countablyadditive measures on B(R) the Borel class of R For ] isinM(R) the Fourier transform ] of ] is a complex-valuedfunction defined on R by the following formula
] (119906) = intR
exp 119894119906V 119889] (V) (17)
Let G be the set of all complex-valued functions on[0 119879]timesR of the form 120579(119904 119906) =
119904(119906) where 120590
119904 0 le 119904 le 119879 is
a family fromM(R) satisfying the following two conditions
(i) for every 119864 isinB(R) 120590119904(119864) is Borel measurable in 119904
(ii) int1198790
120590119904119889119887(119904) lt +infin
Let 120579 isin G and let119867 be given by
119867(119909) = expint119879
0
120579 (119905 119909 (119905)) 119889119905 (18)
for s-ae 119909 isin 119862119886119887[0 119879] Then using the methods similar
to those used in [18] we can show that the function 120579(119905 119906)is Borel-measurable and that 120579(119905 119909(119905)) int119879
0
120579(119905 119909(119905))119889119905 and119867(119909) are elements of F(119862
119886119887[0 119879]) These facts are relevant
to quantum mechanics where exponential functions play aprominent role
Let119860 be a nonnegative self-adjoint operator on1198621015840119886119887[0 119879]
and 119891 any complex measure on 1198621015840119886119887[0 119879] Then the func-
tional
119865 (119909) = int
1198621015840
119886119887[0119879]
exp 119894(11986012119908 119909)sim
119889119891 (119908) (19)
belongs to F(119862119886119887[0 119879]) because it can be rewritten as
int1198621015840
119886119887[0119879]
exp119894(119908 119909)sim119889119891119860(119908) for 119891
119860= 119891 ∘ (119860
12
)minus1 Let 119860 be
self-adjoint but not nonnegative Then 119860 has the form
119860 = 119860+
minus 119860minus
(20)
4 Journal of Function Spaces and Applications
where both 119860+ and 119860minus are bounded nonnegative and self-adjoint operators
In this section we will extend the ideas of [1] to obtainexpressions of the generalized analytic Feynman integral andthe GFFT of functionals of the form (19) when119860 is no longerrequired to be nonnegative To do this we will introducedefinitions and notations analogous to those in [7 12 13]
Let W(1198622119886119887[0 119879]) denote the class of all Wiener mea-
surable subsets of the product function space 1198622119886119887[0 119879] A
subset 119861 of 1198622119886119887[0 119879] is said to be scale-invariant measurable
provided (12058811199091 12058821199092) (119909
1 1199092) isin 119861 is W(1198622
119886119887[0 119879])-
measurable for every 1205881gt 0 and 120588
2gt 0 and a scale-
invariantmeasurable subset119873 of1198622119886119887[0 119879] is said to be scale-
invariant null provided (120583 times 120583)((12058811199091 12058821199092) (119909
1 1199092) isin
119873) = 0 for every 1205881gt 0 and 120588
2gt 0 A property that
holds except on a scale-invariant null set is said to hold s-ae on 1198622
119886119887[0 119879] A functional 119865 on 1198622
119886119887[0 119879] is said to be
scale-invariant measurable provided 119865 is defined on a scale-invariant measurable set and 119865(120588
1sdot 1205882sdot) is W(1198622
119886119887[0 119879])-
measurable for every 1205881gt 0 and 120588
2gt 0 If two functionals
119865 and 119866 defined on 1198622119886119887[0 119879] are equal s-ae then we write
119865 asymp 119866We denote the product function space integral of a
W(1198622119886119887[0 119879])-measurable functional 119865 by
119864 [119865] equiv 119864119909[119865 (1199091 1199092)]
= int
1198622
119886119887[0119879]
119865 (1199091 1199092) 119889 (120583 times 120583) (119909
1 1199092)
(21)
whenever the integral existsThroughout this paper letCC
+and C
+denote the set of
complex numbers complex numbers with positive real partand nonzero complex numbers with nonnegative real partrespectively Furthermore for all 120582 isin C
+ 120582minus12 (or 12058212) is
always chosen to have positive real part We also assume thatevery functional 119865 on 1198622
119886119887[0 119879] we consider is s-ae defined
and is scale-invariant measurable
Definition 3 Let C2+equiv C+times C+and let C2
+equivC+timesC+ Let
119865 1198622
119886119887[0 119879] rarr C be such that for each 120582
1gt 0 and 120582
2gt 0
the function space integral
119869 (1205821 1205822)
= int
1198622
119886119887[0119879]
119865 (120582minus12
11199091 120582minus12
21199092) 119889 (120583 times 120583) (119909
1 1199092)
(22)
exists If there exists a function 119869lowast(1205821 1205822) analytic inC2
+such
that 119869lowast(1205821 1205822) = 119869(120582
1 1205822) for all 120582
1gt 0 and 120582
2gt 0 then
119869lowast
(1205821 1205822) is defined to be the analytic function space integral
of 119865 over 1198622119886119887[0 119879] with parameter 120582 = (120582
1 1205822) and for 120582 isin
C2+we write
119864an[119865] equiv 119864
an
119909[119865 (1199091 1199092)]
equiv 119864
an(12058211205822)
11990911199092[119865 (1199091 1199092)] = 119869
lowast
(1205821 1205822)
(23)
Let 1199021and 1199022be nonzero real numbers Let 119865 be a functional
such that 119864an[119865] exists for all 120582 isin C2+ If the following limit
exists we call it the generalized analytic Feynman integral of119865 with parameter 119902 = (119902
1 1199022) and we write
119864anf 119902[119865] equiv 119864
anf 119902119909[119865 (1199091 1199092)]
equiv 119864
anf(11990211199022)
11990911199092[119865 (1199091 1199092)] = lim
120582rarrminus119894 119902
119864an[119865]
(24)
where 120582 = (1205821 1205822) rarr minus119894 119902 = (minus119894119902
1 minus1198941199022) through values in
C2+
Definition 4 Let 1199021and 1199022be nonzero real numbers For 120582 =
(1205821 1205822) isin C2+and (119910
1 1199102) isin 1198622
119886119887[0 119879] let
119879 120582(119865) (119910
1 1199102) equiv 119879(12058211205822)(119865) (119910
1 1199102)
= 119864
an
119909[119865 (1199101+ 1199091 1199102+ 1199092)]
(25)
For 119901 isin (1 2] we define the 119871119901analytic GFFT 119879(119901)
119902(119865) of 119865
by the formula ( 120582 isin C2+)
119879(119901)
119902(119865) (119910
1 1199102) equiv 119879(119901)
(1199021 1199022)(119865) (119910
1 1199102)
= lim120582rarrminus119894 119902
119879 120582(119865) (119910
1 1199102)
(26)
if it exists that is for each 1205881gt 0 and 120588
2gt 0
lim120582rarrminus119894 119902
int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816
119879 120582(119865) (120588
11199101 12058821199102)
minus119879(119901)
119902(119865) (120588
11199101 12058821199102)
100381610038161003816100381610038161003816
1199011015840
119889 (120583 times 120583) (1199101 1199102)
= 0
(27)
where 1119901+11199011015840 = 1We define the 1198711analytic GFFT119879(1)
119902(119865)
of 119865 by the formula ( 120582 isin C2+)
119879(1)
119902(119865) (119910
1 1199102) = lim120582rarrminus119894 119902
119879 120582(119865) (119910
1 1199102) (28)
if it exists
We note that for 1 le 119901 le 2 119879(119901)119902(119865) is defined only s-ae
We also note that if 119879(119901)119902(119865) exists and if 119865 asymp 119866 then 119879(119901)
119902(119866)
exists and 119879(119901)119902(119866) asymp 119879
(119901)
119902(119865) Moreover from Definition 4
we see that for 1199021 1199022isin R minus 0
119864
anf 119902119909[119865 (1199091 1199092)] = 119879
(1)
119902(119865) (0 0) (29)
Next we give the definition of the generalized Fresnel typeclassF119886119887
11986011198602
Journal of Function Spaces and Applications 5
Definition 5 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] The generalized Fresnel
type classF11988611988711986011198602
of functionals on1198622119886119887[0 119879] is defined as the
space of all functionals 119865 on 1198622119886119887[0 119879] of the following form
119865 (1199091 1199092) = int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119909119895)
sim
119889119891 (119908) (30)
for some119891 isinM(1198621015840119886119887[0 119879]) More precisely since we identify
functionals which coincide s-ae on 1198622119886119887[0 119879] F119886119887
11986011198602
canbe regarded as the space of all 119904-equivalence classes offunctionals of the form (30)
Remark 6 (1) In Definition 5 let 1198601be the identity operator
on 1198621015840119886119887[0 119879] and 119860
2equiv 0 Then F 119886119887
11986011198602
is essentially theFresnel type class F(119862
119886119887[0 119879]) and for 119901 isin [1 2] and
nonzero real numbers 1199021and 1199022
119879(119901)
(1199021 1199022)(119865) (119910
1 1199102) = 119879(119901)
1199021
(1198650) (1199101) (31)
if it exists where 1198650(1199091) = 119865(119909
1 1199092) for all (119909
1 1199092) isin
1198622
119886119887[0 119879] and 119879(119901)
1199021
(1198650) means the 119871
119901analytic GFFT on
119862119886119887[0 119879] see [6 12](2) The map 119891 997891rarr 119865 defined by (30) sets up an algebra
isomorphism betweenM(1198621015840119886119887[0 119879]) andF119886119887
11986011198602
if Ran(1198601+
1198602) is dense in1198621015840
119886119887[0 119879] where Ran indicates the range of an
operator In this caseF 11988611988711986011198602
becomes a Banach algebra underthe norm 119865 = 119891 For more details see [1]
Remark 7 Let 119865 be given by (30) In evaluating119864119909[119865(120582minus12
11199091 120582minus12
21199092)] and 119879
(12058211205822)(119865)(1199101 1199102) = 119864
119909[119865(1199101+
120582minus12
11199091 1199102+ 120582minus12
21199092)] for 120582
1gt 0 and 120582
2gt 0 the expression
120595 (120582 119908)
equiv 120595 (1205821 1205822 1198601 1198602 119908)
= exp
2
sum
119895=1
[
[
minus
(119860119895119908119908)
119887
2120582119895
+ 119894120582minus12
119895(11986012
119895119908 119886)119887
]
]
(32)
occurs Clearly for 120582119895gt 0 119895 isin 1 2 |120595( 120582 119908)| le 1 for all
119908 isin 1198621015840
119886119887[0 119879] But for 120582 isin C2
+ |120595( 120582 119908)| is not necessarily
bounded by 1Note that for each 120582
119895isinC+with 120582
119895= 120572119895+ 119894120573119895 119895 isin 1 2
12058212
119895=
radicradic1205722
119895+ 1205732
119895+ 120572119895
2
+ 119894
120573119895
10038161003816100381610038161003816120573119895
10038161003816100381610038161003816
radicradic1205722
119895+ 1205732
119895minus 120572119895
2
120582minus12
119895=radic
radic1205722
119895+ 1205732
119895+ 120572119895
2 (1205722
119895+ 1205732
119895)
minus 119894
120573119895
10038161003816100381610038161003816120573119895
10038161003816100381610038161003816
radic
radic1205722
119895+ 1205732
119895minus 120572119895
2 (1205722
119895+ 1205732
119895)
(33)
Hence for 120582119895isinC+with 120582
119895= 120572119895+ 119894120573119895 119895 isin 1 2
10038161003816100381610038161003816120595 (120582 119908)
10038161003816100381610038161003816
= exp
2
sum
119895=1
[
[
[
[
minus
120572119895
2 (1205722
119895+ 1205732
119895)
(119860119895119908119908)
119887
+
120573119895
10038161003816100381610038161003816120573119895
10038161003816100381610038161003816
radic
radic1205722
119895+ 1205732
119895minus 120572119895
2 (1205722
119895+ 1205732
119895)
(11986012
119895119908 119886)119887
]
]
]
]
(34)
The right hand side of (34) is an unbounded functionof 119908 for 119908 isin 1198621015840
119886119887[0 119879] Thus 119864an[119865] 119864anf 119902[119865] 119879
120582(119865) and
119879(119901)
119902(119865) might not exist Thus throughout this paper we will
need to put additional restrictions on the complex measure119891 corresponding to 119865 in order to obtain our results for theGFFT and the generalized analytic Feynman integral of 119865
In view of Remark 7 we clearly need to impose additionalrestrictions on the functionals 119865 inF119886119887
11986011198602
For a positive real number 119902
0 let
Γ1199020
=
120582 = (120582
1 1205822) isinC2
+| 120582119895= 120572119895+ 119894120573119895
10038161003816100381610038161003816Im (120582minus12
119895)
10038161003816100381610038161003816=radic
radic1205722
119895+ 1205732
119895minus 120572119895
2 (1205722
119895+ 1205732
119895)
lt
1
radic21199020
119895 = 1 2
(35)
and let119896 (1199020 119908) equiv 119896 (119902
0 1198601 1198602 119908)
= exp
2
sum
119895=1
(21199020)minus121003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817119887119886119887
(36)
Then for all 120582 = (1205821 1205822) isin Γ1199020
10038161003816100381610038161003816120595 (120582 119908)
10038161003816100381610038161003816le exp
2
sum
119895=1
radic
radic1205722
119895+ 1205732
119895minus 120572119895
2 (1205722
119895+ 1205732
119895)
100381610038161003816100381610038161003816
(11986012
119895119908 119886)119887
100381610038161003816100381610038161003816
le exp
2
sum
119895=1
radic
radic1205722
119895+ 1205732
119895minus 120572119895
2 (1205722
119895+ 1205732
119895)
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817119887119886119887
lt 119896 (1199020 119908)
(37)
6 Journal of Function Spaces and Applications
We note that for all real 119902119895with |119902
119895| gt 1199020 119895 isin 1 2
(minus119894119902119895)
minus12
=
1
radic
100381610038161003816100381610038162119902119895
10038161003816100381610038161003816
+ sign (119902119895)
119894
radic
100381610038161003816100381610038162119902119895
10038161003816100381610038161003816
(38)
and (minus1198941199021 minus1198941199022) isin Γ1199020
For the existence of the GFFT of 119865 we define a subclass
F1199020
11986011198602
ofF 11988611988711986011198602
by 119865 isin F 119902011986011198602
if and only if
int
1198621015840
119886119887[0119879]
119896 (1199020 119908) 119889
10038161003816100381610038161198911003816100381610038161003816(119908) lt +infin (39)
where 119891 and 119865 are related by (30) and 119896 is given by (36)
Remark 8 Note that in case 119886(119905) equiv 0 and 119887(119905) = 119905 on [0 119879]the function space 119862
119886119887[0 119879] reduces to the classical Wiener
space 1198620[0 119879] and (119908 119886)
119887= 0 for all 119908 isin 1198621015840
119886119887[0 119879] =
1198621015840
0[0 119879] Hence for all 120582 isin C2
+ |120595( 120582 119908)| le 1 and for any
positive real number 1199020F 119902011986011198602
= F11986011198602
theKallianpur andBromleyrsquos class introduced in Section 1
Theorem 9 Let 1199020be a positive real number and let 119865 be an
element ofF 119902011986011198602
Then for any nonzero real numbers 1199021and
1199022with |119902
119895| gt 1199020 119895 isin 1 2 the 119871
1analytic GFFT of 119865 119879(1)
119902(119865)
exists and is given by the following formula
119879(1)
119902(119865) (119910
1 1199102)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
120595(minus119894 119902 119908) 119889119891 (119908)
(40)
for s-ae (1199101 1199102) isin 1198622
119886119887[0 119879] where 120595 is given by (32)
Proof We first note that for 119895 isin 1 2 the PWZ stochasticintegral (11986012
119895119908 119909)sim is a Gaussian random variable withmean
(11986012
119895119908 119886)119887and variance 11986012
119895119908
2
119887
= (119860119895119908119908)119887
Hence using(30) the Fubini theorem the change of variables theorem and(14) we have that for all 120582
1gt 0 and 120582
2gt 0
119869 (1199101 1199102 1205821 1205822)
equiv 119864119909[119865 (1199101+ 120582minus12
11199091 1199102+ 120582minus12
21199092)]
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
times(
2
prod
119895=1
119864119909119895
[exp 119894120582minus12119895(11986012
119895119908 119909119895)
sim
])119889119891 (119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
times[
[
[
2
prod
119895=1
(2120587(119860119895119908119908)
119887
)
minus12
times int
R
exp
119894120582minus12
119895119906119895
minus
[119906119895minus (11986012
119895119908 119886)119887
]
2
2(119860119895119908119908)
119887
119889119906119895
]
]
]
119889119891(119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
120595(120582 119908) 119889119891 (119908)
(41)
Let
119879 120582(119865) (119910
1 1199102)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
120595(120582 119908) 119889119891 (119908)
(42)
for each 120582 isin C2+ Clearly
119879 120582(119865) (119910
1 1199102) = 119869 (119910
1 1199102 1205821 1205822) (43)
for all 1205821gt 0 and 120582
2gt 0 Let Γ
1199020
be given by (35)Then for all120582 isin Int(Γ
1199020
)
1003816100381610038161003816119879 120582(119865) (119910
1 1199102)1003816100381610038161003816lt int
1198621015840
119886119887[0119879]
119896 (1199020 119908) 119889
10038161003816100381610038161198911003816100381610038161003816(119908) lt +infin
(44)
Using this fact and the dominated convergence theoremwe see that 119879
120582(119865)(1199101 1199102) is a continuous function of 120582 =
(1205821 1205822) on Int(Γ
1199020
) For each 119908 isin 1198621015840119886119887[0 119879] 120595( 120582 119908) is an
analytic function of 120582 throughout the domain Int(Γ1199020
) so thatintΔ
120595(120582 119908)119889
120582 = 0 for every rectifiable simple closed curve
Δ in Int(Γ1199020
) By (42) the Fubini theorem and the Moreratheorem we see that 119879
120582(119865)(1199101 1199102) is an analytic function of
120582 throughout the domain Int(Γ
1199020
) Finally using (28) withthe dominated convergence theorem we obtain the desiredresult
Theorem 10 Let 1199020and 119865 be as inTheorem 9Then for all 119901 isin
(1 2] and all nonzero real numbers 1199021and 119902
2with |119902
119895| gt 1199020
119895 isin 1 2 the119871119901analytic GFFT of119865119879(119901)
119902(119865) exists and is given
by the right hand side of (40) for s-ae (1199101 1199102) isin 1198622
119886119887[0 119879]
Journal of Function Spaces and Applications 7
Proof Let Γ1199020
be given by (35) It was shown in the proofof Theorem 9 that 119879
120582(119865)(1199101 1199102) is an analytic function of 120582
throughout the domain Int(Γ1199020
) In viewofDefinition 4 it willsuffice to show that for each 120588
1gt 0 and 120588
2gt 0
lim120582rarrminus119894 119902
int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816
119879 120582(119865) (120588
11199101 12058821199102)
minus119879(119901)
119902(119865) (120588
11199101 12058821199102)
100381610038161003816100381610038161003816
1199011015840
times 119889 (120583 times 120583) (1199101 1199102) = 0
(45)
Fixing 119901 isin (1 2] and using the inequalities (37) and (39)we obtain that for all 120588
119895gt 0 119895 isin 1 2 and all 120582 isin Γ
1199020
100381610038161003816100381610038161003816
119879 120582(119865) (120588
11199101 12058821199102) minus 119879(119901)
119902(119865) (120588
11199101 12058821199102)
100381610038161003816100381610038161003816
1199011015840
le
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894120588119895(11986012
119895119908 119910119895)
sim
times[120595 (120582 119908) minus 120595 (minus119894 119902 119908)] 119889119891 (119908)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199011015840
le (int
1198621015840
119886119887[0119879]
[
10038161003816100381610038161003816120595 (120582 119908)
10038161003816100381610038161003816+
10038161003816100381610038161003816120595 (minus119894 119902 119908)
10038161003816100381610038161003816] 11988910038161003816100381610038161198911003816100381610038161003816(119908))
1199011015840
le (2int
1198621015840
119886119887[0119879]
119896 (1199020 119908) 119889
10038161003816100381610038161198911003816100381610038161003816(119908))
1199011015840
lt +infin
(46)
Hence by the dominated convergence theorem we see thatfor each 119901 isin (1 2] and each 120588
1gt 0 and 120588
2gt 0
lim120582rarrminus119894 119902
int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816
119879 120582(119865) (120588
11199101 12058821199102)
minus119879(119901)
119902(119865) (120588
11199101 12058821199102)
100381610038161003816100381610038161003816
1199011015840
119889 (120583 times 120583) (1199101 1199102)
= lim120582rarrminus119894 119902
int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119910119895)
sim
times 120595(120582 119908) 119889119891 (119908)
minus int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119910119895)
sim
times120595(minus119894 119902 119908) 119889119891 (119908)
10038161003816100381610038161003816100381610038161003816100381610038161003816
1199011015840
times 119889 (120583 times 120583) (1199101 1199102)
= int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119910119895)
sim
times lim120582rarrminus119894 119902
[120595 (120582 119908)
minus120595 (minus119894 119902 119908)] 119889119891 (119908)
10038161003816100381610038161003816100381610038161003816100381610038161003816
1199011015840
times 119889 (120583 times 120583) (1199101 1199102)
= 0
(47)
which concludes the proof of Theorem 10
Remark 11 (1) In view of Theorems 9 and 10 we see thatfor each 119901 isin [1 2] the 119871
119901analytic GFFT of 119865 119879(119901)
119902(119865) is
given by the right hand side of (40) for 1199020 1199021 1199022 and 119865 as
in Theorem 9 and for s-ae (1199101 1199102) isin 1198622
119886119887[0 119879]
119879(119901)
119902(119865) (119910
1 1199102) = 119864
anf 119902119909[119865 (1199101+ 1199091 1199102+ 1199092)] 119901 isin [1 2]
(48)
In particular using this fact and (29) we have that for all 119901 isin[1 2]
119879(119901)
119902(119865) (0 0) = 119864
anf 119902119909[119865 (1199091 1199092)] (49)
(2) For nonzero real numbers 1199021and 119902
2with |119902
119895| gt 119902
0
119895 isin 1 2 define a set function 119891 119860119902B(1198621015840
119886119887[0 119879]) rarr C by
119891
119860
119902(119861) = int
119861
120595 (minus119894 119902 119908) 119889119891 (119908) 119861 isinB (1198621015840
119886119887[0 119879])
(50)
where 119891 and 119865 are related by (30) and B(1198621015840119886119887[0 119879]) is the
Borel 120590-algebra of 1198621015840119886119887[0 119879] Then it is obvious that 119891 119860
119902
belongs to M(1198621015840119886119887[0 119879]) and for all 119901 isin [1 2] 119879(119901)
119902(119865) can
be expressed as
119879(119901)
119902(119865) (119910
1 1199102) = int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
119889119891
119860
119902(119908)
(51)
for s-ae (1199101 1199102) isin 119862
2
119886119887[0 119879] Hence 119879(119901)
119902(119865) belongs to
F 11988611988711986011198602
for all 119901 isin [1 2]
4 Relationships between the GFFT and theFunction Space Integral of Functionals inF11988611988711986011198602
In this section we establish a relationship between the GFFTand the function space integral of functionals in the Fresneltype classF119886119887
11986011198602
8 Journal of Function Spaces and Applications
Throughout this section for convenience we use thefollowing notation for given 120582 isin C
+and 119899 = 1 2 let
119866119899(120582 119909)
= exp[1 minus 1205822
]
119899
sum
119896=1
[(119890119896 119909)sim
]
2
+ (12058212
minus 1)
119899
sum
119896=1
(119890119896 119886)1198621015840
119886119887
(119890119896 119909)sim
(52)
where 119890119899infin
119899=1is a complete orthonormal set in (1198621015840
119886119887[0 119879]
sdot 119887)To obtain our main results Theorems 14 and 17 below
we state a fundamental integration formula for the functionspace 119862
119886119887[0 119879]
Let 1198901 119890
119899 be an orthonormal set in (1198621015840
119886119887[0 119879] sdot
119887)
let 119896 R119899 rarr C be a Lebesgue measurable function and let119870 119862119886119887[0 119879] rarr C be given by
119870 (119909) = 119896 ((1198901 119909)sim
(119890119899 119909)sim
) (53)
Then
119864 [119870] = int
119862119886119887[0119879]
119896 ((1198901 119909)sim
(119890119899 119909)sim
) 119889120583 (119909)
= (2120587)minus1198992
int
R119899119896 (1199061 119906
119899)
times exp
minus
119899
sum
119895=1
[119906119895minus (119890119895 119886)119887
]
2
2
1198891199061 119889119906
119899
(54)
in the sense that if either side of (54) exists both sides existand equality holds
We also need the following lemma to obtain our maintheorem in this section
Lemma 12 Let 1198901 119890
119899 be an orthonormal subset of
(1198621015840
119886119887[0 119879] sdot
119887) and let 119908 be an element of 1198621015840
119886119887[0 119879] Then
for each 120582 isin C+ the function space integral
119864119909[119866119899(120582 119909) exp 119894(119908 119909)sim] (55)
exists and is given by the formula
119864119909[119866119899(120582 119909) exp 119894(119908 119909)sim]
= 120582minus1198992 exp
[
120582 minus 1
2120582
]
119899
sum
119896=1
(119890119896 119908)2
119887minus
1
2
1199082
119887
+ 119894120582minus12
119899
sum
119896=1
(119890119896 119886)119887(119890119896 119908)119887
+ 119894(119890119899+1 119886)119887[1199082
119887minus
119899
sum
119896=1
(119890119896 119908)2
119887]
12
(56)
where 119866119899is given by (52) above and
119890119899+1=[
[
1199082
119887minus
119899
sum
119895=1
(119890119895 119908)
2
119887
]
]
minus12
119908 minus
119899
sum
119895=1
(119890119895 119908)119887
119890119895
(57)
Proof (Outline) Using the Gram-Schmidt process for any119908 isin 119862
1015840
119886119887[0 119879] we can write 119908 = sum
119899+1
119896=1119888119896119890119896 where
1198901 119890
119899 119890119899+1 is an orthonormal set in (1198621015840
119886119887[0 119879] sdot
119887)
and
119888119896=
(119890119896 119908)119887 119896 = 1 119899
[1199082
119887minus
119899
sum
119895=1
(119890119895 119908)
2
119887
]
12
119896 = 119899 + 1
(58)
Then using (52) (54) the Fubini theorem and (14) it followsthat (56) holds for all 120582 isin C
+
The following remark will be very useful in the proof ofour main theorem in this section
Remark 13 Let 1199020be a positive real number and let Γ
1199020
begiven by (35) For real numbers 119902
1and 1199022with |119902
119895| gt 1199020 119895 isin
1 2 let 120582119899infin
119899=1= (1205821119899 1205822119899)infin
119899=1be a sequence in C2
+such
that
120582119899= (1205821119899 1205822119899) 997888rarr minus119894 119902 = (minus119894119902
1 minus1198941199022) (59)
Let 120582119895119899= 120572119895119899+ 119894120573119895119899
for 119895 isin 1 2 and 119899 isin N Then for119895 isin 1 2 Re(120582
119895119899) = 120572119895119899gt 0 and
120582minus1
119895119899= (120572119895119899+ 119894120573119895119899)
minus1
=
120572119895119899minus 119894120573119895119899
1205722
119895119899+ 1205732
119895119899
(60)
for each 119899 isin N Since |Im ((minus119894119902119895)minus12
)| = 1radic2|119902119895| lt 1radic2119902
0
for 119895 isin 1 2 there exists a sufficiently large 119871 isin N such thatfor any 119899 ge 119871 120582
1119899and 120582
2119899are in Int(Γ
1199020
) and
120575 (1199021 1199022) equiv sup ( 1003816100381610038161003816
1003816Im (120582minus12
1119899)
10038161003816100381610038161003816 119899 ge 119871
cup
10038161003816100381610038161003816Im (120582minus12
2119899)
10038161003816100381610038161003816 119899 ge 119871
cup
100381610038161003816100381610038161003816
Im ((minus1198941199021)minus12
)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
Im ((minus1198941199022)minus12
)
100381610038161003816100381610038161003816
)
lt
1
radic21199020
(61)
Thus there exists a positive real number 120576 gt 1 such that120575(1199021 1199022) lt 1(120576radic2119902
0)
Let 119890119899infin
119899=1be a complete orthonormal set in (1198621015840
119886119887[0 119879]
sdot 119887) Using Parsevalrsquos identity it follows that
(1198921 1198922)119887=
infin
sum
119899=1
(119890119899 1198921)119887(119890119899 1198922)119887
(62)
Journal of Function Spaces and Applications 9
for all 1198921 1198922isin 1198621015840
119886119887[0 119879] In addition for each 119892 isin 1198621015840
119886119887[0 119879]
10038171003817100381710038171198921003817100381710038171003817
2
119887minus
119899
sum
119896=1
(119890119896 119892)2
119887=
infin
sum
119896=119899+1
(119890119896 119892)2
119887ge 0 (63)
for every 119899 isin NSince
(119892 119886)119887=
infin
sum
119899=1
(119890119899 119892)119887(119890119899 119886)119887
(64)
and for 120576 gt 1
minus12057610038171003817100381710038171198921003817100381710038171003817119887119886119887lt minus10038171003817100381710038171198921003817100381710038171003817119887119886119887le (119892 119886)
119887
le10038171003817100381710038171198921003817100381710038171003817119887119886119887lt 12057610038171003817100381710038171198921003817100381710038171003817119887119886119887
(65)
there exists a sufficiently large119870119895isin N such that for any 119899 ge 119870
119895
1003816100381610038161003816100381610038161003816100381610038161003816
119899
sum
119896=1
(119890119896 11986012
119895119908)119887
(119890119896 119886)119887
1003816100381610038161003816100381610038161003816100381610038161003816
lt 120576
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817119887119886119887
(66)
for 119895 isin 1 2Using these and a long and tedious calculation we can
show that for every 119899 ge max119871 1198701 1198702
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
exp
2
sum
119895=1
([
120582119895119899minus 1
2120582119895119899
]
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
minus
1
2
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
+ 119894120582minus12
119895119899
119899
sum
119896=1
(119890119896 11986012
119895119908)119887
(119890119896 119886)119887+ 119894(119890119899+1 119886)119887
times [
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
minus
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
]
12
)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119896 (1199020 119908)
(67)
where 119896(1199020 119908) is given by (36)
In our next theorem for119865 isin F11988611988711986011198602
we express theGFFTof 119865 as the limit of a sequence of function space integrals on1198622
119886119887[0 119879]
Theorem 14 Let 1199020and 119865 be as in Theorem 10 Let 119890
119899infin
119899=1
be a complete orthonormal set in (1198621015840119886119887[0 119879] sdot
119887) and let
(1205821119899 1205822119899)infin
119899=1be a sequence in C2
+such that 120582
119895119899rarr minus119894119902
119895
where 119902119895is a real number with |119902
119895| gt 1199020 119895 isin 1 2 Then for
119901 isin [1 2] and for s-ae (1199101 1199102) isin 1198622
119886119887[0 119879]
119879(119901)
119902(119865) (119910
1 1199102)
= lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865 (119910
1+ 1199091 1199102+ 1199092)]
(68)
where 119866119899is given by (52)
Proof ByTheorems9 and 10we know that for each119901 isin [1 2]the 119871119901analytic GFFT of 119865 119879(119901)
119902(119865) exists and is given by the
right hand side of (40) Thus it suffices to show that
119879(1)
119902(119865) (119910
1 1199102) = 119864
anf 119902119909[119865 (1199101+ 1199091 1199102+ 1199092)]
= lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times 119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092)
times 119865 (1199101+ 1199091 1199102+ 1199092)] sdot
(69)
Using (30) the Fubini theorem and (56) with 120582 and 119908replaced with 120582
119895119899and 11986012
119895119908 119895 isin 1 2 respectively we see
that
1205821198992
11198991205821198992
2119899119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865 (119910
1+ 1199091 1199102+ 1199092)]
= 1205821198992
11198991205821198992
2119899int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
times(
2
prod
119895=1
119864119909119895
[119866119899(120582minus12
119895119899 119909119895)
times exp 119894(11986012119895119908 119909119895)
sim
])119889119891 (119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
(119894(11986012
119895119908 119910119895)
sim
+ [
120582119895119899minus 1
2120582119895119899
]
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
minus
1
2
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
+ 119894120582minus12
119895119899
119899
sum
119896=1
(119890119896 119886)119887(119890119896 11986012
119895119908)119887
+ 119894(119890119899+1 119886)119887[
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
minus
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
]
12
)
119889119891 (119908)
(70)
But by Remark 13 we see that the last expression of (70) isdominated by (39) on the region Γ
1199020
given by (35) for allbut a finite number of values of 119899 Next using the dominatedconvergence theorem Parsevalrsquos relation and (40) we obtainthe desired result
Corollary 15 Let 1199020 119865 119890
119899infin
119899=1 (1205821119899 1205822119899)infin
119899=1and (119902
1 1199022)
be as in Theorem 14 Then
119864
anf119902
119909[119865 (1199091 1199092)]
= lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times 119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865 (119909
1 1199092)]
(71)
where 119866119899is given by (52)
10 Journal of Function Spaces and Applications
Corollary 16 Let 1199020 119865 and 119890
119899infin
119899=1be as in Theorem 14 and
let Γ1199020
be given by (35) Let 120582 = (1205821 1205822) isin Int (Γ
1199020
) and(1205821119899 1205822119899)infin
119899=1be a sequence in C2
+such that 120582
119895119899rarr 120582
119895
119895 isin 1 2 Then
119864
an
119909[119865 (1199091 1199092)] = lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865(1199091 1199092)]
(72)
where 119866119899is given by (52)
Our another result namely a change of scale formula forfunction space integrals now follows fromCorollary 16 above
Theorem 17 Let 119865 isin F11988611988711986011198602
and let 119890119899infin
119899=1be a complete
orthonormal set in (1198621015840119886119887[0 119879] sdot
119887) Then for any 120588
1gt 0 and
1205882gt 0
119864119909[119865 (12058811199091 12058821199092)]
= lim119899rarrinfin
120588minus119899
1120588minus119899
2
times 119864119909[119866119899(120588minus2
1 1199091)119866119899(120588minus2
2 1199092) 119865 (119909
1 1199092)]
(73)
where 119866119899is given by (52)
Proof Simply choose 120582119895= 120588minus2
119895for 119895 isin 1 2 and 120582
119895119899= 120588minus2
119895
for 119895 isin 1 2 and 119899 isin N in (72)
Remark 18 Of course if we choose 119886(119905) equiv 0 119887(119905) = 1199051198601= 119868 (identity operator) and 119860
2= 0 (zero operator) then
the function space 119862119886119887[0 119879] reduces to the classical Wiener
space 1198620[0 119879] and the generalized Fresnel type class F 119886119887
11986011198602
reduces to the Fresnel class F(1198620[0 119879]) It is known that
F(1198620[0 119879]) forms a Banach algebra over the complex field
In this case we have the relationships between the analyticFeynman integral and theWiener integral on classicalWienerspace as discussed in [14 15]
In recent paper [19] Yoo et al have studied a change ofscale formula for function space integral of the functionalsin the Banach algebra S(1198712
119886119887[0 119879]) the Banach algebra
S(1198712119886119887[0 119879]) is introduced in [12]
5 Functionals in F11988611988711986011198602
In this section we prove a theorem ensuring that variousfunctionals are inF119886119887
11986011198602
Theorem 19 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] Let (119884Y 120574) be a 120590-finite
measure space and let 120593119897 119884 rarr 119862
1015840
119886119887[0 119879] beYndashB(1198621015840
119886119887[0 119879])
measurable for 119897 isin 1 119889 Let 120579 119884 times R119889 rarr C be given by120579(120578 sdot) = ]
120578(sdot) where ]
120578isin M(R119889) for every 120578 isin 119884 and where
the family ]120578 120578 isin 119884 satisfies
(i) ]120578(119864) is a Y-measurable function of 120578 for every 119864 isin
B(R119889)(ii) ]
120578 isin 1198711
(119884Y 120574)
Under these hypothesis the functional 119865 1198622119886119887[0 119879] rarr C
given by
119865 (1199091 1199092) = int
119884
120579(120578
2
sum
119895=1
(11986012
1198951205931(120578) 119909
119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 119909
119895)
sim
)119889120574 (120578)
(74)
belongs toF11988611988711986011198602
and satisfies the inequality
119865 le int
119884
10038171003817100381710038171003817]120578
10038171003817100381710038171003817119889120574 (120578) (75)
Proof Using the techniques similar to those used in [20] wecan show that ]
120578 is measurable as a function of 120578 that 120579 is
Y-measurable and that the integrand in (74) is a measurablefunction of 120578 for every (119909
1 1199092) isin 1198622
119886119887[0 119879]
We define a measure 120591 onY timesB(R119889) by
120591 (119864) = int
119884
]120578(119864(120578)
) 119889120574 (120578) for 119864 isin Y timesB (R119889) (76)
Then by the first assertion of Theorem 31 in [17] 120591 satisfies120591 le int
119884
]120578119889120574(120578) Now let Φ 119884 times R119889 rarr 119862
1015840
119886119887[0 119879] be
defined by Φ(120578 V1 V
119889) = sum
119889
119897=1V119897120593119897(120578) Then Φ is Y times
B(R119889) ndashB(1198621015840119886119887[0 119879])-measurable on the hypothesis for 120593
119897
119897 isin 1 119889 Let 120590 = 120591 ∘Φminus1 Then clearly 120590 isinM(1198621015840119886119887[0 119879])
and satisfies 120590 le 120591From the change of variables theorem and the second
assertion of Theorem 31 in [17] it follows that for ae(1199091 1199092) isin 1198622
119886119887[0 119879] and for every 120588
1gt 0 and 120588
2gt 0
119865 (12058811199091 12058821199092)
= int
119884
]120578(
2
sum
119895=1
(11986012
1198951205931(120578) 120588
119895119909119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 120588
119895119909119895)
sim
)119889120574 (120578)
= int
119884
[
[
int
R119889exp
119894
119889
sum
119897=1
V119897
[
[
2
sum
119895=1
(11986012
119895120593119897(120578)
120588119895119909119895)
sim
]
]
119889]120578
times (V1 V
119889)]
]
119889120574 (120578)
= int
119884timesR119889exp
119894
119889
sum
119897=1
V119897
[
[
2
sum
119895=1
(11986012
119895120593119897(120578) 120588
119895119909119895)
sim
]
]
119889120591
times (120578 V1 V
119889)
Journal of Function Spaces and Applications 11
= int
119884timesR119889exp
2
sum
119895=1
119894(11986012
119895Φ(120578 V
1 V
119889) 120588119895119909119895)
sim
119889120591
times (120578 V1 V
119889)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119909119895)
sim
119889120591 ∘ Φminus1
(119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119909119895)
sim
119889120590 (119908)
(77)
Thus the functional 119865 given by (74) belongs to F11988611988711986011198602
andsatisfies the inequality
119865 = 120590 le 120591 le int
119884
10038171003817100381710038171003817]120578
10038171003817100381710038171003817119889120574 (120578) (78)
As mentioned in (2) of Remark 6 F 11988611988711986011198602
is a Banachalgebra if Ran(119860
1+ 1198602) is dense in 1198621015840
119886119887[0 119879] In this case
many analytic functionals of 119865 can be formed The followingcorollary is relevant to Feynman integration theories andquantum mechanics where exponential functions play animportant role
Corollary 20 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] such that Ran(119860
1+ 1198602)
is dense in 1198621015840119886119887[0 119879] Let 119865 be given by (74) with 120579 as in
Theorem 19 and let 120573 C rarr C be an entire function Then(120573 ∘ 119865)(119909
1 1199092) is in F119886119887
11986011198602
In particular exp119865(1199091 1199092) isin
F11988611988711986011198602
Corollary 21 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] and let 119892
1 119892
119889 be a
finite subset of 1198621015840119886119887[0 119879] Given 120573 = ] where ] isin M(R119889)
define 119865 1198622119886119887[0 119879] rarr C by
119865 (1199091 1199092) = 120573(
2
sum
119895=1
(11986012
1198951198921 119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 119909119895)
sim
)
(79)
Then 119865 is an element ofF11988611988711986011198602
Proof Let (119884Y 120574) be a probability space and for 119897 isin1 119889 let 120593
119897(120578) equiv 119892
119897 Take 120579(120578 sdot) = 120573(sdot) = ](sdot) Then for
all 1205881gt 0 and 120588
2gt 0 and for ae (119909
1 1199092) isin 1198622
119886119887[0 119879]
int
119884
120579(120578
2
sum
119895=1
(11986012
1198951205931(120578) 120588
119895119909119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 120588
119895119909119895)
sim
)119889120574 (120578)
= int
119884
120573(
2
sum
119895=1
(11986012
1198951198921 120588119895119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 120588119895119909119895)
sim
)119889120574 (120578)
= 120573(
2
sum
119895=1
(11986012
1198951198921 120588119895119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 120588119895119909119895)
sim
)
= 119865 (12058811199091 12058821199092)
(80)
Hence 119865 isin F11988611988711986011198602
Remark 22 Let 119889 = 1 and let (119884Y 120574) = ([0 119879]B([0 119879])119898119871) in Theorem 19 where 119898
119871denotes the Lebesgue measure
on [0 119879] ThenTheorems 46 47 and 49 in [18] follow fromthe results in this section by letting119860
1be the identity operator
and letting 1198602equiv 0 on 1198621015840
119886119887[0 119879] The function 120579 studied in
[18] (and mentioned in Remark 2 above) is interpreted as thepotential energy in quantum mechanics
Acknowledgments
The authors would like to express their gratitude to the refer-ees for their valuable comments and suggestions which haveimproved the original paper This research was supported bythe Basic Science Research Program through the NationalResearch Foundation of Korea (NRF) funded by theMinistryof Education (2011-0014552)
References
[1] G Kallianpur and C Bromley ldquoGeneralized Feynman integralsusing analytic continuation in several complex variablesrdquo inStochastic Analysis and Applications M A Pinsky Ed vol 7pp 217ndash267 Marcel Dekker New York NY USA 1984
[2] G Kallianpur D Kannan and R L Karandikar ldquoAnalytic andsequential Feynman integrals on abstract Wiener and Hilbertspaces and a Cameron-Martin formulardquo Annales de lrsquoInstitutHenri Poincare vol 21 no 4 pp 323ndash361 1985
[3] G W Johnson and M L Lapidus The Feynman Integral andFeynmanrsquos Operational Calculus Clarendon Press Oxford UK2000
[4] J Yeh ldquoSingularity of Gaussian measures on function spacesinduced by Brownian motion processes with non-stationaryincrementsrdquo Illinois Journal of Mathematics vol 15 pp 37ndash461971
[5] J Yeh Stochastic Processes and the Wiener Integral MarcelDekker New York NY USA 1973
[6] S J Chang J G Choi and D Skoug ldquoIntegration by partsformulas involving generalized Fourier-Feynman transformson function spacerdquo Transactions of the American MathematicalSociety vol 355 no 7 pp 2925ndash2948 2003
[7] S J Chang J G Choi and D Skoug ldquoParts formulas involvingconditional generalized Feynman integrals and conditionalgeneralized Fourier-Feynman transforms on function spacerdquo
12 Journal of Function Spaces and Applications
Integral Transforms and Special Functions vol 15 no 6 pp 491ndash512 2004
[8] S J Chang J G Choi and D Skoug ldquoEvaluation formulas forconditional function space integralsmdashIrdquo Stochastic Analysis andApplications vol 25 no 1 pp 141ndash168 2007
[9] S J Chang J G Choi and D Skoug ldquoGeneralized Fourier-Feynman transforms convolution products and first variationson function spacerdquoTheRockyMountain Journal ofMathematicsvol 40 no 3 pp 761ndash788 2010
[10] S J Chang and D M Chung ldquoConditional function spaceintegrals with applicationsrdquo The Rocky Mountain Journal ofMathematics vol 26 no 1 pp 37ndash62 1996
[11] S J Chang H S Chung and D Skoug ldquoIntegral transformsof functionals in 119871
2(119862119886119887[0 119879])rdquoThe Journal of Fourier Analysis
and Applications vol 15 no 4 pp 441ndash462 2009[12] S J Chang and D Skoug ldquoGeneralized Fourier-Feynman
transforms and a first variation on function spacerdquo IntegralTransforms and Special Functions vol 14 pp 375ndash393 2003
[13] J G Choi and S J Chang ldquoGeneralized Fourier-Feynmantransform and sequential transforms on function spacerdquo Jour-nal of the Korean Mathematical Society vol 49 pp 1065ndash10822012
[14] R H Cameron and D A Storvick ldquoRelationships between theWiener integral and the analytic Feynman integralrdquo Rendicontidel Circolo Matematico di Palermo no 17 supplement pp 117ndash133 1987
[15] RHCameron andDA Storvick ldquoChange of scale formulas forWiener integralrdquo Rendiconti del Circolo Matematico di Palermono 17 pp 105ndash115 1987
[16] G W Johnson ldquoThe equivalence of two approaches to theFeynman integralrdquo Journal of Mathematical Physics vol 23 pp2090ndash2096 1982
[17] G W Johnson and D L Skoug ldquoNotes on the FeynmanintegralmdashIII the Schroedinger equationrdquo Pacific Journal ofMathematics vol 105 pp 321ndash358 1983
[18] S J Chang J G Choi and S D Lee ldquoA Fresnel type class onfunction spacerdquo Journal of the Korean Society of MathematicalEducation Series B vol 16 no 1 pp 107ndash119 2009
[19] I Yoo B J Kim and B S Kim ldquoA change of scale formulafor a function space integral on 119862
119886119887[0 119879]rdquo Proceedings of the
American Mathematical Society vol 141 no 8 pp 2729ndash27392013
[20] K S Chang G W Johnson and D L Skoug ldquoFunctions inthe Fresnel classrdquo Proceedings of the American MathematicalSociety vol 100 no 2 pp 309ndash318 1987
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Function Spaces and Applications
where both 119860+ and 119860minus are bounded nonnegative and self-adjoint operators
In this section we will extend the ideas of [1] to obtainexpressions of the generalized analytic Feynman integral andthe GFFT of functionals of the form (19) when119860 is no longerrequired to be nonnegative To do this we will introducedefinitions and notations analogous to those in [7 12 13]
Let W(1198622119886119887[0 119879]) denote the class of all Wiener mea-
surable subsets of the product function space 1198622119886119887[0 119879] A
subset 119861 of 1198622119886119887[0 119879] is said to be scale-invariant measurable
provided (12058811199091 12058821199092) (119909
1 1199092) isin 119861 is W(1198622
119886119887[0 119879])-
measurable for every 1205881gt 0 and 120588
2gt 0 and a scale-
invariantmeasurable subset119873 of1198622119886119887[0 119879] is said to be scale-
invariant null provided (120583 times 120583)((12058811199091 12058821199092) (119909
1 1199092) isin
119873) = 0 for every 1205881gt 0 and 120588
2gt 0 A property that
holds except on a scale-invariant null set is said to hold s-ae on 1198622
119886119887[0 119879] A functional 119865 on 1198622
119886119887[0 119879] is said to be
scale-invariant measurable provided 119865 is defined on a scale-invariant measurable set and 119865(120588
1sdot 1205882sdot) is W(1198622
119886119887[0 119879])-
measurable for every 1205881gt 0 and 120588
2gt 0 If two functionals
119865 and 119866 defined on 1198622119886119887[0 119879] are equal s-ae then we write
119865 asymp 119866We denote the product function space integral of a
W(1198622119886119887[0 119879])-measurable functional 119865 by
119864 [119865] equiv 119864119909[119865 (1199091 1199092)]
= int
1198622
119886119887[0119879]
119865 (1199091 1199092) 119889 (120583 times 120583) (119909
1 1199092)
(21)
whenever the integral existsThroughout this paper letCC
+and C
+denote the set of
complex numbers complex numbers with positive real partand nonzero complex numbers with nonnegative real partrespectively Furthermore for all 120582 isin C
+ 120582minus12 (or 12058212) is
always chosen to have positive real part We also assume thatevery functional 119865 on 1198622
119886119887[0 119879] we consider is s-ae defined
and is scale-invariant measurable
Definition 3 Let C2+equiv C+times C+and let C2
+equivC+timesC+ Let
119865 1198622
119886119887[0 119879] rarr C be such that for each 120582
1gt 0 and 120582
2gt 0
the function space integral
119869 (1205821 1205822)
= int
1198622
119886119887[0119879]
119865 (120582minus12
11199091 120582minus12
21199092) 119889 (120583 times 120583) (119909
1 1199092)
(22)
exists If there exists a function 119869lowast(1205821 1205822) analytic inC2
+such
that 119869lowast(1205821 1205822) = 119869(120582
1 1205822) for all 120582
1gt 0 and 120582
2gt 0 then
119869lowast
(1205821 1205822) is defined to be the analytic function space integral
of 119865 over 1198622119886119887[0 119879] with parameter 120582 = (120582
1 1205822) and for 120582 isin
C2+we write
119864an[119865] equiv 119864
an
119909[119865 (1199091 1199092)]
equiv 119864
an(12058211205822)
11990911199092[119865 (1199091 1199092)] = 119869
lowast
(1205821 1205822)
(23)
Let 1199021and 1199022be nonzero real numbers Let 119865 be a functional
such that 119864an[119865] exists for all 120582 isin C2+ If the following limit
exists we call it the generalized analytic Feynman integral of119865 with parameter 119902 = (119902
1 1199022) and we write
119864anf 119902[119865] equiv 119864
anf 119902119909[119865 (1199091 1199092)]
equiv 119864
anf(11990211199022)
11990911199092[119865 (1199091 1199092)] = lim
120582rarrminus119894 119902
119864an[119865]
(24)
where 120582 = (1205821 1205822) rarr minus119894 119902 = (minus119894119902
1 minus1198941199022) through values in
C2+
Definition 4 Let 1199021and 1199022be nonzero real numbers For 120582 =
(1205821 1205822) isin C2+and (119910
1 1199102) isin 1198622
119886119887[0 119879] let
119879 120582(119865) (119910
1 1199102) equiv 119879(12058211205822)(119865) (119910
1 1199102)
= 119864
an
119909[119865 (1199101+ 1199091 1199102+ 1199092)]
(25)
For 119901 isin (1 2] we define the 119871119901analytic GFFT 119879(119901)
119902(119865) of 119865
by the formula ( 120582 isin C2+)
119879(119901)
119902(119865) (119910
1 1199102) equiv 119879(119901)
(1199021 1199022)(119865) (119910
1 1199102)
= lim120582rarrminus119894 119902
119879 120582(119865) (119910
1 1199102)
(26)
if it exists that is for each 1205881gt 0 and 120588
2gt 0
lim120582rarrminus119894 119902
int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816
119879 120582(119865) (120588
11199101 12058821199102)
minus119879(119901)
119902(119865) (120588
11199101 12058821199102)
100381610038161003816100381610038161003816
1199011015840
119889 (120583 times 120583) (1199101 1199102)
= 0
(27)
where 1119901+11199011015840 = 1We define the 1198711analytic GFFT119879(1)
119902(119865)
of 119865 by the formula ( 120582 isin C2+)
119879(1)
119902(119865) (119910
1 1199102) = lim120582rarrminus119894 119902
119879 120582(119865) (119910
1 1199102) (28)
if it exists
We note that for 1 le 119901 le 2 119879(119901)119902(119865) is defined only s-ae
We also note that if 119879(119901)119902(119865) exists and if 119865 asymp 119866 then 119879(119901)
119902(119866)
exists and 119879(119901)119902(119866) asymp 119879
(119901)
119902(119865) Moreover from Definition 4
we see that for 1199021 1199022isin R minus 0
119864
anf 119902119909[119865 (1199091 1199092)] = 119879
(1)
119902(119865) (0 0) (29)
Next we give the definition of the generalized Fresnel typeclassF119886119887
11986011198602
Journal of Function Spaces and Applications 5
Definition 5 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] The generalized Fresnel
type classF11988611988711986011198602
of functionals on1198622119886119887[0 119879] is defined as the
space of all functionals 119865 on 1198622119886119887[0 119879] of the following form
119865 (1199091 1199092) = int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119909119895)
sim
119889119891 (119908) (30)
for some119891 isinM(1198621015840119886119887[0 119879]) More precisely since we identify
functionals which coincide s-ae on 1198622119886119887[0 119879] F119886119887
11986011198602
canbe regarded as the space of all 119904-equivalence classes offunctionals of the form (30)
Remark 6 (1) In Definition 5 let 1198601be the identity operator
on 1198621015840119886119887[0 119879] and 119860
2equiv 0 Then F 119886119887
11986011198602
is essentially theFresnel type class F(119862
119886119887[0 119879]) and for 119901 isin [1 2] and
nonzero real numbers 1199021and 1199022
119879(119901)
(1199021 1199022)(119865) (119910
1 1199102) = 119879(119901)
1199021
(1198650) (1199101) (31)
if it exists where 1198650(1199091) = 119865(119909
1 1199092) for all (119909
1 1199092) isin
1198622
119886119887[0 119879] and 119879(119901)
1199021
(1198650) means the 119871
119901analytic GFFT on
119862119886119887[0 119879] see [6 12](2) The map 119891 997891rarr 119865 defined by (30) sets up an algebra
isomorphism betweenM(1198621015840119886119887[0 119879]) andF119886119887
11986011198602
if Ran(1198601+
1198602) is dense in1198621015840
119886119887[0 119879] where Ran indicates the range of an
operator In this caseF 11988611988711986011198602
becomes a Banach algebra underthe norm 119865 = 119891 For more details see [1]
Remark 7 Let 119865 be given by (30) In evaluating119864119909[119865(120582minus12
11199091 120582minus12
21199092)] and 119879
(12058211205822)(119865)(1199101 1199102) = 119864
119909[119865(1199101+
120582minus12
11199091 1199102+ 120582minus12
21199092)] for 120582
1gt 0 and 120582
2gt 0 the expression
120595 (120582 119908)
equiv 120595 (1205821 1205822 1198601 1198602 119908)
= exp
2
sum
119895=1
[
[
minus
(119860119895119908119908)
119887
2120582119895
+ 119894120582minus12
119895(11986012
119895119908 119886)119887
]
]
(32)
occurs Clearly for 120582119895gt 0 119895 isin 1 2 |120595( 120582 119908)| le 1 for all
119908 isin 1198621015840
119886119887[0 119879] But for 120582 isin C2
+ |120595( 120582 119908)| is not necessarily
bounded by 1Note that for each 120582
119895isinC+with 120582
119895= 120572119895+ 119894120573119895 119895 isin 1 2
12058212
119895=
radicradic1205722
119895+ 1205732
119895+ 120572119895
2
+ 119894
120573119895
10038161003816100381610038161003816120573119895
10038161003816100381610038161003816
radicradic1205722
119895+ 1205732
119895minus 120572119895
2
120582minus12
119895=radic
radic1205722
119895+ 1205732
119895+ 120572119895
2 (1205722
119895+ 1205732
119895)
minus 119894
120573119895
10038161003816100381610038161003816120573119895
10038161003816100381610038161003816
radic
radic1205722
119895+ 1205732
119895minus 120572119895
2 (1205722
119895+ 1205732
119895)
(33)
Hence for 120582119895isinC+with 120582
119895= 120572119895+ 119894120573119895 119895 isin 1 2
10038161003816100381610038161003816120595 (120582 119908)
10038161003816100381610038161003816
= exp
2
sum
119895=1
[
[
[
[
minus
120572119895
2 (1205722
119895+ 1205732
119895)
(119860119895119908119908)
119887
+
120573119895
10038161003816100381610038161003816120573119895
10038161003816100381610038161003816
radic
radic1205722
119895+ 1205732
119895minus 120572119895
2 (1205722
119895+ 1205732
119895)
(11986012
119895119908 119886)119887
]
]
]
]
(34)
The right hand side of (34) is an unbounded functionof 119908 for 119908 isin 1198621015840
119886119887[0 119879] Thus 119864an[119865] 119864anf 119902[119865] 119879
120582(119865) and
119879(119901)
119902(119865) might not exist Thus throughout this paper we will
need to put additional restrictions on the complex measure119891 corresponding to 119865 in order to obtain our results for theGFFT and the generalized analytic Feynman integral of 119865
In view of Remark 7 we clearly need to impose additionalrestrictions on the functionals 119865 inF119886119887
11986011198602
For a positive real number 119902
0 let
Γ1199020
=
120582 = (120582
1 1205822) isinC2
+| 120582119895= 120572119895+ 119894120573119895
10038161003816100381610038161003816Im (120582minus12
119895)
10038161003816100381610038161003816=radic
radic1205722
119895+ 1205732
119895minus 120572119895
2 (1205722
119895+ 1205732
119895)
lt
1
radic21199020
119895 = 1 2
(35)
and let119896 (1199020 119908) equiv 119896 (119902
0 1198601 1198602 119908)
= exp
2
sum
119895=1
(21199020)minus121003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817119887119886119887
(36)
Then for all 120582 = (1205821 1205822) isin Γ1199020
10038161003816100381610038161003816120595 (120582 119908)
10038161003816100381610038161003816le exp
2
sum
119895=1
radic
radic1205722
119895+ 1205732
119895minus 120572119895
2 (1205722
119895+ 1205732
119895)
100381610038161003816100381610038161003816
(11986012
119895119908 119886)119887
100381610038161003816100381610038161003816
le exp
2
sum
119895=1
radic
radic1205722
119895+ 1205732
119895minus 120572119895
2 (1205722
119895+ 1205732
119895)
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817119887119886119887
lt 119896 (1199020 119908)
(37)
6 Journal of Function Spaces and Applications
We note that for all real 119902119895with |119902
119895| gt 1199020 119895 isin 1 2
(minus119894119902119895)
minus12
=
1
radic
100381610038161003816100381610038162119902119895
10038161003816100381610038161003816
+ sign (119902119895)
119894
radic
100381610038161003816100381610038162119902119895
10038161003816100381610038161003816
(38)
and (minus1198941199021 minus1198941199022) isin Γ1199020
For the existence of the GFFT of 119865 we define a subclass
F1199020
11986011198602
ofF 11988611988711986011198602
by 119865 isin F 119902011986011198602
if and only if
int
1198621015840
119886119887[0119879]
119896 (1199020 119908) 119889
10038161003816100381610038161198911003816100381610038161003816(119908) lt +infin (39)
where 119891 and 119865 are related by (30) and 119896 is given by (36)
Remark 8 Note that in case 119886(119905) equiv 0 and 119887(119905) = 119905 on [0 119879]the function space 119862
119886119887[0 119879] reduces to the classical Wiener
space 1198620[0 119879] and (119908 119886)
119887= 0 for all 119908 isin 1198621015840
119886119887[0 119879] =
1198621015840
0[0 119879] Hence for all 120582 isin C2
+ |120595( 120582 119908)| le 1 and for any
positive real number 1199020F 119902011986011198602
= F11986011198602
theKallianpur andBromleyrsquos class introduced in Section 1
Theorem 9 Let 1199020be a positive real number and let 119865 be an
element ofF 119902011986011198602
Then for any nonzero real numbers 1199021and
1199022with |119902
119895| gt 1199020 119895 isin 1 2 the 119871
1analytic GFFT of 119865 119879(1)
119902(119865)
exists and is given by the following formula
119879(1)
119902(119865) (119910
1 1199102)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
120595(minus119894 119902 119908) 119889119891 (119908)
(40)
for s-ae (1199101 1199102) isin 1198622
119886119887[0 119879] where 120595 is given by (32)
Proof We first note that for 119895 isin 1 2 the PWZ stochasticintegral (11986012
119895119908 119909)sim is a Gaussian random variable withmean
(11986012
119895119908 119886)119887and variance 11986012
119895119908
2
119887
= (119860119895119908119908)119887
Hence using(30) the Fubini theorem the change of variables theorem and(14) we have that for all 120582
1gt 0 and 120582
2gt 0
119869 (1199101 1199102 1205821 1205822)
equiv 119864119909[119865 (1199101+ 120582minus12
11199091 1199102+ 120582minus12
21199092)]
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
times(
2
prod
119895=1
119864119909119895
[exp 119894120582minus12119895(11986012
119895119908 119909119895)
sim
])119889119891 (119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
times[
[
[
2
prod
119895=1
(2120587(119860119895119908119908)
119887
)
minus12
times int
R
exp
119894120582minus12
119895119906119895
minus
[119906119895minus (11986012
119895119908 119886)119887
]
2
2(119860119895119908119908)
119887
119889119906119895
]
]
]
119889119891(119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
120595(120582 119908) 119889119891 (119908)
(41)
Let
119879 120582(119865) (119910
1 1199102)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
120595(120582 119908) 119889119891 (119908)
(42)
for each 120582 isin C2+ Clearly
119879 120582(119865) (119910
1 1199102) = 119869 (119910
1 1199102 1205821 1205822) (43)
for all 1205821gt 0 and 120582
2gt 0 Let Γ
1199020
be given by (35)Then for all120582 isin Int(Γ
1199020
)
1003816100381610038161003816119879 120582(119865) (119910
1 1199102)1003816100381610038161003816lt int
1198621015840
119886119887[0119879]
119896 (1199020 119908) 119889
10038161003816100381610038161198911003816100381610038161003816(119908) lt +infin
(44)
Using this fact and the dominated convergence theoremwe see that 119879
120582(119865)(1199101 1199102) is a continuous function of 120582 =
(1205821 1205822) on Int(Γ
1199020
) For each 119908 isin 1198621015840119886119887[0 119879] 120595( 120582 119908) is an
analytic function of 120582 throughout the domain Int(Γ1199020
) so thatintΔ
120595(120582 119908)119889
120582 = 0 for every rectifiable simple closed curve
Δ in Int(Γ1199020
) By (42) the Fubini theorem and the Moreratheorem we see that 119879
120582(119865)(1199101 1199102) is an analytic function of
120582 throughout the domain Int(Γ
1199020
) Finally using (28) withthe dominated convergence theorem we obtain the desiredresult
Theorem 10 Let 1199020and 119865 be as inTheorem 9Then for all 119901 isin
(1 2] and all nonzero real numbers 1199021and 119902
2with |119902
119895| gt 1199020
119895 isin 1 2 the119871119901analytic GFFT of119865119879(119901)
119902(119865) exists and is given
by the right hand side of (40) for s-ae (1199101 1199102) isin 1198622
119886119887[0 119879]
Journal of Function Spaces and Applications 7
Proof Let Γ1199020
be given by (35) It was shown in the proofof Theorem 9 that 119879
120582(119865)(1199101 1199102) is an analytic function of 120582
throughout the domain Int(Γ1199020
) In viewofDefinition 4 it willsuffice to show that for each 120588
1gt 0 and 120588
2gt 0
lim120582rarrminus119894 119902
int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816
119879 120582(119865) (120588
11199101 12058821199102)
minus119879(119901)
119902(119865) (120588
11199101 12058821199102)
100381610038161003816100381610038161003816
1199011015840
times 119889 (120583 times 120583) (1199101 1199102) = 0
(45)
Fixing 119901 isin (1 2] and using the inequalities (37) and (39)we obtain that for all 120588
119895gt 0 119895 isin 1 2 and all 120582 isin Γ
1199020
100381610038161003816100381610038161003816
119879 120582(119865) (120588
11199101 12058821199102) minus 119879(119901)
119902(119865) (120588
11199101 12058821199102)
100381610038161003816100381610038161003816
1199011015840
le
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894120588119895(11986012
119895119908 119910119895)
sim
times[120595 (120582 119908) minus 120595 (minus119894 119902 119908)] 119889119891 (119908)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199011015840
le (int
1198621015840
119886119887[0119879]
[
10038161003816100381610038161003816120595 (120582 119908)
10038161003816100381610038161003816+
10038161003816100381610038161003816120595 (minus119894 119902 119908)
10038161003816100381610038161003816] 11988910038161003816100381610038161198911003816100381610038161003816(119908))
1199011015840
le (2int
1198621015840
119886119887[0119879]
119896 (1199020 119908) 119889
10038161003816100381610038161198911003816100381610038161003816(119908))
1199011015840
lt +infin
(46)
Hence by the dominated convergence theorem we see thatfor each 119901 isin (1 2] and each 120588
1gt 0 and 120588
2gt 0
lim120582rarrminus119894 119902
int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816
119879 120582(119865) (120588
11199101 12058821199102)
minus119879(119901)
119902(119865) (120588
11199101 12058821199102)
100381610038161003816100381610038161003816
1199011015840
119889 (120583 times 120583) (1199101 1199102)
= lim120582rarrminus119894 119902
int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119910119895)
sim
times 120595(120582 119908) 119889119891 (119908)
minus int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119910119895)
sim
times120595(minus119894 119902 119908) 119889119891 (119908)
10038161003816100381610038161003816100381610038161003816100381610038161003816
1199011015840
times 119889 (120583 times 120583) (1199101 1199102)
= int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119910119895)
sim
times lim120582rarrminus119894 119902
[120595 (120582 119908)
minus120595 (minus119894 119902 119908)] 119889119891 (119908)
10038161003816100381610038161003816100381610038161003816100381610038161003816
1199011015840
times 119889 (120583 times 120583) (1199101 1199102)
= 0
(47)
which concludes the proof of Theorem 10
Remark 11 (1) In view of Theorems 9 and 10 we see thatfor each 119901 isin [1 2] the 119871
119901analytic GFFT of 119865 119879(119901)
119902(119865) is
given by the right hand side of (40) for 1199020 1199021 1199022 and 119865 as
in Theorem 9 and for s-ae (1199101 1199102) isin 1198622
119886119887[0 119879]
119879(119901)
119902(119865) (119910
1 1199102) = 119864
anf 119902119909[119865 (1199101+ 1199091 1199102+ 1199092)] 119901 isin [1 2]
(48)
In particular using this fact and (29) we have that for all 119901 isin[1 2]
119879(119901)
119902(119865) (0 0) = 119864
anf 119902119909[119865 (1199091 1199092)] (49)
(2) For nonzero real numbers 1199021and 119902
2with |119902
119895| gt 119902
0
119895 isin 1 2 define a set function 119891 119860119902B(1198621015840
119886119887[0 119879]) rarr C by
119891
119860
119902(119861) = int
119861
120595 (minus119894 119902 119908) 119889119891 (119908) 119861 isinB (1198621015840
119886119887[0 119879])
(50)
where 119891 and 119865 are related by (30) and B(1198621015840119886119887[0 119879]) is the
Borel 120590-algebra of 1198621015840119886119887[0 119879] Then it is obvious that 119891 119860
119902
belongs to M(1198621015840119886119887[0 119879]) and for all 119901 isin [1 2] 119879(119901)
119902(119865) can
be expressed as
119879(119901)
119902(119865) (119910
1 1199102) = int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
119889119891
119860
119902(119908)
(51)
for s-ae (1199101 1199102) isin 119862
2
119886119887[0 119879] Hence 119879(119901)
119902(119865) belongs to
F 11988611988711986011198602
for all 119901 isin [1 2]
4 Relationships between the GFFT and theFunction Space Integral of Functionals inF11988611988711986011198602
In this section we establish a relationship between the GFFTand the function space integral of functionals in the Fresneltype classF119886119887
11986011198602
8 Journal of Function Spaces and Applications
Throughout this section for convenience we use thefollowing notation for given 120582 isin C
+and 119899 = 1 2 let
119866119899(120582 119909)
= exp[1 minus 1205822
]
119899
sum
119896=1
[(119890119896 119909)sim
]
2
+ (12058212
minus 1)
119899
sum
119896=1
(119890119896 119886)1198621015840
119886119887
(119890119896 119909)sim
(52)
where 119890119899infin
119899=1is a complete orthonormal set in (1198621015840
119886119887[0 119879]
sdot 119887)To obtain our main results Theorems 14 and 17 below
we state a fundamental integration formula for the functionspace 119862
119886119887[0 119879]
Let 1198901 119890
119899 be an orthonormal set in (1198621015840
119886119887[0 119879] sdot
119887)
let 119896 R119899 rarr C be a Lebesgue measurable function and let119870 119862119886119887[0 119879] rarr C be given by
119870 (119909) = 119896 ((1198901 119909)sim
(119890119899 119909)sim
) (53)
Then
119864 [119870] = int
119862119886119887[0119879]
119896 ((1198901 119909)sim
(119890119899 119909)sim
) 119889120583 (119909)
= (2120587)minus1198992
int
R119899119896 (1199061 119906
119899)
times exp
minus
119899
sum
119895=1
[119906119895minus (119890119895 119886)119887
]
2
2
1198891199061 119889119906
119899
(54)
in the sense that if either side of (54) exists both sides existand equality holds
We also need the following lemma to obtain our maintheorem in this section
Lemma 12 Let 1198901 119890
119899 be an orthonormal subset of
(1198621015840
119886119887[0 119879] sdot
119887) and let 119908 be an element of 1198621015840
119886119887[0 119879] Then
for each 120582 isin C+ the function space integral
119864119909[119866119899(120582 119909) exp 119894(119908 119909)sim] (55)
exists and is given by the formula
119864119909[119866119899(120582 119909) exp 119894(119908 119909)sim]
= 120582minus1198992 exp
[
120582 minus 1
2120582
]
119899
sum
119896=1
(119890119896 119908)2
119887minus
1
2
1199082
119887
+ 119894120582minus12
119899
sum
119896=1
(119890119896 119886)119887(119890119896 119908)119887
+ 119894(119890119899+1 119886)119887[1199082
119887minus
119899
sum
119896=1
(119890119896 119908)2
119887]
12
(56)
where 119866119899is given by (52) above and
119890119899+1=[
[
1199082
119887minus
119899
sum
119895=1
(119890119895 119908)
2
119887
]
]
minus12
119908 minus
119899
sum
119895=1
(119890119895 119908)119887
119890119895
(57)
Proof (Outline) Using the Gram-Schmidt process for any119908 isin 119862
1015840
119886119887[0 119879] we can write 119908 = sum
119899+1
119896=1119888119896119890119896 where
1198901 119890
119899 119890119899+1 is an orthonormal set in (1198621015840
119886119887[0 119879] sdot
119887)
and
119888119896=
(119890119896 119908)119887 119896 = 1 119899
[1199082
119887minus
119899
sum
119895=1
(119890119895 119908)
2
119887
]
12
119896 = 119899 + 1
(58)
Then using (52) (54) the Fubini theorem and (14) it followsthat (56) holds for all 120582 isin C
+
The following remark will be very useful in the proof ofour main theorem in this section
Remark 13 Let 1199020be a positive real number and let Γ
1199020
begiven by (35) For real numbers 119902
1and 1199022with |119902
119895| gt 1199020 119895 isin
1 2 let 120582119899infin
119899=1= (1205821119899 1205822119899)infin
119899=1be a sequence in C2
+such
that
120582119899= (1205821119899 1205822119899) 997888rarr minus119894 119902 = (minus119894119902
1 minus1198941199022) (59)
Let 120582119895119899= 120572119895119899+ 119894120573119895119899
for 119895 isin 1 2 and 119899 isin N Then for119895 isin 1 2 Re(120582
119895119899) = 120572119895119899gt 0 and
120582minus1
119895119899= (120572119895119899+ 119894120573119895119899)
minus1
=
120572119895119899minus 119894120573119895119899
1205722
119895119899+ 1205732
119895119899
(60)
for each 119899 isin N Since |Im ((minus119894119902119895)minus12
)| = 1radic2|119902119895| lt 1radic2119902
0
for 119895 isin 1 2 there exists a sufficiently large 119871 isin N such thatfor any 119899 ge 119871 120582
1119899and 120582
2119899are in Int(Γ
1199020
) and
120575 (1199021 1199022) equiv sup ( 1003816100381610038161003816
1003816Im (120582minus12
1119899)
10038161003816100381610038161003816 119899 ge 119871
cup
10038161003816100381610038161003816Im (120582minus12
2119899)
10038161003816100381610038161003816 119899 ge 119871
cup
100381610038161003816100381610038161003816
Im ((minus1198941199021)minus12
)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
Im ((minus1198941199022)minus12
)
100381610038161003816100381610038161003816
)
lt
1
radic21199020
(61)
Thus there exists a positive real number 120576 gt 1 such that120575(1199021 1199022) lt 1(120576radic2119902
0)
Let 119890119899infin
119899=1be a complete orthonormal set in (1198621015840
119886119887[0 119879]
sdot 119887) Using Parsevalrsquos identity it follows that
(1198921 1198922)119887=
infin
sum
119899=1
(119890119899 1198921)119887(119890119899 1198922)119887
(62)
Journal of Function Spaces and Applications 9
for all 1198921 1198922isin 1198621015840
119886119887[0 119879] In addition for each 119892 isin 1198621015840
119886119887[0 119879]
10038171003817100381710038171198921003817100381710038171003817
2
119887minus
119899
sum
119896=1
(119890119896 119892)2
119887=
infin
sum
119896=119899+1
(119890119896 119892)2
119887ge 0 (63)
for every 119899 isin NSince
(119892 119886)119887=
infin
sum
119899=1
(119890119899 119892)119887(119890119899 119886)119887
(64)
and for 120576 gt 1
minus12057610038171003817100381710038171198921003817100381710038171003817119887119886119887lt minus10038171003817100381710038171198921003817100381710038171003817119887119886119887le (119892 119886)
119887
le10038171003817100381710038171198921003817100381710038171003817119887119886119887lt 12057610038171003817100381710038171198921003817100381710038171003817119887119886119887
(65)
there exists a sufficiently large119870119895isin N such that for any 119899 ge 119870
119895
1003816100381610038161003816100381610038161003816100381610038161003816
119899
sum
119896=1
(119890119896 11986012
119895119908)119887
(119890119896 119886)119887
1003816100381610038161003816100381610038161003816100381610038161003816
lt 120576
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817119887119886119887
(66)
for 119895 isin 1 2Using these and a long and tedious calculation we can
show that for every 119899 ge max119871 1198701 1198702
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
exp
2
sum
119895=1
([
120582119895119899minus 1
2120582119895119899
]
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
minus
1
2
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
+ 119894120582minus12
119895119899
119899
sum
119896=1
(119890119896 11986012
119895119908)119887
(119890119896 119886)119887+ 119894(119890119899+1 119886)119887
times [
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
minus
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
]
12
)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119896 (1199020 119908)
(67)
where 119896(1199020 119908) is given by (36)
In our next theorem for119865 isin F11988611988711986011198602
we express theGFFTof 119865 as the limit of a sequence of function space integrals on1198622
119886119887[0 119879]
Theorem 14 Let 1199020and 119865 be as in Theorem 10 Let 119890
119899infin
119899=1
be a complete orthonormal set in (1198621015840119886119887[0 119879] sdot
119887) and let
(1205821119899 1205822119899)infin
119899=1be a sequence in C2
+such that 120582
119895119899rarr minus119894119902
119895
where 119902119895is a real number with |119902
119895| gt 1199020 119895 isin 1 2 Then for
119901 isin [1 2] and for s-ae (1199101 1199102) isin 1198622
119886119887[0 119879]
119879(119901)
119902(119865) (119910
1 1199102)
= lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865 (119910
1+ 1199091 1199102+ 1199092)]
(68)
where 119866119899is given by (52)
Proof ByTheorems9 and 10we know that for each119901 isin [1 2]the 119871119901analytic GFFT of 119865 119879(119901)
119902(119865) exists and is given by the
right hand side of (40) Thus it suffices to show that
119879(1)
119902(119865) (119910
1 1199102) = 119864
anf 119902119909[119865 (1199101+ 1199091 1199102+ 1199092)]
= lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times 119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092)
times 119865 (1199101+ 1199091 1199102+ 1199092)] sdot
(69)
Using (30) the Fubini theorem and (56) with 120582 and 119908replaced with 120582
119895119899and 11986012
119895119908 119895 isin 1 2 respectively we see
that
1205821198992
11198991205821198992
2119899119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865 (119910
1+ 1199091 1199102+ 1199092)]
= 1205821198992
11198991205821198992
2119899int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
times(
2
prod
119895=1
119864119909119895
[119866119899(120582minus12
119895119899 119909119895)
times exp 119894(11986012119895119908 119909119895)
sim
])119889119891 (119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
(119894(11986012
119895119908 119910119895)
sim
+ [
120582119895119899minus 1
2120582119895119899
]
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
minus
1
2
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
+ 119894120582minus12
119895119899
119899
sum
119896=1
(119890119896 119886)119887(119890119896 11986012
119895119908)119887
+ 119894(119890119899+1 119886)119887[
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
minus
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
]
12
)
119889119891 (119908)
(70)
But by Remark 13 we see that the last expression of (70) isdominated by (39) on the region Γ
1199020
given by (35) for allbut a finite number of values of 119899 Next using the dominatedconvergence theorem Parsevalrsquos relation and (40) we obtainthe desired result
Corollary 15 Let 1199020 119865 119890
119899infin
119899=1 (1205821119899 1205822119899)infin
119899=1and (119902
1 1199022)
be as in Theorem 14 Then
119864
anf119902
119909[119865 (1199091 1199092)]
= lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times 119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865 (119909
1 1199092)]
(71)
where 119866119899is given by (52)
10 Journal of Function Spaces and Applications
Corollary 16 Let 1199020 119865 and 119890
119899infin
119899=1be as in Theorem 14 and
let Γ1199020
be given by (35) Let 120582 = (1205821 1205822) isin Int (Γ
1199020
) and(1205821119899 1205822119899)infin
119899=1be a sequence in C2
+such that 120582
119895119899rarr 120582
119895
119895 isin 1 2 Then
119864
an
119909[119865 (1199091 1199092)] = lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865(1199091 1199092)]
(72)
where 119866119899is given by (52)
Our another result namely a change of scale formula forfunction space integrals now follows fromCorollary 16 above
Theorem 17 Let 119865 isin F11988611988711986011198602
and let 119890119899infin
119899=1be a complete
orthonormal set in (1198621015840119886119887[0 119879] sdot
119887) Then for any 120588
1gt 0 and
1205882gt 0
119864119909[119865 (12058811199091 12058821199092)]
= lim119899rarrinfin
120588minus119899
1120588minus119899
2
times 119864119909[119866119899(120588minus2
1 1199091)119866119899(120588minus2
2 1199092) 119865 (119909
1 1199092)]
(73)
where 119866119899is given by (52)
Proof Simply choose 120582119895= 120588minus2
119895for 119895 isin 1 2 and 120582
119895119899= 120588minus2
119895
for 119895 isin 1 2 and 119899 isin N in (72)
Remark 18 Of course if we choose 119886(119905) equiv 0 119887(119905) = 1199051198601= 119868 (identity operator) and 119860
2= 0 (zero operator) then
the function space 119862119886119887[0 119879] reduces to the classical Wiener
space 1198620[0 119879] and the generalized Fresnel type class F 119886119887
11986011198602
reduces to the Fresnel class F(1198620[0 119879]) It is known that
F(1198620[0 119879]) forms a Banach algebra over the complex field
In this case we have the relationships between the analyticFeynman integral and theWiener integral on classicalWienerspace as discussed in [14 15]
In recent paper [19] Yoo et al have studied a change ofscale formula for function space integral of the functionalsin the Banach algebra S(1198712
119886119887[0 119879]) the Banach algebra
S(1198712119886119887[0 119879]) is introduced in [12]
5 Functionals in F11988611988711986011198602
In this section we prove a theorem ensuring that variousfunctionals are inF119886119887
11986011198602
Theorem 19 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] Let (119884Y 120574) be a 120590-finite
measure space and let 120593119897 119884 rarr 119862
1015840
119886119887[0 119879] beYndashB(1198621015840
119886119887[0 119879])
measurable for 119897 isin 1 119889 Let 120579 119884 times R119889 rarr C be given by120579(120578 sdot) = ]
120578(sdot) where ]
120578isin M(R119889) for every 120578 isin 119884 and where
the family ]120578 120578 isin 119884 satisfies
(i) ]120578(119864) is a Y-measurable function of 120578 for every 119864 isin
B(R119889)(ii) ]
120578 isin 1198711
(119884Y 120574)
Under these hypothesis the functional 119865 1198622119886119887[0 119879] rarr C
given by
119865 (1199091 1199092) = int
119884
120579(120578
2
sum
119895=1
(11986012
1198951205931(120578) 119909
119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 119909
119895)
sim
)119889120574 (120578)
(74)
belongs toF11988611988711986011198602
and satisfies the inequality
119865 le int
119884
10038171003817100381710038171003817]120578
10038171003817100381710038171003817119889120574 (120578) (75)
Proof Using the techniques similar to those used in [20] wecan show that ]
120578 is measurable as a function of 120578 that 120579 is
Y-measurable and that the integrand in (74) is a measurablefunction of 120578 for every (119909
1 1199092) isin 1198622
119886119887[0 119879]
We define a measure 120591 onY timesB(R119889) by
120591 (119864) = int
119884
]120578(119864(120578)
) 119889120574 (120578) for 119864 isin Y timesB (R119889) (76)
Then by the first assertion of Theorem 31 in [17] 120591 satisfies120591 le int
119884
]120578119889120574(120578) Now let Φ 119884 times R119889 rarr 119862
1015840
119886119887[0 119879] be
defined by Φ(120578 V1 V
119889) = sum
119889
119897=1V119897120593119897(120578) Then Φ is Y times
B(R119889) ndashB(1198621015840119886119887[0 119879])-measurable on the hypothesis for 120593
119897
119897 isin 1 119889 Let 120590 = 120591 ∘Φminus1 Then clearly 120590 isinM(1198621015840119886119887[0 119879])
and satisfies 120590 le 120591From the change of variables theorem and the second
assertion of Theorem 31 in [17] it follows that for ae(1199091 1199092) isin 1198622
119886119887[0 119879] and for every 120588
1gt 0 and 120588
2gt 0
119865 (12058811199091 12058821199092)
= int
119884
]120578(
2
sum
119895=1
(11986012
1198951205931(120578) 120588
119895119909119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 120588
119895119909119895)
sim
)119889120574 (120578)
= int
119884
[
[
int
R119889exp
119894
119889
sum
119897=1
V119897
[
[
2
sum
119895=1
(11986012
119895120593119897(120578)
120588119895119909119895)
sim
]
]
119889]120578
times (V1 V
119889)]
]
119889120574 (120578)
= int
119884timesR119889exp
119894
119889
sum
119897=1
V119897
[
[
2
sum
119895=1
(11986012
119895120593119897(120578) 120588
119895119909119895)
sim
]
]
119889120591
times (120578 V1 V
119889)
Journal of Function Spaces and Applications 11
= int
119884timesR119889exp
2
sum
119895=1
119894(11986012
119895Φ(120578 V
1 V
119889) 120588119895119909119895)
sim
119889120591
times (120578 V1 V
119889)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119909119895)
sim
119889120591 ∘ Φminus1
(119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119909119895)
sim
119889120590 (119908)
(77)
Thus the functional 119865 given by (74) belongs to F11988611988711986011198602
andsatisfies the inequality
119865 = 120590 le 120591 le int
119884
10038171003817100381710038171003817]120578
10038171003817100381710038171003817119889120574 (120578) (78)
As mentioned in (2) of Remark 6 F 11988611988711986011198602
is a Banachalgebra if Ran(119860
1+ 1198602) is dense in 1198621015840
119886119887[0 119879] In this case
many analytic functionals of 119865 can be formed The followingcorollary is relevant to Feynman integration theories andquantum mechanics where exponential functions play animportant role
Corollary 20 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] such that Ran(119860
1+ 1198602)
is dense in 1198621015840119886119887[0 119879] Let 119865 be given by (74) with 120579 as in
Theorem 19 and let 120573 C rarr C be an entire function Then(120573 ∘ 119865)(119909
1 1199092) is in F119886119887
11986011198602
In particular exp119865(1199091 1199092) isin
F11988611988711986011198602
Corollary 21 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] and let 119892
1 119892
119889 be a
finite subset of 1198621015840119886119887[0 119879] Given 120573 = ] where ] isin M(R119889)
define 119865 1198622119886119887[0 119879] rarr C by
119865 (1199091 1199092) = 120573(
2
sum
119895=1
(11986012
1198951198921 119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 119909119895)
sim
)
(79)
Then 119865 is an element ofF11988611988711986011198602
Proof Let (119884Y 120574) be a probability space and for 119897 isin1 119889 let 120593
119897(120578) equiv 119892
119897 Take 120579(120578 sdot) = 120573(sdot) = ](sdot) Then for
all 1205881gt 0 and 120588
2gt 0 and for ae (119909
1 1199092) isin 1198622
119886119887[0 119879]
int
119884
120579(120578
2
sum
119895=1
(11986012
1198951205931(120578) 120588
119895119909119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 120588
119895119909119895)
sim
)119889120574 (120578)
= int
119884
120573(
2
sum
119895=1
(11986012
1198951198921 120588119895119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 120588119895119909119895)
sim
)119889120574 (120578)
= 120573(
2
sum
119895=1
(11986012
1198951198921 120588119895119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 120588119895119909119895)
sim
)
= 119865 (12058811199091 12058821199092)
(80)
Hence 119865 isin F11988611988711986011198602
Remark 22 Let 119889 = 1 and let (119884Y 120574) = ([0 119879]B([0 119879])119898119871) in Theorem 19 where 119898
119871denotes the Lebesgue measure
on [0 119879] ThenTheorems 46 47 and 49 in [18] follow fromthe results in this section by letting119860
1be the identity operator
and letting 1198602equiv 0 on 1198621015840
119886119887[0 119879] The function 120579 studied in
[18] (and mentioned in Remark 2 above) is interpreted as thepotential energy in quantum mechanics
Acknowledgments
The authors would like to express their gratitude to the refer-ees for their valuable comments and suggestions which haveimproved the original paper This research was supported bythe Basic Science Research Program through the NationalResearch Foundation of Korea (NRF) funded by theMinistryof Education (2011-0014552)
References
[1] G Kallianpur and C Bromley ldquoGeneralized Feynman integralsusing analytic continuation in several complex variablesrdquo inStochastic Analysis and Applications M A Pinsky Ed vol 7pp 217ndash267 Marcel Dekker New York NY USA 1984
[2] G Kallianpur D Kannan and R L Karandikar ldquoAnalytic andsequential Feynman integrals on abstract Wiener and Hilbertspaces and a Cameron-Martin formulardquo Annales de lrsquoInstitutHenri Poincare vol 21 no 4 pp 323ndash361 1985
[3] G W Johnson and M L Lapidus The Feynman Integral andFeynmanrsquos Operational Calculus Clarendon Press Oxford UK2000
[4] J Yeh ldquoSingularity of Gaussian measures on function spacesinduced by Brownian motion processes with non-stationaryincrementsrdquo Illinois Journal of Mathematics vol 15 pp 37ndash461971
[5] J Yeh Stochastic Processes and the Wiener Integral MarcelDekker New York NY USA 1973
[6] S J Chang J G Choi and D Skoug ldquoIntegration by partsformulas involving generalized Fourier-Feynman transformson function spacerdquo Transactions of the American MathematicalSociety vol 355 no 7 pp 2925ndash2948 2003
[7] S J Chang J G Choi and D Skoug ldquoParts formulas involvingconditional generalized Feynman integrals and conditionalgeneralized Fourier-Feynman transforms on function spacerdquo
12 Journal of Function Spaces and Applications
Integral Transforms and Special Functions vol 15 no 6 pp 491ndash512 2004
[8] S J Chang J G Choi and D Skoug ldquoEvaluation formulas forconditional function space integralsmdashIrdquo Stochastic Analysis andApplications vol 25 no 1 pp 141ndash168 2007
[9] S J Chang J G Choi and D Skoug ldquoGeneralized Fourier-Feynman transforms convolution products and first variationson function spacerdquoTheRockyMountain Journal ofMathematicsvol 40 no 3 pp 761ndash788 2010
[10] S J Chang and D M Chung ldquoConditional function spaceintegrals with applicationsrdquo The Rocky Mountain Journal ofMathematics vol 26 no 1 pp 37ndash62 1996
[11] S J Chang H S Chung and D Skoug ldquoIntegral transformsof functionals in 119871
2(119862119886119887[0 119879])rdquoThe Journal of Fourier Analysis
and Applications vol 15 no 4 pp 441ndash462 2009[12] S J Chang and D Skoug ldquoGeneralized Fourier-Feynman
transforms and a first variation on function spacerdquo IntegralTransforms and Special Functions vol 14 pp 375ndash393 2003
[13] J G Choi and S J Chang ldquoGeneralized Fourier-Feynmantransform and sequential transforms on function spacerdquo Jour-nal of the Korean Mathematical Society vol 49 pp 1065ndash10822012
[14] R H Cameron and D A Storvick ldquoRelationships between theWiener integral and the analytic Feynman integralrdquo Rendicontidel Circolo Matematico di Palermo no 17 supplement pp 117ndash133 1987
[15] RHCameron andDA Storvick ldquoChange of scale formulas forWiener integralrdquo Rendiconti del Circolo Matematico di Palermono 17 pp 105ndash115 1987
[16] G W Johnson ldquoThe equivalence of two approaches to theFeynman integralrdquo Journal of Mathematical Physics vol 23 pp2090ndash2096 1982
[17] G W Johnson and D L Skoug ldquoNotes on the FeynmanintegralmdashIII the Schroedinger equationrdquo Pacific Journal ofMathematics vol 105 pp 321ndash358 1983
[18] S J Chang J G Choi and S D Lee ldquoA Fresnel type class onfunction spacerdquo Journal of the Korean Society of MathematicalEducation Series B vol 16 no 1 pp 107ndash119 2009
[19] I Yoo B J Kim and B S Kim ldquoA change of scale formulafor a function space integral on 119862
119886119887[0 119879]rdquo Proceedings of the
American Mathematical Society vol 141 no 8 pp 2729ndash27392013
[20] K S Chang G W Johnson and D L Skoug ldquoFunctions inthe Fresnel classrdquo Proceedings of the American MathematicalSociety vol 100 no 2 pp 309ndash318 1987
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces and Applications 5
Definition 5 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] The generalized Fresnel
type classF11988611988711986011198602
of functionals on1198622119886119887[0 119879] is defined as the
space of all functionals 119865 on 1198622119886119887[0 119879] of the following form
119865 (1199091 1199092) = int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119909119895)
sim
119889119891 (119908) (30)
for some119891 isinM(1198621015840119886119887[0 119879]) More precisely since we identify
functionals which coincide s-ae on 1198622119886119887[0 119879] F119886119887
11986011198602
canbe regarded as the space of all 119904-equivalence classes offunctionals of the form (30)
Remark 6 (1) In Definition 5 let 1198601be the identity operator
on 1198621015840119886119887[0 119879] and 119860
2equiv 0 Then F 119886119887
11986011198602
is essentially theFresnel type class F(119862
119886119887[0 119879]) and for 119901 isin [1 2] and
nonzero real numbers 1199021and 1199022
119879(119901)
(1199021 1199022)(119865) (119910
1 1199102) = 119879(119901)
1199021
(1198650) (1199101) (31)
if it exists where 1198650(1199091) = 119865(119909
1 1199092) for all (119909
1 1199092) isin
1198622
119886119887[0 119879] and 119879(119901)
1199021
(1198650) means the 119871
119901analytic GFFT on
119862119886119887[0 119879] see [6 12](2) The map 119891 997891rarr 119865 defined by (30) sets up an algebra
isomorphism betweenM(1198621015840119886119887[0 119879]) andF119886119887
11986011198602
if Ran(1198601+
1198602) is dense in1198621015840
119886119887[0 119879] where Ran indicates the range of an
operator In this caseF 11988611988711986011198602
becomes a Banach algebra underthe norm 119865 = 119891 For more details see [1]
Remark 7 Let 119865 be given by (30) In evaluating119864119909[119865(120582minus12
11199091 120582minus12
21199092)] and 119879
(12058211205822)(119865)(1199101 1199102) = 119864
119909[119865(1199101+
120582minus12
11199091 1199102+ 120582minus12
21199092)] for 120582
1gt 0 and 120582
2gt 0 the expression
120595 (120582 119908)
equiv 120595 (1205821 1205822 1198601 1198602 119908)
= exp
2
sum
119895=1
[
[
minus
(119860119895119908119908)
119887
2120582119895
+ 119894120582minus12
119895(11986012
119895119908 119886)119887
]
]
(32)
occurs Clearly for 120582119895gt 0 119895 isin 1 2 |120595( 120582 119908)| le 1 for all
119908 isin 1198621015840
119886119887[0 119879] But for 120582 isin C2
+ |120595( 120582 119908)| is not necessarily
bounded by 1Note that for each 120582
119895isinC+with 120582
119895= 120572119895+ 119894120573119895 119895 isin 1 2
12058212
119895=
radicradic1205722
119895+ 1205732
119895+ 120572119895
2
+ 119894
120573119895
10038161003816100381610038161003816120573119895
10038161003816100381610038161003816
radicradic1205722
119895+ 1205732
119895minus 120572119895
2
120582minus12
119895=radic
radic1205722
119895+ 1205732
119895+ 120572119895
2 (1205722
119895+ 1205732
119895)
minus 119894
120573119895
10038161003816100381610038161003816120573119895
10038161003816100381610038161003816
radic
radic1205722
119895+ 1205732
119895minus 120572119895
2 (1205722
119895+ 1205732
119895)
(33)
Hence for 120582119895isinC+with 120582
119895= 120572119895+ 119894120573119895 119895 isin 1 2
10038161003816100381610038161003816120595 (120582 119908)
10038161003816100381610038161003816
= exp
2
sum
119895=1
[
[
[
[
minus
120572119895
2 (1205722
119895+ 1205732
119895)
(119860119895119908119908)
119887
+
120573119895
10038161003816100381610038161003816120573119895
10038161003816100381610038161003816
radic
radic1205722
119895+ 1205732
119895minus 120572119895
2 (1205722
119895+ 1205732
119895)
(11986012
119895119908 119886)119887
]
]
]
]
(34)
The right hand side of (34) is an unbounded functionof 119908 for 119908 isin 1198621015840
119886119887[0 119879] Thus 119864an[119865] 119864anf 119902[119865] 119879
120582(119865) and
119879(119901)
119902(119865) might not exist Thus throughout this paper we will
need to put additional restrictions on the complex measure119891 corresponding to 119865 in order to obtain our results for theGFFT and the generalized analytic Feynman integral of 119865
In view of Remark 7 we clearly need to impose additionalrestrictions on the functionals 119865 inF119886119887
11986011198602
For a positive real number 119902
0 let
Γ1199020
=
120582 = (120582
1 1205822) isinC2
+| 120582119895= 120572119895+ 119894120573119895
10038161003816100381610038161003816Im (120582minus12
119895)
10038161003816100381610038161003816=radic
radic1205722
119895+ 1205732
119895minus 120572119895
2 (1205722
119895+ 1205732
119895)
lt
1
radic21199020
119895 = 1 2
(35)
and let119896 (1199020 119908) equiv 119896 (119902
0 1198601 1198602 119908)
= exp
2
sum
119895=1
(21199020)minus121003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817119887119886119887
(36)
Then for all 120582 = (1205821 1205822) isin Γ1199020
10038161003816100381610038161003816120595 (120582 119908)
10038161003816100381610038161003816le exp
2
sum
119895=1
radic
radic1205722
119895+ 1205732
119895minus 120572119895
2 (1205722
119895+ 1205732
119895)
100381610038161003816100381610038161003816
(11986012
119895119908 119886)119887
100381610038161003816100381610038161003816
le exp
2
sum
119895=1
radic
radic1205722
119895+ 1205732
119895minus 120572119895
2 (1205722
119895+ 1205732
119895)
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817119887119886119887
lt 119896 (1199020 119908)
(37)
6 Journal of Function Spaces and Applications
We note that for all real 119902119895with |119902
119895| gt 1199020 119895 isin 1 2
(minus119894119902119895)
minus12
=
1
radic
100381610038161003816100381610038162119902119895
10038161003816100381610038161003816
+ sign (119902119895)
119894
radic
100381610038161003816100381610038162119902119895
10038161003816100381610038161003816
(38)
and (minus1198941199021 minus1198941199022) isin Γ1199020
For the existence of the GFFT of 119865 we define a subclass
F1199020
11986011198602
ofF 11988611988711986011198602
by 119865 isin F 119902011986011198602
if and only if
int
1198621015840
119886119887[0119879]
119896 (1199020 119908) 119889
10038161003816100381610038161198911003816100381610038161003816(119908) lt +infin (39)
where 119891 and 119865 are related by (30) and 119896 is given by (36)
Remark 8 Note that in case 119886(119905) equiv 0 and 119887(119905) = 119905 on [0 119879]the function space 119862
119886119887[0 119879] reduces to the classical Wiener
space 1198620[0 119879] and (119908 119886)
119887= 0 for all 119908 isin 1198621015840
119886119887[0 119879] =
1198621015840
0[0 119879] Hence for all 120582 isin C2
+ |120595( 120582 119908)| le 1 and for any
positive real number 1199020F 119902011986011198602
= F11986011198602
theKallianpur andBromleyrsquos class introduced in Section 1
Theorem 9 Let 1199020be a positive real number and let 119865 be an
element ofF 119902011986011198602
Then for any nonzero real numbers 1199021and
1199022with |119902
119895| gt 1199020 119895 isin 1 2 the 119871
1analytic GFFT of 119865 119879(1)
119902(119865)
exists and is given by the following formula
119879(1)
119902(119865) (119910
1 1199102)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
120595(minus119894 119902 119908) 119889119891 (119908)
(40)
for s-ae (1199101 1199102) isin 1198622
119886119887[0 119879] where 120595 is given by (32)
Proof We first note that for 119895 isin 1 2 the PWZ stochasticintegral (11986012
119895119908 119909)sim is a Gaussian random variable withmean
(11986012
119895119908 119886)119887and variance 11986012
119895119908
2
119887
= (119860119895119908119908)119887
Hence using(30) the Fubini theorem the change of variables theorem and(14) we have that for all 120582
1gt 0 and 120582
2gt 0
119869 (1199101 1199102 1205821 1205822)
equiv 119864119909[119865 (1199101+ 120582minus12
11199091 1199102+ 120582minus12
21199092)]
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
times(
2
prod
119895=1
119864119909119895
[exp 119894120582minus12119895(11986012
119895119908 119909119895)
sim
])119889119891 (119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
times[
[
[
2
prod
119895=1
(2120587(119860119895119908119908)
119887
)
minus12
times int
R
exp
119894120582minus12
119895119906119895
minus
[119906119895minus (11986012
119895119908 119886)119887
]
2
2(119860119895119908119908)
119887
119889119906119895
]
]
]
119889119891(119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
120595(120582 119908) 119889119891 (119908)
(41)
Let
119879 120582(119865) (119910
1 1199102)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
120595(120582 119908) 119889119891 (119908)
(42)
for each 120582 isin C2+ Clearly
119879 120582(119865) (119910
1 1199102) = 119869 (119910
1 1199102 1205821 1205822) (43)
for all 1205821gt 0 and 120582
2gt 0 Let Γ
1199020
be given by (35)Then for all120582 isin Int(Γ
1199020
)
1003816100381610038161003816119879 120582(119865) (119910
1 1199102)1003816100381610038161003816lt int
1198621015840
119886119887[0119879]
119896 (1199020 119908) 119889
10038161003816100381610038161198911003816100381610038161003816(119908) lt +infin
(44)
Using this fact and the dominated convergence theoremwe see that 119879
120582(119865)(1199101 1199102) is a continuous function of 120582 =
(1205821 1205822) on Int(Γ
1199020
) For each 119908 isin 1198621015840119886119887[0 119879] 120595( 120582 119908) is an
analytic function of 120582 throughout the domain Int(Γ1199020
) so thatintΔ
120595(120582 119908)119889
120582 = 0 for every rectifiable simple closed curve
Δ in Int(Γ1199020
) By (42) the Fubini theorem and the Moreratheorem we see that 119879
120582(119865)(1199101 1199102) is an analytic function of
120582 throughout the domain Int(Γ
1199020
) Finally using (28) withthe dominated convergence theorem we obtain the desiredresult
Theorem 10 Let 1199020and 119865 be as inTheorem 9Then for all 119901 isin
(1 2] and all nonzero real numbers 1199021and 119902
2with |119902
119895| gt 1199020
119895 isin 1 2 the119871119901analytic GFFT of119865119879(119901)
119902(119865) exists and is given
by the right hand side of (40) for s-ae (1199101 1199102) isin 1198622
119886119887[0 119879]
Journal of Function Spaces and Applications 7
Proof Let Γ1199020
be given by (35) It was shown in the proofof Theorem 9 that 119879
120582(119865)(1199101 1199102) is an analytic function of 120582
throughout the domain Int(Γ1199020
) In viewofDefinition 4 it willsuffice to show that for each 120588
1gt 0 and 120588
2gt 0
lim120582rarrminus119894 119902
int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816
119879 120582(119865) (120588
11199101 12058821199102)
minus119879(119901)
119902(119865) (120588
11199101 12058821199102)
100381610038161003816100381610038161003816
1199011015840
times 119889 (120583 times 120583) (1199101 1199102) = 0
(45)
Fixing 119901 isin (1 2] and using the inequalities (37) and (39)we obtain that for all 120588
119895gt 0 119895 isin 1 2 and all 120582 isin Γ
1199020
100381610038161003816100381610038161003816
119879 120582(119865) (120588
11199101 12058821199102) minus 119879(119901)
119902(119865) (120588
11199101 12058821199102)
100381610038161003816100381610038161003816
1199011015840
le
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894120588119895(11986012
119895119908 119910119895)
sim
times[120595 (120582 119908) minus 120595 (minus119894 119902 119908)] 119889119891 (119908)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199011015840
le (int
1198621015840
119886119887[0119879]
[
10038161003816100381610038161003816120595 (120582 119908)
10038161003816100381610038161003816+
10038161003816100381610038161003816120595 (minus119894 119902 119908)
10038161003816100381610038161003816] 11988910038161003816100381610038161198911003816100381610038161003816(119908))
1199011015840
le (2int
1198621015840
119886119887[0119879]
119896 (1199020 119908) 119889
10038161003816100381610038161198911003816100381610038161003816(119908))
1199011015840
lt +infin
(46)
Hence by the dominated convergence theorem we see thatfor each 119901 isin (1 2] and each 120588
1gt 0 and 120588
2gt 0
lim120582rarrminus119894 119902
int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816
119879 120582(119865) (120588
11199101 12058821199102)
minus119879(119901)
119902(119865) (120588
11199101 12058821199102)
100381610038161003816100381610038161003816
1199011015840
119889 (120583 times 120583) (1199101 1199102)
= lim120582rarrminus119894 119902
int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119910119895)
sim
times 120595(120582 119908) 119889119891 (119908)
minus int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119910119895)
sim
times120595(minus119894 119902 119908) 119889119891 (119908)
10038161003816100381610038161003816100381610038161003816100381610038161003816
1199011015840
times 119889 (120583 times 120583) (1199101 1199102)
= int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119910119895)
sim
times lim120582rarrminus119894 119902
[120595 (120582 119908)
minus120595 (minus119894 119902 119908)] 119889119891 (119908)
10038161003816100381610038161003816100381610038161003816100381610038161003816
1199011015840
times 119889 (120583 times 120583) (1199101 1199102)
= 0
(47)
which concludes the proof of Theorem 10
Remark 11 (1) In view of Theorems 9 and 10 we see thatfor each 119901 isin [1 2] the 119871
119901analytic GFFT of 119865 119879(119901)
119902(119865) is
given by the right hand side of (40) for 1199020 1199021 1199022 and 119865 as
in Theorem 9 and for s-ae (1199101 1199102) isin 1198622
119886119887[0 119879]
119879(119901)
119902(119865) (119910
1 1199102) = 119864
anf 119902119909[119865 (1199101+ 1199091 1199102+ 1199092)] 119901 isin [1 2]
(48)
In particular using this fact and (29) we have that for all 119901 isin[1 2]
119879(119901)
119902(119865) (0 0) = 119864
anf 119902119909[119865 (1199091 1199092)] (49)
(2) For nonzero real numbers 1199021and 119902
2with |119902
119895| gt 119902
0
119895 isin 1 2 define a set function 119891 119860119902B(1198621015840
119886119887[0 119879]) rarr C by
119891
119860
119902(119861) = int
119861
120595 (minus119894 119902 119908) 119889119891 (119908) 119861 isinB (1198621015840
119886119887[0 119879])
(50)
where 119891 and 119865 are related by (30) and B(1198621015840119886119887[0 119879]) is the
Borel 120590-algebra of 1198621015840119886119887[0 119879] Then it is obvious that 119891 119860
119902
belongs to M(1198621015840119886119887[0 119879]) and for all 119901 isin [1 2] 119879(119901)
119902(119865) can
be expressed as
119879(119901)
119902(119865) (119910
1 1199102) = int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
119889119891
119860
119902(119908)
(51)
for s-ae (1199101 1199102) isin 119862
2
119886119887[0 119879] Hence 119879(119901)
119902(119865) belongs to
F 11988611988711986011198602
for all 119901 isin [1 2]
4 Relationships between the GFFT and theFunction Space Integral of Functionals inF11988611988711986011198602
In this section we establish a relationship between the GFFTand the function space integral of functionals in the Fresneltype classF119886119887
11986011198602
8 Journal of Function Spaces and Applications
Throughout this section for convenience we use thefollowing notation for given 120582 isin C
+and 119899 = 1 2 let
119866119899(120582 119909)
= exp[1 minus 1205822
]
119899
sum
119896=1
[(119890119896 119909)sim
]
2
+ (12058212
minus 1)
119899
sum
119896=1
(119890119896 119886)1198621015840
119886119887
(119890119896 119909)sim
(52)
where 119890119899infin
119899=1is a complete orthonormal set in (1198621015840
119886119887[0 119879]
sdot 119887)To obtain our main results Theorems 14 and 17 below
we state a fundamental integration formula for the functionspace 119862
119886119887[0 119879]
Let 1198901 119890
119899 be an orthonormal set in (1198621015840
119886119887[0 119879] sdot
119887)
let 119896 R119899 rarr C be a Lebesgue measurable function and let119870 119862119886119887[0 119879] rarr C be given by
119870 (119909) = 119896 ((1198901 119909)sim
(119890119899 119909)sim
) (53)
Then
119864 [119870] = int
119862119886119887[0119879]
119896 ((1198901 119909)sim
(119890119899 119909)sim
) 119889120583 (119909)
= (2120587)minus1198992
int
R119899119896 (1199061 119906
119899)
times exp
minus
119899
sum
119895=1
[119906119895minus (119890119895 119886)119887
]
2
2
1198891199061 119889119906
119899
(54)
in the sense that if either side of (54) exists both sides existand equality holds
We also need the following lemma to obtain our maintheorem in this section
Lemma 12 Let 1198901 119890
119899 be an orthonormal subset of
(1198621015840
119886119887[0 119879] sdot
119887) and let 119908 be an element of 1198621015840
119886119887[0 119879] Then
for each 120582 isin C+ the function space integral
119864119909[119866119899(120582 119909) exp 119894(119908 119909)sim] (55)
exists and is given by the formula
119864119909[119866119899(120582 119909) exp 119894(119908 119909)sim]
= 120582minus1198992 exp
[
120582 minus 1
2120582
]
119899
sum
119896=1
(119890119896 119908)2
119887minus
1
2
1199082
119887
+ 119894120582minus12
119899
sum
119896=1
(119890119896 119886)119887(119890119896 119908)119887
+ 119894(119890119899+1 119886)119887[1199082
119887minus
119899
sum
119896=1
(119890119896 119908)2
119887]
12
(56)
where 119866119899is given by (52) above and
119890119899+1=[
[
1199082
119887minus
119899
sum
119895=1
(119890119895 119908)
2
119887
]
]
minus12
119908 minus
119899
sum
119895=1
(119890119895 119908)119887
119890119895
(57)
Proof (Outline) Using the Gram-Schmidt process for any119908 isin 119862
1015840
119886119887[0 119879] we can write 119908 = sum
119899+1
119896=1119888119896119890119896 where
1198901 119890
119899 119890119899+1 is an orthonormal set in (1198621015840
119886119887[0 119879] sdot
119887)
and
119888119896=
(119890119896 119908)119887 119896 = 1 119899
[1199082
119887minus
119899
sum
119895=1
(119890119895 119908)
2
119887
]
12
119896 = 119899 + 1
(58)
Then using (52) (54) the Fubini theorem and (14) it followsthat (56) holds for all 120582 isin C
+
The following remark will be very useful in the proof ofour main theorem in this section
Remark 13 Let 1199020be a positive real number and let Γ
1199020
begiven by (35) For real numbers 119902
1and 1199022with |119902
119895| gt 1199020 119895 isin
1 2 let 120582119899infin
119899=1= (1205821119899 1205822119899)infin
119899=1be a sequence in C2
+such
that
120582119899= (1205821119899 1205822119899) 997888rarr minus119894 119902 = (minus119894119902
1 minus1198941199022) (59)
Let 120582119895119899= 120572119895119899+ 119894120573119895119899
for 119895 isin 1 2 and 119899 isin N Then for119895 isin 1 2 Re(120582
119895119899) = 120572119895119899gt 0 and
120582minus1
119895119899= (120572119895119899+ 119894120573119895119899)
minus1
=
120572119895119899minus 119894120573119895119899
1205722
119895119899+ 1205732
119895119899
(60)
for each 119899 isin N Since |Im ((minus119894119902119895)minus12
)| = 1radic2|119902119895| lt 1radic2119902
0
for 119895 isin 1 2 there exists a sufficiently large 119871 isin N such thatfor any 119899 ge 119871 120582
1119899and 120582
2119899are in Int(Γ
1199020
) and
120575 (1199021 1199022) equiv sup ( 1003816100381610038161003816
1003816Im (120582minus12
1119899)
10038161003816100381610038161003816 119899 ge 119871
cup
10038161003816100381610038161003816Im (120582minus12
2119899)
10038161003816100381610038161003816 119899 ge 119871
cup
100381610038161003816100381610038161003816
Im ((minus1198941199021)minus12
)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
Im ((minus1198941199022)minus12
)
100381610038161003816100381610038161003816
)
lt
1
radic21199020
(61)
Thus there exists a positive real number 120576 gt 1 such that120575(1199021 1199022) lt 1(120576radic2119902
0)
Let 119890119899infin
119899=1be a complete orthonormal set in (1198621015840
119886119887[0 119879]
sdot 119887) Using Parsevalrsquos identity it follows that
(1198921 1198922)119887=
infin
sum
119899=1
(119890119899 1198921)119887(119890119899 1198922)119887
(62)
Journal of Function Spaces and Applications 9
for all 1198921 1198922isin 1198621015840
119886119887[0 119879] In addition for each 119892 isin 1198621015840
119886119887[0 119879]
10038171003817100381710038171198921003817100381710038171003817
2
119887minus
119899
sum
119896=1
(119890119896 119892)2
119887=
infin
sum
119896=119899+1
(119890119896 119892)2
119887ge 0 (63)
for every 119899 isin NSince
(119892 119886)119887=
infin
sum
119899=1
(119890119899 119892)119887(119890119899 119886)119887
(64)
and for 120576 gt 1
minus12057610038171003817100381710038171198921003817100381710038171003817119887119886119887lt minus10038171003817100381710038171198921003817100381710038171003817119887119886119887le (119892 119886)
119887
le10038171003817100381710038171198921003817100381710038171003817119887119886119887lt 12057610038171003817100381710038171198921003817100381710038171003817119887119886119887
(65)
there exists a sufficiently large119870119895isin N such that for any 119899 ge 119870
119895
1003816100381610038161003816100381610038161003816100381610038161003816
119899
sum
119896=1
(119890119896 11986012
119895119908)119887
(119890119896 119886)119887
1003816100381610038161003816100381610038161003816100381610038161003816
lt 120576
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817119887119886119887
(66)
for 119895 isin 1 2Using these and a long and tedious calculation we can
show that for every 119899 ge max119871 1198701 1198702
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
exp
2
sum
119895=1
([
120582119895119899minus 1
2120582119895119899
]
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
minus
1
2
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
+ 119894120582minus12
119895119899
119899
sum
119896=1
(119890119896 11986012
119895119908)119887
(119890119896 119886)119887+ 119894(119890119899+1 119886)119887
times [
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
minus
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
]
12
)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119896 (1199020 119908)
(67)
where 119896(1199020 119908) is given by (36)
In our next theorem for119865 isin F11988611988711986011198602
we express theGFFTof 119865 as the limit of a sequence of function space integrals on1198622
119886119887[0 119879]
Theorem 14 Let 1199020and 119865 be as in Theorem 10 Let 119890
119899infin
119899=1
be a complete orthonormal set in (1198621015840119886119887[0 119879] sdot
119887) and let
(1205821119899 1205822119899)infin
119899=1be a sequence in C2
+such that 120582
119895119899rarr minus119894119902
119895
where 119902119895is a real number with |119902
119895| gt 1199020 119895 isin 1 2 Then for
119901 isin [1 2] and for s-ae (1199101 1199102) isin 1198622
119886119887[0 119879]
119879(119901)
119902(119865) (119910
1 1199102)
= lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865 (119910
1+ 1199091 1199102+ 1199092)]
(68)
where 119866119899is given by (52)
Proof ByTheorems9 and 10we know that for each119901 isin [1 2]the 119871119901analytic GFFT of 119865 119879(119901)
119902(119865) exists and is given by the
right hand side of (40) Thus it suffices to show that
119879(1)
119902(119865) (119910
1 1199102) = 119864
anf 119902119909[119865 (1199101+ 1199091 1199102+ 1199092)]
= lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times 119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092)
times 119865 (1199101+ 1199091 1199102+ 1199092)] sdot
(69)
Using (30) the Fubini theorem and (56) with 120582 and 119908replaced with 120582
119895119899and 11986012
119895119908 119895 isin 1 2 respectively we see
that
1205821198992
11198991205821198992
2119899119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865 (119910
1+ 1199091 1199102+ 1199092)]
= 1205821198992
11198991205821198992
2119899int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
times(
2
prod
119895=1
119864119909119895
[119866119899(120582minus12
119895119899 119909119895)
times exp 119894(11986012119895119908 119909119895)
sim
])119889119891 (119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
(119894(11986012
119895119908 119910119895)
sim
+ [
120582119895119899minus 1
2120582119895119899
]
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
minus
1
2
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
+ 119894120582minus12
119895119899
119899
sum
119896=1
(119890119896 119886)119887(119890119896 11986012
119895119908)119887
+ 119894(119890119899+1 119886)119887[
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
minus
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
]
12
)
119889119891 (119908)
(70)
But by Remark 13 we see that the last expression of (70) isdominated by (39) on the region Γ
1199020
given by (35) for allbut a finite number of values of 119899 Next using the dominatedconvergence theorem Parsevalrsquos relation and (40) we obtainthe desired result
Corollary 15 Let 1199020 119865 119890
119899infin
119899=1 (1205821119899 1205822119899)infin
119899=1and (119902
1 1199022)
be as in Theorem 14 Then
119864
anf119902
119909[119865 (1199091 1199092)]
= lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times 119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865 (119909
1 1199092)]
(71)
where 119866119899is given by (52)
10 Journal of Function Spaces and Applications
Corollary 16 Let 1199020 119865 and 119890
119899infin
119899=1be as in Theorem 14 and
let Γ1199020
be given by (35) Let 120582 = (1205821 1205822) isin Int (Γ
1199020
) and(1205821119899 1205822119899)infin
119899=1be a sequence in C2
+such that 120582
119895119899rarr 120582
119895
119895 isin 1 2 Then
119864
an
119909[119865 (1199091 1199092)] = lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865(1199091 1199092)]
(72)
where 119866119899is given by (52)
Our another result namely a change of scale formula forfunction space integrals now follows fromCorollary 16 above
Theorem 17 Let 119865 isin F11988611988711986011198602
and let 119890119899infin
119899=1be a complete
orthonormal set in (1198621015840119886119887[0 119879] sdot
119887) Then for any 120588
1gt 0 and
1205882gt 0
119864119909[119865 (12058811199091 12058821199092)]
= lim119899rarrinfin
120588minus119899
1120588minus119899
2
times 119864119909[119866119899(120588minus2
1 1199091)119866119899(120588minus2
2 1199092) 119865 (119909
1 1199092)]
(73)
where 119866119899is given by (52)
Proof Simply choose 120582119895= 120588minus2
119895for 119895 isin 1 2 and 120582
119895119899= 120588minus2
119895
for 119895 isin 1 2 and 119899 isin N in (72)
Remark 18 Of course if we choose 119886(119905) equiv 0 119887(119905) = 1199051198601= 119868 (identity operator) and 119860
2= 0 (zero operator) then
the function space 119862119886119887[0 119879] reduces to the classical Wiener
space 1198620[0 119879] and the generalized Fresnel type class F 119886119887
11986011198602
reduces to the Fresnel class F(1198620[0 119879]) It is known that
F(1198620[0 119879]) forms a Banach algebra over the complex field
In this case we have the relationships between the analyticFeynman integral and theWiener integral on classicalWienerspace as discussed in [14 15]
In recent paper [19] Yoo et al have studied a change ofscale formula for function space integral of the functionalsin the Banach algebra S(1198712
119886119887[0 119879]) the Banach algebra
S(1198712119886119887[0 119879]) is introduced in [12]
5 Functionals in F11988611988711986011198602
In this section we prove a theorem ensuring that variousfunctionals are inF119886119887
11986011198602
Theorem 19 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] Let (119884Y 120574) be a 120590-finite
measure space and let 120593119897 119884 rarr 119862
1015840
119886119887[0 119879] beYndashB(1198621015840
119886119887[0 119879])
measurable for 119897 isin 1 119889 Let 120579 119884 times R119889 rarr C be given by120579(120578 sdot) = ]
120578(sdot) where ]
120578isin M(R119889) for every 120578 isin 119884 and where
the family ]120578 120578 isin 119884 satisfies
(i) ]120578(119864) is a Y-measurable function of 120578 for every 119864 isin
B(R119889)(ii) ]
120578 isin 1198711
(119884Y 120574)
Under these hypothesis the functional 119865 1198622119886119887[0 119879] rarr C
given by
119865 (1199091 1199092) = int
119884
120579(120578
2
sum
119895=1
(11986012
1198951205931(120578) 119909
119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 119909
119895)
sim
)119889120574 (120578)
(74)
belongs toF11988611988711986011198602
and satisfies the inequality
119865 le int
119884
10038171003817100381710038171003817]120578
10038171003817100381710038171003817119889120574 (120578) (75)
Proof Using the techniques similar to those used in [20] wecan show that ]
120578 is measurable as a function of 120578 that 120579 is
Y-measurable and that the integrand in (74) is a measurablefunction of 120578 for every (119909
1 1199092) isin 1198622
119886119887[0 119879]
We define a measure 120591 onY timesB(R119889) by
120591 (119864) = int
119884
]120578(119864(120578)
) 119889120574 (120578) for 119864 isin Y timesB (R119889) (76)
Then by the first assertion of Theorem 31 in [17] 120591 satisfies120591 le int
119884
]120578119889120574(120578) Now let Φ 119884 times R119889 rarr 119862
1015840
119886119887[0 119879] be
defined by Φ(120578 V1 V
119889) = sum
119889
119897=1V119897120593119897(120578) Then Φ is Y times
B(R119889) ndashB(1198621015840119886119887[0 119879])-measurable on the hypothesis for 120593
119897
119897 isin 1 119889 Let 120590 = 120591 ∘Φminus1 Then clearly 120590 isinM(1198621015840119886119887[0 119879])
and satisfies 120590 le 120591From the change of variables theorem and the second
assertion of Theorem 31 in [17] it follows that for ae(1199091 1199092) isin 1198622
119886119887[0 119879] and for every 120588
1gt 0 and 120588
2gt 0
119865 (12058811199091 12058821199092)
= int
119884
]120578(
2
sum
119895=1
(11986012
1198951205931(120578) 120588
119895119909119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 120588
119895119909119895)
sim
)119889120574 (120578)
= int
119884
[
[
int
R119889exp
119894
119889
sum
119897=1
V119897
[
[
2
sum
119895=1
(11986012
119895120593119897(120578)
120588119895119909119895)
sim
]
]
119889]120578
times (V1 V
119889)]
]
119889120574 (120578)
= int
119884timesR119889exp
119894
119889
sum
119897=1
V119897
[
[
2
sum
119895=1
(11986012
119895120593119897(120578) 120588
119895119909119895)
sim
]
]
119889120591
times (120578 V1 V
119889)
Journal of Function Spaces and Applications 11
= int
119884timesR119889exp
2
sum
119895=1
119894(11986012
119895Φ(120578 V
1 V
119889) 120588119895119909119895)
sim
119889120591
times (120578 V1 V
119889)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119909119895)
sim
119889120591 ∘ Φminus1
(119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119909119895)
sim
119889120590 (119908)
(77)
Thus the functional 119865 given by (74) belongs to F11988611988711986011198602
andsatisfies the inequality
119865 = 120590 le 120591 le int
119884
10038171003817100381710038171003817]120578
10038171003817100381710038171003817119889120574 (120578) (78)
As mentioned in (2) of Remark 6 F 11988611988711986011198602
is a Banachalgebra if Ran(119860
1+ 1198602) is dense in 1198621015840
119886119887[0 119879] In this case
many analytic functionals of 119865 can be formed The followingcorollary is relevant to Feynman integration theories andquantum mechanics where exponential functions play animportant role
Corollary 20 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] such that Ran(119860
1+ 1198602)
is dense in 1198621015840119886119887[0 119879] Let 119865 be given by (74) with 120579 as in
Theorem 19 and let 120573 C rarr C be an entire function Then(120573 ∘ 119865)(119909
1 1199092) is in F119886119887
11986011198602
In particular exp119865(1199091 1199092) isin
F11988611988711986011198602
Corollary 21 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] and let 119892
1 119892
119889 be a
finite subset of 1198621015840119886119887[0 119879] Given 120573 = ] where ] isin M(R119889)
define 119865 1198622119886119887[0 119879] rarr C by
119865 (1199091 1199092) = 120573(
2
sum
119895=1
(11986012
1198951198921 119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 119909119895)
sim
)
(79)
Then 119865 is an element ofF11988611988711986011198602
Proof Let (119884Y 120574) be a probability space and for 119897 isin1 119889 let 120593
119897(120578) equiv 119892
119897 Take 120579(120578 sdot) = 120573(sdot) = ](sdot) Then for
all 1205881gt 0 and 120588
2gt 0 and for ae (119909
1 1199092) isin 1198622
119886119887[0 119879]
int
119884
120579(120578
2
sum
119895=1
(11986012
1198951205931(120578) 120588
119895119909119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 120588
119895119909119895)
sim
)119889120574 (120578)
= int
119884
120573(
2
sum
119895=1
(11986012
1198951198921 120588119895119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 120588119895119909119895)
sim
)119889120574 (120578)
= 120573(
2
sum
119895=1
(11986012
1198951198921 120588119895119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 120588119895119909119895)
sim
)
= 119865 (12058811199091 12058821199092)
(80)
Hence 119865 isin F11988611988711986011198602
Remark 22 Let 119889 = 1 and let (119884Y 120574) = ([0 119879]B([0 119879])119898119871) in Theorem 19 where 119898
119871denotes the Lebesgue measure
on [0 119879] ThenTheorems 46 47 and 49 in [18] follow fromthe results in this section by letting119860
1be the identity operator
and letting 1198602equiv 0 on 1198621015840
119886119887[0 119879] The function 120579 studied in
[18] (and mentioned in Remark 2 above) is interpreted as thepotential energy in quantum mechanics
Acknowledgments
The authors would like to express their gratitude to the refer-ees for their valuable comments and suggestions which haveimproved the original paper This research was supported bythe Basic Science Research Program through the NationalResearch Foundation of Korea (NRF) funded by theMinistryof Education (2011-0014552)
References
[1] G Kallianpur and C Bromley ldquoGeneralized Feynman integralsusing analytic continuation in several complex variablesrdquo inStochastic Analysis and Applications M A Pinsky Ed vol 7pp 217ndash267 Marcel Dekker New York NY USA 1984
[2] G Kallianpur D Kannan and R L Karandikar ldquoAnalytic andsequential Feynman integrals on abstract Wiener and Hilbertspaces and a Cameron-Martin formulardquo Annales de lrsquoInstitutHenri Poincare vol 21 no 4 pp 323ndash361 1985
[3] G W Johnson and M L Lapidus The Feynman Integral andFeynmanrsquos Operational Calculus Clarendon Press Oxford UK2000
[4] J Yeh ldquoSingularity of Gaussian measures on function spacesinduced by Brownian motion processes with non-stationaryincrementsrdquo Illinois Journal of Mathematics vol 15 pp 37ndash461971
[5] J Yeh Stochastic Processes and the Wiener Integral MarcelDekker New York NY USA 1973
[6] S J Chang J G Choi and D Skoug ldquoIntegration by partsformulas involving generalized Fourier-Feynman transformson function spacerdquo Transactions of the American MathematicalSociety vol 355 no 7 pp 2925ndash2948 2003
[7] S J Chang J G Choi and D Skoug ldquoParts formulas involvingconditional generalized Feynman integrals and conditionalgeneralized Fourier-Feynman transforms on function spacerdquo
12 Journal of Function Spaces and Applications
Integral Transforms and Special Functions vol 15 no 6 pp 491ndash512 2004
[8] S J Chang J G Choi and D Skoug ldquoEvaluation formulas forconditional function space integralsmdashIrdquo Stochastic Analysis andApplications vol 25 no 1 pp 141ndash168 2007
[9] S J Chang J G Choi and D Skoug ldquoGeneralized Fourier-Feynman transforms convolution products and first variationson function spacerdquoTheRockyMountain Journal ofMathematicsvol 40 no 3 pp 761ndash788 2010
[10] S J Chang and D M Chung ldquoConditional function spaceintegrals with applicationsrdquo The Rocky Mountain Journal ofMathematics vol 26 no 1 pp 37ndash62 1996
[11] S J Chang H S Chung and D Skoug ldquoIntegral transformsof functionals in 119871
2(119862119886119887[0 119879])rdquoThe Journal of Fourier Analysis
and Applications vol 15 no 4 pp 441ndash462 2009[12] S J Chang and D Skoug ldquoGeneralized Fourier-Feynman
transforms and a first variation on function spacerdquo IntegralTransforms and Special Functions vol 14 pp 375ndash393 2003
[13] J G Choi and S J Chang ldquoGeneralized Fourier-Feynmantransform and sequential transforms on function spacerdquo Jour-nal of the Korean Mathematical Society vol 49 pp 1065ndash10822012
[14] R H Cameron and D A Storvick ldquoRelationships between theWiener integral and the analytic Feynman integralrdquo Rendicontidel Circolo Matematico di Palermo no 17 supplement pp 117ndash133 1987
[15] RHCameron andDA Storvick ldquoChange of scale formulas forWiener integralrdquo Rendiconti del Circolo Matematico di Palermono 17 pp 105ndash115 1987
[16] G W Johnson ldquoThe equivalence of two approaches to theFeynman integralrdquo Journal of Mathematical Physics vol 23 pp2090ndash2096 1982
[17] G W Johnson and D L Skoug ldquoNotes on the FeynmanintegralmdashIII the Schroedinger equationrdquo Pacific Journal ofMathematics vol 105 pp 321ndash358 1983
[18] S J Chang J G Choi and S D Lee ldquoA Fresnel type class onfunction spacerdquo Journal of the Korean Society of MathematicalEducation Series B vol 16 no 1 pp 107ndash119 2009
[19] I Yoo B J Kim and B S Kim ldquoA change of scale formulafor a function space integral on 119862
119886119887[0 119879]rdquo Proceedings of the
American Mathematical Society vol 141 no 8 pp 2729ndash27392013
[20] K S Chang G W Johnson and D L Skoug ldquoFunctions inthe Fresnel classrdquo Proceedings of the American MathematicalSociety vol 100 no 2 pp 309ndash318 1987
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Function Spaces and Applications
We note that for all real 119902119895with |119902
119895| gt 1199020 119895 isin 1 2
(minus119894119902119895)
minus12
=
1
radic
100381610038161003816100381610038162119902119895
10038161003816100381610038161003816
+ sign (119902119895)
119894
radic
100381610038161003816100381610038162119902119895
10038161003816100381610038161003816
(38)
and (minus1198941199021 minus1198941199022) isin Γ1199020
For the existence of the GFFT of 119865 we define a subclass
F1199020
11986011198602
ofF 11988611988711986011198602
by 119865 isin F 119902011986011198602
if and only if
int
1198621015840
119886119887[0119879]
119896 (1199020 119908) 119889
10038161003816100381610038161198911003816100381610038161003816(119908) lt +infin (39)
where 119891 and 119865 are related by (30) and 119896 is given by (36)
Remark 8 Note that in case 119886(119905) equiv 0 and 119887(119905) = 119905 on [0 119879]the function space 119862
119886119887[0 119879] reduces to the classical Wiener
space 1198620[0 119879] and (119908 119886)
119887= 0 for all 119908 isin 1198621015840
119886119887[0 119879] =
1198621015840
0[0 119879] Hence for all 120582 isin C2
+ |120595( 120582 119908)| le 1 and for any
positive real number 1199020F 119902011986011198602
= F11986011198602
theKallianpur andBromleyrsquos class introduced in Section 1
Theorem 9 Let 1199020be a positive real number and let 119865 be an
element ofF 119902011986011198602
Then for any nonzero real numbers 1199021and
1199022with |119902
119895| gt 1199020 119895 isin 1 2 the 119871
1analytic GFFT of 119865 119879(1)
119902(119865)
exists and is given by the following formula
119879(1)
119902(119865) (119910
1 1199102)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
120595(minus119894 119902 119908) 119889119891 (119908)
(40)
for s-ae (1199101 1199102) isin 1198622
119886119887[0 119879] where 120595 is given by (32)
Proof We first note that for 119895 isin 1 2 the PWZ stochasticintegral (11986012
119895119908 119909)sim is a Gaussian random variable withmean
(11986012
119895119908 119886)119887and variance 11986012
119895119908
2
119887
= (119860119895119908119908)119887
Hence using(30) the Fubini theorem the change of variables theorem and(14) we have that for all 120582
1gt 0 and 120582
2gt 0
119869 (1199101 1199102 1205821 1205822)
equiv 119864119909[119865 (1199101+ 120582minus12
11199091 1199102+ 120582minus12
21199092)]
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
times(
2
prod
119895=1
119864119909119895
[exp 119894120582minus12119895(11986012
119895119908 119909119895)
sim
])119889119891 (119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
times[
[
[
2
prod
119895=1
(2120587(119860119895119908119908)
119887
)
minus12
times int
R
exp
119894120582minus12
119895119906119895
minus
[119906119895minus (11986012
119895119908 119886)119887
]
2
2(119860119895119908119908)
119887
119889119906119895
]
]
]
119889119891(119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
120595(120582 119908) 119889119891 (119908)
(41)
Let
119879 120582(119865) (119910
1 1199102)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
120595(120582 119908) 119889119891 (119908)
(42)
for each 120582 isin C2+ Clearly
119879 120582(119865) (119910
1 1199102) = 119869 (119910
1 1199102 1205821 1205822) (43)
for all 1205821gt 0 and 120582
2gt 0 Let Γ
1199020
be given by (35)Then for all120582 isin Int(Γ
1199020
)
1003816100381610038161003816119879 120582(119865) (119910
1 1199102)1003816100381610038161003816lt int
1198621015840
119886119887[0119879]
119896 (1199020 119908) 119889
10038161003816100381610038161198911003816100381610038161003816(119908) lt +infin
(44)
Using this fact and the dominated convergence theoremwe see that 119879
120582(119865)(1199101 1199102) is a continuous function of 120582 =
(1205821 1205822) on Int(Γ
1199020
) For each 119908 isin 1198621015840119886119887[0 119879] 120595( 120582 119908) is an
analytic function of 120582 throughout the domain Int(Γ1199020
) so thatintΔ
120595(120582 119908)119889
120582 = 0 for every rectifiable simple closed curve
Δ in Int(Γ1199020
) By (42) the Fubini theorem and the Moreratheorem we see that 119879
120582(119865)(1199101 1199102) is an analytic function of
120582 throughout the domain Int(Γ
1199020
) Finally using (28) withthe dominated convergence theorem we obtain the desiredresult
Theorem 10 Let 1199020and 119865 be as inTheorem 9Then for all 119901 isin
(1 2] and all nonzero real numbers 1199021and 119902
2with |119902
119895| gt 1199020
119895 isin 1 2 the119871119901analytic GFFT of119865119879(119901)
119902(119865) exists and is given
by the right hand side of (40) for s-ae (1199101 1199102) isin 1198622
119886119887[0 119879]
Journal of Function Spaces and Applications 7
Proof Let Γ1199020
be given by (35) It was shown in the proofof Theorem 9 that 119879
120582(119865)(1199101 1199102) is an analytic function of 120582
throughout the domain Int(Γ1199020
) In viewofDefinition 4 it willsuffice to show that for each 120588
1gt 0 and 120588
2gt 0
lim120582rarrminus119894 119902
int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816
119879 120582(119865) (120588
11199101 12058821199102)
minus119879(119901)
119902(119865) (120588
11199101 12058821199102)
100381610038161003816100381610038161003816
1199011015840
times 119889 (120583 times 120583) (1199101 1199102) = 0
(45)
Fixing 119901 isin (1 2] and using the inequalities (37) and (39)we obtain that for all 120588
119895gt 0 119895 isin 1 2 and all 120582 isin Γ
1199020
100381610038161003816100381610038161003816
119879 120582(119865) (120588
11199101 12058821199102) minus 119879(119901)
119902(119865) (120588
11199101 12058821199102)
100381610038161003816100381610038161003816
1199011015840
le
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894120588119895(11986012
119895119908 119910119895)
sim
times[120595 (120582 119908) minus 120595 (minus119894 119902 119908)] 119889119891 (119908)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199011015840
le (int
1198621015840
119886119887[0119879]
[
10038161003816100381610038161003816120595 (120582 119908)
10038161003816100381610038161003816+
10038161003816100381610038161003816120595 (minus119894 119902 119908)
10038161003816100381610038161003816] 11988910038161003816100381610038161198911003816100381610038161003816(119908))
1199011015840
le (2int
1198621015840
119886119887[0119879]
119896 (1199020 119908) 119889
10038161003816100381610038161198911003816100381610038161003816(119908))
1199011015840
lt +infin
(46)
Hence by the dominated convergence theorem we see thatfor each 119901 isin (1 2] and each 120588
1gt 0 and 120588
2gt 0
lim120582rarrminus119894 119902
int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816
119879 120582(119865) (120588
11199101 12058821199102)
minus119879(119901)
119902(119865) (120588
11199101 12058821199102)
100381610038161003816100381610038161003816
1199011015840
119889 (120583 times 120583) (1199101 1199102)
= lim120582rarrminus119894 119902
int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119910119895)
sim
times 120595(120582 119908) 119889119891 (119908)
minus int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119910119895)
sim
times120595(minus119894 119902 119908) 119889119891 (119908)
10038161003816100381610038161003816100381610038161003816100381610038161003816
1199011015840
times 119889 (120583 times 120583) (1199101 1199102)
= int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119910119895)
sim
times lim120582rarrminus119894 119902
[120595 (120582 119908)
minus120595 (minus119894 119902 119908)] 119889119891 (119908)
10038161003816100381610038161003816100381610038161003816100381610038161003816
1199011015840
times 119889 (120583 times 120583) (1199101 1199102)
= 0
(47)
which concludes the proof of Theorem 10
Remark 11 (1) In view of Theorems 9 and 10 we see thatfor each 119901 isin [1 2] the 119871
119901analytic GFFT of 119865 119879(119901)
119902(119865) is
given by the right hand side of (40) for 1199020 1199021 1199022 and 119865 as
in Theorem 9 and for s-ae (1199101 1199102) isin 1198622
119886119887[0 119879]
119879(119901)
119902(119865) (119910
1 1199102) = 119864
anf 119902119909[119865 (1199101+ 1199091 1199102+ 1199092)] 119901 isin [1 2]
(48)
In particular using this fact and (29) we have that for all 119901 isin[1 2]
119879(119901)
119902(119865) (0 0) = 119864
anf 119902119909[119865 (1199091 1199092)] (49)
(2) For nonzero real numbers 1199021and 119902
2with |119902
119895| gt 119902
0
119895 isin 1 2 define a set function 119891 119860119902B(1198621015840
119886119887[0 119879]) rarr C by
119891
119860
119902(119861) = int
119861
120595 (minus119894 119902 119908) 119889119891 (119908) 119861 isinB (1198621015840
119886119887[0 119879])
(50)
where 119891 and 119865 are related by (30) and B(1198621015840119886119887[0 119879]) is the
Borel 120590-algebra of 1198621015840119886119887[0 119879] Then it is obvious that 119891 119860
119902
belongs to M(1198621015840119886119887[0 119879]) and for all 119901 isin [1 2] 119879(119901)
119902(119865) can
be expressed as
119879(119901)
119902(119865) (119910
1 1199102) = int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
119889119891
119860
119902(119908)
(51)
for s-ae (1199101 1199102) isin 119862
2
119886119887[0 119879] Hence 119879(119901)
119902(119865) belongs to
F 11988611988711986011198602
for all 119901 isin [1 2]
4 Relationships between the GFFT and theFunction Space Integral of Functionals inF11988611988711986011198602
In this section we establish a relationship between the GFFTand the function space integral of functionals in the Fresneltype classF119886119887
11986011198602
8 Journal of Function Spaces and Applications
Throughout this section for convenience we use thefollowing notation for given 120582 isin C
+and 119899 = 1 2 let
119866119899(120582 119909)
= exp[1 minus 1205822
]
119899
sum
119896=1
[(119890119896 119909)sim
]
2
+ (12058212
minus 1)
119899
sum
119896=1
(119890119896 119886)1198621015840
119886119887
(119890119896 119909)sim
(52)
where 119890119899infin
119899=1is a complete orthonormal set in (1198621015840
119886119887[0 119879]
sdot 119887)To obtain our main results Theorems 14 and 17 below
we state a fundamental integration formula for the functionspace 119862
119886119887[0 119879]
Let 1198901 119890
119899 be an orthonormal set in (1198621015840
119886119887[0 119879] sdot
119887)
let 119896 R119899 rarr C be a Lebesgue measurable function and let119870 119862119886119887[0 119879] rarr C be given by
119870 (119909) = 119896 ((1198901 119909)sim
(119890119899 119909)sim
) (53)
Then
119864 [119870] = int
119862119886119887[0119879]
119896 ((1198901 119909)sim
(119890119899 119909)sim
) 119889120583 (119909)
= (2120587)minus1198992
int
R119899119896 (1199061 119906
119899)
times exp
minus
119899
sum
119895=1
[119906119895minus (119890119895 119886)119887
]
2
2
1198891199061 119889119906
119899
(54)
in the sense that if either side of (54) exists both sides existand equality holds
We also need the following lemma to obtain our maintheorem in this section
Lemma 12 Let 1198901 119890
119899 be an orthonormal subset of
(1198621015840
119886119887[0 119879] sdot
119887) and let 119908 be an element of 1198621015840
119886119887[0 119879] Then
for each 120582 isin C+ the function space integral
119864119909[119866119899(120582 119909) exp 119894(119908 119909)sim] (55)
exists and is given by the formula
119864119909[119866119899(120582 119909) exp 119894(119908 119909)sim]
= 120582minus1198992 exp
[
120582 minus 1
2120582
]
119899
sum
119896=1
(119890119896 119908)2
119887minus
1
2
1199082
119887
+ 119894120582minus12
119899
sum
119896=1
(119890119896 119886)119887(119890119896 119908)119887
+ 119894(119890119899+1 119886)119887[1199082
119887minus
119899
sum
119896=1
(119890119896 119908)2
119887]
12
(56)
where 119866119899is given by (52) above and
119890119899+1=[
[
1199082
119887minus
119899
sum
119895=1
(119890119895 119908)
2
119887
]
]
minus12
119908 minus
119899
sum
119895=1
(119890119895 119908)119887
119890119895
(57)
Proof (Outline) Using the Gram-Schmidt process for any119908 isin 119862
1015840
119886119887[0 119879] we can write 119908 = sum
119899+1
119896=1119888119896119890119896 where
1198901 119890
119899 119890119899+1 is an orthonormal set in (1198621015840
119886119887[0 119879] sdot
119887)
and
119888119896=
(119890119896 119908)119887 119896 = 1 119899
[1199082
119887minus
119899
sum
119895=1
(119890119895 119908)
2
119887
]
12
119896 = 119899 + 1
(58)
Then using (52) (54) the Fubini theorem and (14) it followsthat (56) holds for all 120582 isin C
+
The following remark will be very useful in the proof ofour main theorem in this section
Remark 13 Let 1199020be a positive real number and let Γ
1199020
begiven by (35) For real numbers 119902
1and 1199022with |119902
119895| gt 1199020 119895 isin
1 2 let 120582119899infin
119899=1= (1205821119899 1205822119899)infin
119899=1be a sequence in C2
+such
that
120582119899= (1205821119899 1205822119899) 997888rarr minus119894 119902 = (minus119894119902
1 minus1198941199022) (59)
Let 120582119895119899= 120572119895119899+ 119894120573119895119899
for 119895 isin 1 2 and 119899 isin N Then for119895 isin 1 2 Re(120582
119895119899) = 120572119895119899gt 0 and
120582minus1
119895119899= (120572119895119899+ 119894120573119895119899)
minus1
=
120572119895119899minus 119894120573119895119899
1205722
119895119899+ 1205732
119895119899
(60)
for each 119899 isin N Since |Im ((minus119894119902119895)minus12
)| = 1radic2|119902119895| lt 1radic2119902
0
for 119895 isin 1 2 there exists a sufficiently large 119871 isin N such thatfor any 119899 ge 119871 120582
1119899and 120582
2119899are in Int(Γ
1199020
) and
120575 (1199021 1199022) equiv sup ( 1003816100381610038161003816
1003816Im (120582minus12
1119899)
10038161003816100381610038161003816 119899 ge 119871
cup
10038161003816100381610038161003816Im (120582minus12
2119899)
10038161003816100381610038161003816 119899 ge 119871
cup
100381610038161003816100381610038161003816
Im ((minus1198941199021)minus12
)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
Im ((minus1198941199022)minus12
)
100381610038161003816100381610038161003816
)
lt
1
radic21199020
(61)
Thus there exists a positive real number 120576 gt 1 such that120575(1199021 1199022) lt 1(120576radic2119902
0)
Let 119890119899infin
119899=1be a complete orthonormal set in (1198621015840
119886119887[0 119879]
sdot 119887) Using Parsevalrsquos identity it follows that
(1198921 1198922)119887=
infin
sum
119899=1
(119890119899 1198921)119887(119890119899 1198922)119887
(62)
Journal of Function Spaces and Applications 9
for all 1198921 1198922isin 1198621015840
119886119887[0 119879] In addition for each 119892 isin 1198621015840
119886119887[0 119879]
10038171003817100381710038171198921003817100381710038171003817
2
119887minus
119899
sum
119896=1
(119890119896 119892)2
119887=
infin
sum
119896=119899+1
(119890119896 119892)2
119887ge 0 (63)
for every 119899 isin NSince
(119892 119886)119887=
infin
sum
119899=1
(119890119899 119892)119887(119890119899 119886)119887
(64)
and for 120576 gt 1
minus12057610038171003817100381710038171198921003817100381710038171003817119887119886119887lt minus10038171003817100381710038171198921003817100381710038171003817119887119886119887le (119892 119886)
119887
le10038171003817100381710038171198921003817100381710038171003817119887119886119887lt 12057610038171003817100381710038171198921003817100381710038171003817119887119886119887
(65)
there exists a sufficiently large119870119895isin N such that for any 119899 ge 119870
119895
1003816100381610038161003816100381610038161003816100381610038161003816
119899
sum
119896=1
(119890119896 11986012
119895119908)119887
(119890119896 119886)119887
1003816100381610038161003816100381610038161003816100381610038161003816
lt 120576
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817119887119886119887
(66)
for 119895 isin 1 2Using these and a long and tedious calculation we can
show that for every 119899 ge max119871 1198701 1198702
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
exp
2
sum
119895=1
([
120582119895119899minus 1
2120582119895119899
]
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
minus
1
2
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
+ 119894120582minus12
119895119899
119899
sum
119896=1
(119890119896 11986012
119895119908)119887
(119890119896 119886)119887+ 119894(119890119899+1 119886)119887
times [
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
minus
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
]
12
)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119896 (1199020 119908)
(67)
where 119896(1199020 119908) is given by (36)
In our next theorem for119865 isin F11988611988711986011198602
we express theGFFTof 119865 as the limit of a sequence of function space integrals on1198622
119886119887[0 119879]
Theorem 14 Let 1199020and 119865 be as in Theorem 10 Let 119890
119899infin
119899=1
be a complete orthonormal set in (1198621015840119886119887[0 119879] sdot
119887) and let
(1205821119899 1205822119899)infin
119899=1be a sequence in C2
+such that 120582
119895119899rarr minus119894119902
119895
where 119902119895is a real number with |119902
119895| gt 1199020 119895 isin 1 2 Then for
119901 isin [1 2] and for s-ae (1199101 1199102) isin 1198622
119886119887[0 119879]
119879(119901)
119902(119865) (119910
1 1199102)
= lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865 (119910
1+ 1199091 1199102+ 1199092)]
(68)
where 119866119899is given by (52)
Proof ByTheorems9 and 10we know that for each119901 isin [1 2]the 119871119901analytic GFFT of 119865 119879(119901)
119902(119865) exists and is given by the
right hand side of (40) Thus it suffices to show that
119879(1)
119902(119865) (119910
1 1199102) = 119864
anf 119902119909[119865 (1199101+ 1199091 1199102+ 1199092)]
= lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times 119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092)
times 119865 (1199101+ 1199091 1199102+ 1199092)] sdot
(69)
Using (30) the Fubini theorem and (56) with 120582 and 119908replaced with 120582
119895119899and 11986012
119895119908 119895 isin 1 2 respectively we see
that
1205821198992
11198991205821198992
2119899119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865 (119910
1+ 1199091 1199102+ 1199092)]
= 1205821198992
11198991205821198992
2119899int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
times(
2
prod
119895=1
119864119909119895
[119866119899(120582minus12
119895119899 119909119895)
times exp 119894(11986012119895119908 119909119895)
sim
])119889119891 (119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
(119894(11986012
119895119908 119910119895)
sim
+ [
120582119895119899minus 1
2120582119895119899
]
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
minus
1
2
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
+ 119894120582minus12
119895119899
119899
sum
119896=1
(119890119896 119886)119887(119890119896 11986012
119895119908)119887
+ 119894(119890119899+1 119886)119887[
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
minus
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
]
12
)
119889119891 (119908)
(70)
But by Remark 13 we see that the last expression of (70) isdominated by (39) on the region Γ
1199020
given by (35) for allbut a finite number of values of 119899 Next using the dominatedconvergence theorem Parsevalrsquos relation and (40) we obtainthe desired result
Corollary 15 Let 1199020 119865 119890
119899infin
119899=1 (1205821119899 1205822119899)infin
119899=1and (119902
1 1199022)
be as in Theorem 14 Then
119864
anf119902
119909[119865 (1199091 1199092)]
= lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times 119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865 (119909
1 1199092)]
(71)
where 119866119899is given by (52)
10 Journal of Function Spaces and Applications
Corollary 16 Let 1199020 119865 and 119890
119899infin
119899=1be as in Theorem 14 and
let Γ1199020
be given by (35) Let 120582 = (1205821 1205822) isin Int (Γ
1199020
) and(1205821119899 1205822119899)infin
119899=1be a sequence in C2
+such that 120582
119895119899rarr 120582
119895
119895 isin 1 2 Then
119864
an
119909[119865 (1199091 1199092)] = lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865(1199091 1199092)]
(72)
where 119866119899is given by (52)
Our another result namely a change of scale formula forfunction space integrals now follows fromCorollary 16 above
Theorem 17 Let 119865 isin F11988611988711986011198602
and let 119890119899infin
119899=1be a complete
orthonormal set in (1198621015840119886119887[0 119879] sdot
119887) Then for any 120588
1gt 0 and
1205882gt 0
119864119909[119865 (12058811199091 12058821199092)]
= lim119899rarrinfin
120588minus119899
1120588minus119899
2
times 119864119909[119866119899(120588minus2
1 1199091)119866119899(120588minus2
2 1199092) 119865 (119909
1 1199092)]
(73)
where 119866119899is given by (52)
Proof Simply choose 120582119895= 120588minus2
119895for 119895 isin 1 2 and 120582
119895119899= 120588minus2
119895
for 119895 isin 1 2 and 119899 isin N in (72)
Remark 18 Of course if we choose 119886(119905) equiv 0 119887(119905) = 1199051198601= 119868 (identity operator) and 119860
2= 0 (zero operator) then
the function space 119862119886119887[0 119879] reduces to the classical Wiener
space 1198620[0 119879] and the generalized Fresnel type class F 119886119887
11986011198602
reduces to the Fresnel class F(1198620[0 119879]) It is known that
F(1198620[0 119879]) forms a Banach algebra over the complex field
In this case we have the relationships between the analyticFeynman integral and theWiener integral on classicalWienerspace as discussed in [14 15]
In recent paper [19] Yoo et al have studied a change ofscale formula for function space integral of the functionalsin the Banach algebra S(1198712
119886119887[0 119879]) the Banach algebra
S(1198712119886119887[0 119879]) is introduced in [12]
5 Functionals in F11988611988711986011198602
In this section we prove a theorem ensuring that variousfunctionals are inF119886119887
11986011198602
Theorem 19 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] Let (119884Y 120574) be a 120590-finite
measure space and let 120593119897 119884 rarr 119862
1015840
119886119887[0 119879] beYndashB(1198621015840
119886119887[0 119879])
measurable for 119897 isin 1 119889 Let 120579 119884 times R119889 rarr C be given by120579(120578 sdot) = ]
120578(sdot) where ]
120578isin M(R119889) for every 120578 isin 119884 and where
the family ]120578 120578 isin 119884 satisfies
(i) ]120578(119864) is a Y-measurable function of 120578 for every 119864 isin
B(R119889)(ii) ]
120578 isin 1198711
(119884Y 120574)
Under these hypothesis the functional 119865 1198622119886119887[0 119879] rarr C
given by
119865 (1199091 1199092) = int
119884
120579(120578
2
sum
119895=1
(11986012
1198951205931(120578) 119909
119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 119909
119895)
sim
)119889120574 (120578)
(74)
belongs toF11988611988711986011198602
and satisfies the inequality
119865 le int
119884
10038171003817100381710038171003817]120578
10038171003817100381710038171003817119889120574 (120578) (75)
Proof Using the techniques similar to those used in [20] wecan show that ]
120578 is measurable as a function of 120578 that 120579 is
Y-measurable and that the integrand in (74) is a measurablefunction of 120578 for every (119909
1 1199092) isin 1198622
119886119887[0 119879]
We define a measure 120591 onY timesB(R119889) by
120591 (119864) = int
119884
]120578(119864(120578)
) 119889120574 (120578) for 119864 isin Y timesB (R119889) (76)
Then by the first assertion of Theorem 31 in [17] 120591 satisfies120591 le int
119884
]120578119889120574(120578) Now let Φ 119884 times R119889 rarr 119862
1015840
119886119887[0 119879] be
defined by Φ(120578 V1 V
119889) = sum
119889
119897=1V119897120593119897(120578) Then Φ is Y times
B(R119889) ndashB(1198621015840119886119887[0 119879])-measurable on the hypothesis for 120593
119897
119897 isin 1 119889 Let 120590 = 120591 ∘Φminus1 Then clearly 120590 isinM(1198621015840119886119887[0 119879])
and satisfies 120590 le 120591From the change of variables theorem and the second
assertion of Theorem 31 in [17] it follows that for ae(1199091 1199092) isin 1198622
119886119887[0 119879] and for every 120588
1gt 0 and 120588
2gt 0
119865 (12058811199091 12058821199092)
= int
119884
]120578(
2
sum
119895=1
(11986012
1198951205931(120578) 120588
119895119909119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 120588
119895119909119895)
sim
)119889120574 (120578)
= int
119884
[
[
int
R119889exp
119894
119889
sum
119897=1
V119897
[
[
2
sum
119895=1
(11986012
119895120593119897(120578)
120588119895119909119895)
sim
]
]
119889]120578
times (V1 V
119889)]
]
119889120574 (120578)
= int
119884timesR119889exp
119894
119889
sum
119897=1
V119897
[
[
2
sum
119895=1
(11986012
119895120593119897(120578) 120588
119895119909119895)
sim
]
]
119889120591
times (120578 V1 V
119889)
Journal of Function Spaces and Applications 11
= int
119884timesR119889exp
2
sum
119895=1
119894(11986012
119895Φ(120578 V
1 V
119889) 120588119895119909119895)
sim
119889120591
times (120578 V1 V
119889)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119909119895)
sim
119889120591 ∘ Φminus1
(119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119909119895)
sim
119889120590 (119908)
(77)
Thus the functional 119865 given by (74) belongs to F11988611988711986011198602
andsatisfies the inequality
119865 = 120590 le 120591 le int
119884
10038171003817100381710038171003817]120578
10038171003817100381710038171003817119889120574 (120578) (78)
As mentioned in (2) of Remark 6 F 11988611988711986011198602
is a Banachalgebra if Ran(119860
1+ 1198602) is dense in 1198621015840
119886119887[0 119879] In this case
many analytic functionals of 119865 can be formed The followingcorollary is relevant to Feynman integration theories andquantum mechanics where exponential functions play animportant role
Corollary 20 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] such that Ran(119860
1+ 1198602)
is dense in 1198621015840119886119887[0 119879] Let 119865 be given by (74) with 120579 as in
Theorem 19 and let 120573 C rarr C be an entire function Then(120573 ∘ 119865)(119909
1 1199092) is in F119886119887
11986011198602
In particular exp119865(1199091 1199092) isin
F11988611988711986011198602
Corollary 21 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] and let 119892
1 119892
119889 be a
finite subset of 1198621015840119886119887[0 119879] Given 120573 = ] where ] isin M(R119889)
define 119865 1198622119886119887[0 119879] rarr C by
119865 (1199091 1199092) = 120573(
2
sum
119895=1
(11986012
1198951198921 119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 119909119895)
sim
)
(79)
Then 119865 is an element ofF11988611988711986011198602
Proof Let (119884Y 120574) be a probability space and for 119897 isin1 119889 let 120593
119897(120578) equiv 119892
119897 Take 120579(120578 sdot) = 120573(sdot) = ](sdot) Then for
all 1205881gt 0 and 120588
2gt 0 and for ae (119909
1 1199092) isin 1198622
119886119887[0 119879]
int
119884
120579(120578
2
sum
119895=1
(11986012
1198951205931(120578) 120588
119895119909119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 120588
119895119909119895)
sim
)119889120574 (120578)
= int
119884
120573(
2
sum
119895=1
(11986012
1198951198921 120588119895119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 120588119895119909119895)
sim
)119889120574 (120578)
= 120573(
2
sum
119895=1
(11986012
1198951198921 120588119895119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 120588119895119909119895)
sim
)
= 119865 (12058811199091 12058821199092)
(80)
Hence 119865 isin F11988611988711986011198602
Remark 22 Let 119889 = 1 and let (119884Y 120574) = ([0 119879]B([0 119879])119898119871) in Theorem 19 where 119898
119871denotes the Lebesgue measure
on [0 119879] ThenTheorems 46 47 and 49 in [18] follow fromthe results in this section by letting119860
1be the identity operator
and letting 1198602equiv 0 on 1198621015840
119886119887[0 119879] The function 120579 studied in
[18] (and mentioned in Remark 2 above) is interpreted as thepotential energy in quantum mechanics
Acknowledgments
The authors would like to express their gratitude to the refer-ees for their valuable comments and suggestions which haveimproved the original paper This research was supported bythe Basic Science Research Program through the NationalResearch Foundation of Korea (NRF) funded by theMinistryof Education (2011-0014552)
References
[1] G Kallianpur and C Bromley ldquoGeneralized Feynman integralsusing analytic continuation in several complex variablesrdquo inStochastic Analysis and Applications M A Pinsky Ed vol 7pp 217ndash267 Marcel Dekker New York NY USA 1984
[2] G Kallianpur D Kannan and R L Karandikar ldquoAnalytic andsequential Feynman integrals on abstract Wiener and Hilbertspaces and a Cameron-Martin formulardquo Annales de lrsquoInstitutHenri Poincare vol 21 no 4 pp 323ndash361 1985
[3] G W Johnson and M L Lapidus The Feynman Integral andFeynmanrsquos Operational Calculus Clarendon Press Oxford UK2000
[4] J Yeh ldquoSingularity of Gaussian measures on function spacesinduced by Brownian motion processes with non-stationaryincrementsrdquo Illinois Journal of Mathematics vol 15 pp 37ndash461971
[5] J Yeh Stochastic Processes and the Wiener Integral MarcelDekker New York NY USA 1973
[6] S J Chang J G Choi and D Skoug ldquoIntegration by partsformulas involving generalized Fourier-Feynman transformson function spacerdquo Transactions of the American MathematicalSociety vol 355 no 7 pp 2925ndash2948 2003
[7] S J Chang J G Choi and D Skoug ldquoParts formulas involvingconditional generalized Feynman integrals and conditionalgeneralized Fourier-Feynman transforms on function spacerdquo
12 Journal of Function Spaces and Applications
Integral Transforms and Special Functions vol 15 no 6 pp 491ndash512 2004
[8] S J Chang J G Choi and D Skoug ldquoEvaluation formulas forconditional function space integralsmdashIrdquo Stochastic Analysis andApplications vol 25 no 1 pp 141ndash168 2007
[9] S J Chang J G Choi and D Skoug ldquoGeneralized Fourier-Feynman transforms convolution products and first variationson function spacerdquoTheRockyMountain Journal ofMathematicsvol 40 no 3 pp 761ndash788 2010
[10] S J Chang and D M Chung ldquoConditional function spaceintegrals with applicationsrdquo The Rocky Mountain Journal ofMathematics vol 26 no 1 pp 37ndash62 1996
[11] S J Chang H S Chung and D Skoug ldquoIntegral transformsof functionals in 119871
2(119862119886119887[0 119879])rdquoThe Journal of Fourier Analysis
and Applications vol 15 no 4 pp 441ndash462 2009[12] S J Chang and D Skoug ldquoGeneralized Fourier-Feynman
transforms and a first variation on function spacerdquo IntegralTransforms and Special Functions vol 14 pp 375ndash393 2003
[13] J G Choi and S J Chang ldquoGeneralized Fourier-Feynmantransform and sequential transforms on function spacerdquo Jour-nal of the Korean Mathematical Society vol 49 pp 1065ndash10822012
[14] R H Cameron and D A Storvick ldquoRelationships between theWiener integral and the analytic Feynman integralrdquo Rendicontidel Circolo Matematico di Palermo no 17 supplement pp 117ndash133 1987
[15] RHCameron andDA Storvick ldquoChange of scale formulas forWiener integralrdquo Rendiconti del Circolo Matematico di Palermono 17 pp 105ndash115 1987
[16] G W Johnson ldquoThe equivalence of two approaches to theFeynman integralrdquo Journal of Mathematical Physics vol 23 pp2090ndash2096 1982
[17] G W Johnson and D L Skoug ldquoNotes on the FeynmanintegralmdashIII the Schroedinger equationrdquo Pacific Journal ofMathematics vol 105 pp 321ndash358 1983
[18] S J Chang J G Choi and S D Lee ldquoA Fresnel type class onfunction spacerdquo Journal of the Korean Society of MathematicalEducation Series B vol 16 no 1 pp 107ndash119 2009
[19] I Yoo B J Kim and B S Kim ldquoA change of scale formulafor a function space integral on 119862
119886119887[0 119879]rdquo Proceedings of the
American Mathematical Society vol 141 no 8 pp 2729ndash27392013
[20] K S Chang G W Johnson and D L Skoug ldquoFunctions inthe Fresnel classrdquo Proceedings of the American MathematicalSociety vol 100 no 2 pp 309ndash318 1987
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces and Applications 7
Proof Let Γ1199020
be given by (35) It was shown in the proofof Theorem 9 that 119879
120582(119865)(1199101 1199102) is an analytic function of 120582
throughout the domain Int(Γ1199020
) In viewofDefinition 4 it willsuffice to show that for each 120588
1gt 0 and 120588
2gt 0
lim120582rarrminus119894 119902
int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816
119879 120582(119865) (120588
11199101 12058821199102)
minus119879(119901)
119902(119865) (120588
11199101 12058821199102)
100381610038161003816100381610038161003816
1199011015840
times 119889 (120583 times 120583) (1199101 1199102) = 0
(45)
Fixing 119901 isin (1 2] and using the inequalities (37) and (39)we obtain that for all 120588
119895gt 0 119895 isin 1 2 and all 120582 isin Γ
1199020
100381610038161003816100381610038161003816
119879 120582(119865) (120588
11199101 12058821199102) minus 119879(119901)
119902(119865) (120588
11199101 12058821199102)
100381610038161003816100381610038161003816
1199011015840
le
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894120588119895(11986012
119895119908 119910119895)
sim
times[120595 (120582 119908) minus 120595 (minus119894 119902 119908)] 119889119891 (119908)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
1199011015840
le (int
1198621015840
119886119887[0119879]
[
10038161003816100381610038161003816120595 (120582 119908)
10038161003816100381610038161003816+
10038161003816100381610038161003816120595 (minus119894 119902 119908)
10038161003816100381610038161003816] 11988910038161003816100381610038161198911003816100381610038161003816(119908))
1199011015840
le (2int
1198621015840
119886119887[0119879]
119896 (1199020 119908) 119889
10038161003816100381610038161198911003816100381610038161003816(119908))
1199011015840
lt +infin
(46)
Hence by the dominated convergence theorem we see thatfor each 119901 isin (1 2] and each 120588
1gt 0 and 120588
2gt 0
lim120582rarrminus119894 119902
int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816
119879 120582(119865) (120588
11199101 12058821199102)
minus119879(119901)
119902(119865) (120588
11199101 12058821199102)
100381610038161003816100381610038161003816
1199011015840
119889 (120583 times 120583) (1199101 1199102)
= lim120582rarrminus119894 119902
int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119910119895)
sim
times 120595(120582 119908) 119889119891 (119908)
minus int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119910119895)
sim
times120595(minus119894 119902 119908) 119889119891 (119908)
10038161003816100381610038161003816100381610038161003816100381610038161003816
1199011015840
times 119889 (120583 times 120583) (1199101 1199102)
= int
1198622
119886119887[0119879]
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119910119895)
sim
times lim120582rarrminus119894 119902
[120595 (120582 119908)
minus120595 (minus119894 119902 119908)] 119889119891 (119908)
10038161003816100381610038161003816100381610038161003816100381610038161003816
1199011015840
times 119889 (120583 times 120583) (1199101 1199102)
= 0
(47)
which concludes the proof of Theorem 10
Remark 11 (1) In view of Theorems 9 and 10 we see thatfor each 119901 isin [1 2] the 119871
119901analytic GFFT of 119865 119879(119901)
119902(119865) is
given by the right hand side of (40) for 1199020 1199021 1199022 and 119865 as
in Theorem 9 and for s-ae (1199101 1199102) isin 1198622
119886119887[0 119879]
119879(119901)
119902(119865) (119910
1 1199102) = 119864
anf 119902119909[119865 (1199101+ 1199091 1199102+ 1199092)] 119901 isin [1 2]
(48)
In particular using this fact and (29) we have that for all 119901 isin[1 2]
119879(119901)
119902(119865) (0 0) = 119864
anf 119902119909[119865 (1199091 1199092)] (49)
(2) For nonzero real numbers 1199021and 119902
2with |119902
119895| gt 119902
0
119895 isin 1 2 define a set function 119891 119860119902B(1198621015840
119886119887[0 119879]) rarr C by
119891
119860
119902(119861) = int
119861
120595 (minus119894 119902 119908) 119889119891 (119908) 119861 isinB (1198621015840
119886119887[0 119879])
(50)
where 119891 and 119865 are related by (30) and B(1198621015840119886119887[0 119879]) is the
Borel 120590-algebra of 1198621015840119886119887[0 119879] Then it is obvious that 119891 119860
119902
belongs to M(1198621015840119886119887[0 119879]) and for all 119901 isin [1 2] 119879(119901)
119902(119865) can
be expressed as
119879(119901)
119902(119865) (119910
1 1199102) = int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
119889119891
119860
119902(119908)
(51)
for s-ae (1199101 1199102) isin 119862
2
119886119887[0 119879] Hence 119879(119901)
119902(119865) belongs to
F 11988611988711986011198602
for all 119901 isin [1 2]
4 Relationships between the GFFT and theFunction Space Integral of Functionals inF11988611988711986011198602
In this section we establish a relationship between the GFFTand the function space integral of functionals in the Fresneltype classF119886119887
11986011198602
8 Journal of Function Spaces and Applications
Throughout this section for convenience we use thefollowing notation for given 120582 isin C
+and 119899 = 1 2 let
119866119899(120582 119909)
= exp[1 minus 1205822
]
119899
sum
119896=1
[(119890119896 119909)sim
]
2
+ (12058212
minus 1)
119899
sum
119896=1
(119890119896 119886)1198621015840
119886119887
(119890119896 119909)sim
(52)
where 119890119899infin
119899=1is a complete orthonormal set in (1198621015840
119886119887[0 119879]
sdot 119887)To obtain our main results Theorems 14 and 17 below
we state a fundamental integration formula for the functionspace 119862
119886119887[0 119879]
Let 1198901 119890
119899 be an orthonormal set in (1198621015840
119886119887[0 119879] sdot
119887)
let 119896 R119899 rarr C be a Lebesgue measurable function and let119870 119862119886119887[0 119879] rarr C be given by
119870 (119909) = 119896 ((1198901 119909)sim
(119890119899 119909)sim
) (53)
Then
119864 [119870] = int
119862119886119887[0119879]
119896 ((1198901 119909)sim
(119890119899 119909)sim
) 119889120583 (119909)
= (2120587)minus1198992
int
R119899119896 (1199061 119906
119899)
times exp
minus
119899
sum
119895=1
[119906119895minus (119890119895 119886)119887
]
2
2
1198891199061 119889119906
119899
(54)
in the sense that if either side of (54) exists both sides existand equality holds
We also need the following lemma to obtain our maintheorem in this section
Lemma 12 Let 1198901 119890
119899 be an orthonormal subset of
(1198621015840
119886119887[0 119879] sdot
119887) and let 119908 be an element of 1198621015840
119886119887[0 119879] Then
for each 120582 isin C+ the function space integral
119864119909[119866119899(120582 119909) exp 119894(119908 119909)sim] (55)
exists and is given by the formula
119864119909[119866119899(120582 119909) exp 119894(119908 119909)sim]
= 120582minus1198992 exp
[
120582 minus 1
2120582
]
119899
sum
119896=1
(119890119896 119908)2
119887minus
1
2
1199082
119887
+ 119894120582minus12
119899
sum
119896=1
(119890119896 119886)119887(119890119896 119908)119887
+ 119894(119890119899+1 119886)119887[1199082
119887minus
119899
sum
119896=1
(119890119896 119908)2
119887]
12
(56)
where 119866119899is given by (52) above and
119890119899+1=[
[
1199082
119887minus
119899
sum
119895=1
(119890119895 119908)
2
119887
]
]
minus12
119908 minus
119899
sum
119895=1
(119890119895 119908)119887
119890119895
(57)
Proof (Outline) Using the Gram-Schmidt process for any119908 isin 119862
1015840
119886119887[0 119879] we can write 119908 = sum
119899+1
119896=1119888119896119890119896 where
1198901 119890
119899 119890119899+1 is an orthonormal set in (1198621015840
119886119887[0 119879] sdot
119887)
and
119888119896=
(119890119896 119908)119887 119896 = 1 119899
[1199082
119887minus
119899
sum
119895=1
(119890119895 119908)
2
119887
]
12
119896 = 119899 + 1
(58)
Then using (52) (54) the Fubini theorem and (14) it followsthat (56) holds for all 120582 isin C
+
The following remark will be very useful in the proof ofour main theorem in this section
Remark 13 Let 1199020be a positive real number and let Γ
1199020
begiven by (35) For real numbers 119902
1and 1199022with |119902
119895| gt 1199020 119895 isin
1 2 let 120582119899infin
119899=1= (1205821119899 1205822119899)infin
119899=1be a sequence in C2
+such
that
120582119899= (1205821119899 1205822119899) 997888rarr minus119894 119902 = (minus119894119902
1 minus1198941199022) (59)
Let 120582119895119899= 120572119895119899+ 119894120573119895119899
for 119895 isin 1 2 and 119899 isin N Then for119895 isin 1 2 Re(120582
119895119899) = 120572119895119899gt 0 and
120582minus1
119895119899= (120572119895119899+ 119894120573119895119899)
minus1
=
120572119895119899minus 119894120573119895119899
1205722
119895119899+ 1205732
119895119899
(60)
for each 119899 isin N Since |Im ((minus119894119902119895)minus12
)| = 1radic2|119902119895| lt 1radic2119902
0
for 119895 isin 1 2 there exists a sufficiently large 119871 isin N such thatfor any 119899 ge 119871 120582
1119899and 120582
2119899are in Int(Γ
1199020
) and
120575 (1199021 1199022) equiv sup ( 1003816100381610038161003816
1003816Im (120582minus12
1119899)
10038161003816100381610038161003816 119899 ge 119871
cup
10038161003816100381610038161003816Im (120582minus12
2119899)
10038161003816100381610038161003816 119899 ge 119871
cup
100381610038161003816100381610038161003816
Im ((minus1198941199021)minus12
)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
Im ((minus1198941199022)minus12
)
100381610038161003816100381610038161003816
)
lt
1
radic21199020
(61)
Thus there exists a positive real number 120576 gt 1 such that120575(1199021 1199022) lt 1(120576radic2119902
0)
Let 119890119899infin
119899=1be a complete orthonormal set in (1198621015840
119886119887[0 119879]
sdot 119887) Using Parsevalrsquos identity it follows that
(1198921 1198922)119887=
infin
sum
119899=1
(119890119899 1198921)119887(119890119899 1198922)119887
(62)
Journal of Function Spaces and Applications 9
for all 1198921 1198922isin 1198621015840
119886119887[0 119879] In addition for each 119892 isin 1198621015840
119886119887[0 119879]
10038171003817100381710038171198921003817100381710038171003817
2
119887minus
119899
sum
119896=1
(119890119896 119892)2
119887=
infin
sum
119896=119899+1
(119890119896 119892)2
119887ge 0 (63)
for every 119899 isin NSince
(119892 119886)119887=
infin
sum
119899=1
(119890119899 119892)119887(119890119899 119886)119887
(64)
and for 120576 gt 1
minus12057610038171003817100381710038171198921003817100381710038171003817119887119886119887lt minus10038171003817100381710038171198921003817100381710038171003817119887119886119887le (119892 119886)
119887
le10038171003817100381710038171198921003817100381710038171003817119887119886119887lt 12057610038171003817100381710038171198921003817100381710038171003817119887119886119887
(65)
there exists a sufficiently large119870119895isin N such that for any 119899 ge 119870
119895
1003816100381610038161003816100381610038161003816100381610038161003816
119899
sum
119896=1
(119890119896 11986012
119895119908)119887
(119890119896 119886)119887
1003816100381610038161003816100381610038161003816100381610038161003816
lt 120576
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817119887119886119887
(66)
for 119895 isin 1 2Using these and a long and tedious calculation we can
show that for every 119899 ge max119871 1198701 1198702
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
exp
2
sum
119895=1
([
120582119895119899minus 1
2120582119895119899
]
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
minus
1
2
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
+ 119894120582minus12
119895119899
119899
sum
119896=1
(119890119896 11986012
119895119908)119887
(119890119896 119886)119887+ 119894(119890119899+1 119886)119887
times [
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
minus
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
]
12
)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119896 (1199020 119908)
(67)
where 119896(1199020 119908) is given by (36)
In our next theorem for119865 isin F11988611988711986011198602
we express theGFFTof 119865 as the limit of a sequence of function space integrals on1198622
119886119887[0 119879]
Theorem 14 Let 1199020and 119865 be as in Theorem 10 Let 119890
119899infin
119899=1
be a complete orthonormal set in (1198621015840119886119887[0 119879] sdot
119887) and let
(1205821119899 1205822119899)infin
119899=1be a sequence in C2
+such that 120582
119895119899rarr minus119894119902
119895
where 119902119895is a real number with |119902
119895| gt 1199020 119895 isin 1 2 Then for
119901 isin [1 2] and for s-ae (1199101 1199102) isin 1198622
119886119887[0 119879]
119879(119901)
119902(119865) (119910
1 1199102)
= lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865 (119910
1+ 1199091 1199102+ 1199092)]
(68)
where 119866119899is given by (52)
Proof ByTheorems9 and 10we know that for each119901 isin [1 2]the 119871119901analytic GFFT of 119865 119879(119901)
119902(119865) exists and is given by the
right hand side of (40) Thus it suffices to show that
119879(1)
119902(119865) (119910
1 1199102) = 119864
anf 119902119909[119865 (1199101+ 1199091 1199102+ 1199092)]
= lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times 119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092)
times 119865 (1199101+ 1199091 1199102+ 1199092)] sdot
(69)
Using (30) the Fubini theorem and (56) with 120582 and 119908replaced with 120582
119895119899and 11986012
119895119908 119895 isin 1 2 respectively we see
that
1205821198992
11198991205821198992
2119899119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865 (119910
1+ 1199091 1199102+ 1199092)]
= 1205821198992
11198991205821198992
2119899int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
times(
2
prod
119895=1
119864119909119895
[119866119899(120582minus12
119895119899 119909119895)
times exp 119894(11986012119895119908 119909119895)
sim
])119889119891 (119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
(119894(11986012
119895119908 119910119895)
sim
+ [
120582119895119899minus 1
2120582119895119899
]
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
minus
1
2
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
+ 119894120582minus12
119895119899
119899
sum
119896=1
(119890119896 119886)119887(119890119896 11986012
119895119908)119887
+ 119894(119890119899+1 119886)119887[
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
minus
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
]
12
)
119889119891 (119908)
(70)
But by Remark 13 we see that the last expression of (70) isdominated by (39) on the region Γ
1199020
given by (35) for allbut a finite number of values of 119899 Next using the dominatedconvergence theorem Parsevalrsquos relation and (40) we obtainthe desired result
Corollary 15 Let 1199020 119865 119890
119899infin
119899=1 (1205821119899 1205822119899)infin
119899=1and (119902
1 1199022)
be as in Theorem 14 Then
119864
anf119902
119909[119865 (1199091 1199092)]
= lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times 119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865 (119909
1 1199092)]
(71)
where 119866119899is given by (52)
10 Journal of Function Spaces and Applications
Corollary 16 Let 1199020 119865 and 119890
119899infin
119899=1be as in Theorem 14 and
let Γ1199020
be given by (35) Let 120582 = (1205821 1205822) isin Int (Γ
1199020
) and(1205821119899 1205822119899)infin
119899=1be a sequence in C2
+such that 120582
119895119899rarr 120582
119895
119895 isin 1 2 Then
119864
an
119909[119865 (1199091 1199092)] = lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865(1199091 1199092)]
(72)
where 119866119899is given by (52)
Our another result namely a change of scale formula forfunction space integrals now follows fromCorollary 16 above
Theorem 17 Let 119865 isin F11988611988711986011198602
and let 119890119899infin
119899=1be a complete
orthonormal set in (1198621015840119886119887[0 119879] sdot
119887) Then for any 120588
1gt 0 and
1205882gt 0
119864119909[119865 (12058811199091 12058821199092)]
= lim119899rarrinfin
120588minus119899
1120588minus119899
2
times 119864119909[119866119899(120588minus2
1 1199091)119866119899(120588minus2
2 1199092) 119865 (119909
1 1199092)]
(73)
where 119866119899is given by (52)
Proof Simply choose 120582119895= 120588minus2
119895for 119895 isin 1 2 and 120582
119895119899= 120588minus2
119895
for 119895 isin 1 2 and 119899 isin N in (72)
Remark 18 Of course if we choose 119886(119905) equiv 0 119887(119905) = 1199051198601= 119868 (identity operator) and 119860
2= 0 (zero operator) then
the function space 119862119886119887[0 119879] reduces to the classical Wiener
space 1198620[0 119879] and the generalized Fresnel type class F 119886119887
11986011198602
reduces to the Fresnel class F(1198620[0 119879]) It is known that
F(1198620[0 119879]) forms a Banach algebra over the complex field
In this case we have the relationships between the analyticFeynman integral and theWiener integral on classicalWienerspace as discussed in [14 15]
In recent paper [19] Yoo et al have studied a change ofscale formula for function space integral of the functionalsin the Banach algebra S(1198712
119886119887[0 119879]) the Banach algebra
S(1198712119886119887[0 119879]) is introduced in [12]
5 Functionals in F11988611988711986011198602
In this section we prove a theorem ensuring that variousfunctionals are inF119886119887
11986011198602
Theorem 19 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] Let (119884Y 120574) be a 120590-finite
measure space and let 120593119897 119884 rarr 119862
1015840
119886119887[0 119879] beYndashB(1198621015840
119886119887[0 119879])
measurable for 119897 isin 1 119889 Let 120579 119884 times R119889 rarr C be given by120579(120578 sdot) = ]
120578(sdot) where ]
120578isin M(R119889) for every 120578 isin 119884 and where
the family ]120578 120578 isin 119884 satisfies
(i) ]120578(119864) is a Y-measurable function of 120578 for every 119864 isin
B(R119889)(ii) ]
120578 isin 1198711
(119884Y 120574)
Under these hypothesis the functional 119865 1198622119886119887[0 119879] rarr C
given by
119865 (1199091 1199092) = int
119884
120579(120578
2
sum
119895=1
(11986012
1198951205931(120578) 119909
119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 119909
119895)
sim
)119889120574 (120578)
(74)
belongs toF11988611988711986011198602
and satisfies the inequality
119865 le int
119884
10038171003817100381710038171003817]120578
10038171003817100381710038171003817119889120574 (120578) (75)
Proof Using the techniques similar to those used in [20] wecan show that ]
120578 is measurable as a function of 120578 that 120579 is
Y-measurable and that the integrand in (74) is a measurablefunction of 120578 for every (119909
1 1199092) isin 1198622
119886119887[0 119879]
We define a measure 120591 onY timesB(R119889) by
120591 (119864) = int
119884
]120578(119864(120578)
) 119889120574 (120578) for 119864 isin Y timesB (R119889) (76)
Then by the first assertion of Theorem 31 in [17] 120591 satisfies120591 le int
119884
]120578119889120574(120578) Now let Φ 119884 times R119889 rarr 119862
1015840
119886119887[0 119879] be
defined by Φ(120578 V1 V
119889) = sum
119889
119897=1V119897120593119897(120578) Then Φ is Y times
B(R119889) ndashB(1198621015840119886119887[0 119879])-measurable on the hypothesis for 120593
119897
119897 isin 1 119889 Let 120590 = 120591 ∘Φminus1 Then clearly 120590 isinM(1198621015840119886119887[0 119879])
and satisfies 120590 le 120591From the change of variables theorem and the second
assertion of Theorem 31 in [17] it follows that for ae(1199091 1199092) isin 1198622
119886119887[0 119879] and for every 120588
1gt 0 and 120588
2gt 0
119865 (12058811199091 12058821199092)
= int
119884
]120578(
2
sum
119895=1
(11986012
1198951205931(120578) 120588
119895119909119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 120588
119895119909119895)
sim
)119889120574 (120578)
= int
119884
[
[
int
R119889exp
119894
119889
sum
119897=1
V119897
[
[
2
sum
119895=1
(11986012
119895120593119897(120578)
120588119895119909119895)
sim
]
]
119889]120578
times (V1 V
119889)]
]
119889120574 (120578)
= int
119884timesR119889exp
119894
119889
sum
119897=1
V119897
[
[
2
sum
119895=1
(11986012
119895120593119897(120578) 120588
119895119909119895)
sim
]
]
119889120591
times (120578 V1 V
119889)
Journal of Function Spaces and Applications 11
= int
119884timesR119889exp
2
sum
119895=1
119894(11986012
119895Φ(120578 V
1 V
119889) 120588119895119909119895)
sim
119889120591
times (120578 V1 V
119889)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119909119895)
sim
119889120591 ∘ Φminus1
(119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119909119895)
sim
119889120590 (119908)
(77)
Thus the functional 119865 given by (74) belongs to F11988611988711986011198602
andsatisfies the inequality
119865 = 120590 le 120591 le int
119884
10038171003817100381710038171003817]120578
10038171003817100381710038171003817119889120574 (120578) (78)
As mentioned in (2) of Remark 6 F 11988611988711986011198602
is a Banachalgebra if Ran(119860
1+ 1198602) is dense in 1198621015840
119886119887[0 119879] In this case
many analytic functionals of 119865 can be formed The followingcorollary is relevant to Feynman integration theories andquantum mechanics where exponential functions play animportant role
Corollary 20 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] such that Ran(119860
1+ 1198602)
is dense in 1198621015840119886119887[0 119879] Let 119865 be given by (74) with 120579 as in
Theorem 19 and let 120573 C rarr C be an entire function Then(120573 ∘ 119865)(119909
1 1199092) is in F119886119887
11986011198602
In particular exp119865(1199091 1199092) isin
F11988611988711986011198602
Corollary 21 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] and let 119892
1 119892
119889 be a
finite subset of 1198621015840119886119887[0 119879] Given 120573 = ] where ] isin M(R119889)
define 119865 1198622119886119887[0 119879] rarr C by
119865 (1199091 1199092) = 120573(
2
sum
119895=1
(11986012
1198951198921 119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 119909119895)
sim
)
(79)
Then 119865 is an element ofF11988611988711986011198602
Proof Let (119884Y 120574) be a probability space and for 119897 isin1 119889 let 120593
119897(120578) equiv 119892
119897 Take 120579(120578 sdot) = 120573(sdot) = ](sdot) Then for
all 1205881gt 0 and 120588
2gt 0 and for ae (119909
1 1199092) isin 1198622
119886119887[0 119879]
int
119884
120579(120578
2
sum
119895=1
(11986012
1198951205931(120578) 120588
119895119909119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 120588
119895119909119895)
sim
)119889120574 (120578)
= int
119884
120573(
2
sum
119895=1
(11986012
1198951198921 120588119895119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 120588119895119909119895)
sim
)119889120574 (120578)
= 120573(
2
sum
119895=1
(11986012
1198951198921 120588119895119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 120588119895119909119895)
sim
)
= 119865 (12058811199091 12058821199092)
(80)
Hence 119865 isin F11988611988711986011198602
Remark 22 Let 119889 = 1 and let (119884Y 120574) = ([0 119879]B([0 119879])119898119871) in Theorem 19 where 119898
119871denotes the Lebesgue measure
on [0 119879] ThenTheorems 46 47 and 49 in [18] follow fromthe results in this section by letting119860
1be the identity operator
and letting 1198602equiv 0 on 1198621015840
119886119887[0 119879] The function 120579 studied in
[18] (and mentioned in Remark 2 above) is interpreted as thepotential energy in quantum mechanics
Acknowledgments
The authors would like to express their gratitude to the refer-ees for their valuable comments and suggestions which haveimproved the original paper This research was supported bythe Basic Science Research Program through the NationalResearch Foundation of Korea (NRF) funded by theMinistryof Education (2011-0014552)
References
[1] G Kallianpur and C Bromley ldquoGeneralized Feynman integralsusing analytic continuation in several complex variablesrdquo inStochastic Analysis and Applications M A Pinsky Ed vol 7pp 217ndash267 Marcel Dekker New York NY USA 1984
[2] G Kallianpur D Kannan and R L Karandikar ldquoAnalytic andsequential Feynman integrals on abstract Wiener and Hilbertspaces and a Cameron-Martin formulardquo Annales de lrsquoInstitutHenri Poincare vol 21 no 4 pp 323ndash361 1985
[3] G W Johnson and M L Lapidus The Feynman Integral andFeynmanrsquos Operational Calculus Clarendon Press Oxford UK2000
[4] J Yeh ldquoSingularity of Gaussian measures on function spacesinduced by Brownian motion processes with non-stationaryincrementsrdquo Illinois Journal of Mathematics vol 15 pp 37ndash461971
[5] J Yeh Stochastic Processes and the Wiener Integral MarcelDekker New York NY USA 1973
[6] S J Chang J G Choi and D Skoug ldquoIntegration by partsformulas involving generalized Fourier-Feynman transformson function spacerdquo Transactions of the American MathematicalSociety vol 355 no 7 pp 2925ndash2948 2003
[7] S J Chang J G Choi and D Skoug ldquoParts formulas involvingconditional generalized Feynman integrals and conditionalgeneralized Fourier-Feynman transforms on function spacerdquo
12 Journal of Function Spaces and Applications
Integral Transforms and Special Functions vol 15 no 6 pp 491ndash512 2004
[8] S J Chang J G Choi and D Skoug ldquoEvaluation formulas forconditional function space integralsmdashIrdquo Stochastic Analysis andApplications vol 25 no 1 pp 141ndash168 2007
[9] S J Chang J G Choi and D Skoug ldquoGeneralized Fourier-Feynman transforms convolution products and first variationson function spacerdquoTheRockyMountain Journal ofMathematicsvol 40 no 3 pp 761ndash788 2010
[10] S J Chang and D M Chung ldquoConditional function spaceintegrals with applicationsrdquo The Rocky Mountain Journal ofMathematics vol 26 no 1 pp 37ndash62 1996
[11] S J Chang H S Chung and D Skoug ldquoIntegral transformsof functionals in 119871
2(119862119886119887[0 119879])rdquoThe Journal of Fourier Analysis
and Applications vol 15 no 4 pp 441ndash462 2009[12] S J Chang and D Skoug ldquoGeneralized Fourier-Feynman
transforms and a first variation on function spacerdquo IntegralTransforms and Special Functions vol 14 pp 375ndash393 2003
[13] J G Choi and S J Chang ldquoGeneralized Fourier-Feynmantransform and sequential transforms on function spacerdquo Jour-nal of the Korean Mathematical Society vol 49 pp 1065ndash10822012
[14] R H Cameron and D A Storvick ldquoRelationships between theWiener integral and the analytic Feynman integralrdquo Rendicontidel Circolo Matematico di Palermo no 17 supplement pp 117ndash133 1987
[15] RHCameron andDA Storvick ldquoChange of scale formulas forWiener integralrdquo Rendiconti del Circolo Matematico di Palermono 17 pp 105ndash115 1987
[16] G W Johnson ldquoThe equivalence of two approaches to theFeynman integralrdquo Journal of Mathematical Physics vol 23 pp2090ndash2096 1982
[17] G W Johnson and D L Skoug ldquoNotes on the FeynmanintegralmdashIII the Schroedinger equationrdquo Pacific Journal ofMathematics vol 105 pp 321ndash358 1983
[18] S J Chang J G Choi and S D Lee ldquoA Fresnel type class onfunction spacerdquo Journal of the Korean Society of MathematicalEducation Series B vol 16 no 1 pp 107ndash119 2009
[19] I Yoo B J Kim and B S Kim ldquoA change of scale formulafor a function space integral on 119862
119886119887[0 119879]rdquo Proceedings of the
American Mathematical Society vol 141 no 8 pp 2729ndash27392013
[20] K S Chang G W Johnson and D L Skoug ldquoFunctions inthe Fresnel classrdquo Proceedings of the American MathematicalSociety vol 100 no 2 pp 309ndash318 1987
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Journal of Function Spaces and Applications
Throughout this section for convenience we use thefollowing notation for given 120582 isin C
+and 119899 = 1 2 let
119866119899(120582 119909)
= exp[1 minus 1205822
]
119899
sum
119896=1
[(119890119896 119909)sim
]
2
+ (12058212
minus 1)
119899
sum
119896=1
(119890119896 119886)1198621015840
119886119887
(119890119896 119909)sim
(52)
where 119890119899infin
119899=1is a complete orthonormal set in (1198621015840
119886119887[0 119879]
sdot 119887)To obtain our main results Theorems 14 and 17 below
we state a fundamental integration formula for the functionspace 119862
119886119887[0 119879]
Let 1198901 119890
119899 be an orthonormal set in (1198621015840
119886119887[0 119879] sdot
119887)
let 119896 R119899 rarr C be a Lebesgue measurable function and let119870 119862119886119887[0 119879] rarr C be given by
119870 (119909) = 119896 ((1198901 119909)sim
(119890119899 119909)sim
) (53)
Then
119864 [119870] = int
119862119886119887[0119879]
119896 ((1198901 119909)sim
(119890119899 119909)sim
) 119889120583 (119909)
= (2120587)minus1198992
int
R119899119896 (1199061 119906
119899)
times exp
minus
119899
sum
119895=1
[119906119895minus (119890119895 119886)119887
]
2
2
1198891199061 119889119906
119899
(54)
in the sense that if either side of (54) exists both sides existand equality holds
We also need the following lemma to obtain our maintheorem in this section
Lemma 12 Let 1198901 119890
119899 be an orthonormal subset of
(1198621015840
119886119887[0 119879] sdot
119887) and let 119908 be an element of 1198621015840
119886119887[0 119879] Then
for each 120582 isin C+ the function space integral
119864119909[119866119899(120582 119909) exp 119894(119908 119909)sim] (55)
exists and is given by the formula
119864119909[119866119899(120582 119909) exp 119894(119908 119909)sim]
= 120582minus1198992 exp
[
120582 minus 1
2120582
]
119899
sum
119896=1
(119890119896 119908)2
119887minus
1
2
1199082
119887
+ 119894120582minus12
119899
sum
119896=1
(119890119896 119886)119887(119890119896 119908)119887
+ 119894(119890119899+1 119886)119887[1199082
119887minus
119899
sum
119896=1
(119890119896 119908)2
119887]
12
(56)
where 119866119899is given by (52) above and
119890119899+1=[
[
1199082
119887minus
119899
sum
119895=1
(119890119895 119908)
2
119887
]
]
minus12
119908 minus
119899
sum
119895=1
(119890119895 119908)119887
119890119895
(57)
Proof (Outline) Using the Gram-Schmidt process for any119908 isin 119862
1015840
119886119887[0 119879] we can write 119908 = sum
119899+1
119896=1119888119896119890119896 where
1198901 119890
119899 119890119899+1 is an orthonormal set in (1198621015840
119886119887[0 119879] sdot
119887)
and
119888119896=
(119890119896 119908)119887 119896 = 1 119899
[1199082
119887minus
119899
sum
119895=1
(119890119895 119908)
2
119887
]
12
119896 = 119899 + 1
(58)
Then using (52) (54) the Fubini theorem and (14) it followsthat (56) holds for all 120582 isin C
+
The following remark will be very useful in the proof ofour main theorem in this section
Remark 13 Let 1199020be a positive real number and let Γ
1199020
begiven by (35) For real numbers 119902
1and 1199022with |119902
119895| gt 1199020 119895 isin
1 2 let 120582119899infin
119899=1= (1205821119899 1205822119899)infin
119899=1be a sequence in C2
+such
that
120582119899= (1205821119899 1205822119899) 997888rarr minus119894 119902 = (minus119894119902
1 minus1198941199022) (59)
Let 120582119895119899= 120572119895119899+ 119894120573119895119899
for 119895 isin 1 2 and 119899 isin N Then for119895 isin 1 2 Re(120582
119895119899) = 120572119895119899gt 0 and
120582minus1
119895119899= (120572119895119899+ 119894120573119895119899)
minus1
=
120572119895119899minus 119894120573119895119899
1205722
119895119899+ 1205732
119895119899
(60)
for each 119899 isin N Since |Im ((minus119894119902119895)minus12
)| = 1radic2|119902119895| lt 1radic2119902
0
for 119895 isin 1 2 there exists a sufficiently large 119871 isin N such thatfor any 119899 ge 119871 120582
1119899and 120582
2119899are in Int(Γ
1199020
) and
120575 (1199021 1199022) equiv sup ( 1003816100381610038161003816
1003816Im (120582minus12
1119899)
10038161003816100381610038161003816 119899 ge 119871
cup
10038161003816100381610038161003816Im (120582minus12
2119899)
10038161003816100381610038161003816 119899 ge 119871
cup
100381610038161003816100381610038161003816
Im ((minus1198941199021)minus12
)
100381610038161003816100381610038161003816
100381610038161003816100381610038161003816
Im ((minus1198941199022)minus12
)
100381610038161003816100381610038161003816
)
lt
1
radic21199020
(61)
Thus there exists a positive real number 120576 gt 1 such that120575(1199021 1199022) lt 1(120576radic2119902
0)
Let 119890119899infin
119899=1be a complete orthonormal set in (1198621015840
119886119887[0 119879]
sdot 119887) Using Parsevalrsquos identity it follows that
(1198921 1198922)119887=
infin
sum
119899=1
(119890119899 1198921)119887(119890119899 1198922)119887
(62)
Journal of Function Spaces and Applications 9
for all 1198921 1198922isin 1198621015840
119886119887[0 119879] In addition for each 119892 isin 1198621015840
119886119887[0 119879]
10038171003817100381710038171198921003817100381710038171003817
2
119887minus
119899
sum
119896=1
(119890119896 119892)2
119887=
infin
sum
119896=119899+1
(119890119896 119892)2
119887ge 0 (63)
for every 119899 isin NSince
(119892 119886)119887=
infin
sum
119899=1
(119890119899 119892)119887(119890119899 119886)119887
(64)
and for 120576 gt 1
minus12057610038171003817100381710038171198921003817100381710038171003817119887119886119887lt minus10038171003817100381710038171198921003817100381710038171003817119887119886119887le (119892 119886)
119887
le10038171003817100381710038171198921003817100381710038171003817119887119886119887lt 12057610038171003817100381710038171198921003817100381710038171003817119887119886119887
(65)
there exists a sufficiently large119870119895isin N such that for any 119899 ge 119870
119895
1003816100381610038161003816100381610038161003816100381610038161003816
119899
sum
119896=1
(119890119896 11986012
119895119908)119887
(119890119896 119886)119887
1003816100381610038161003816100381610038161003816100381610038161003816
lt 120576
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817119887119886119887
(66)
for 119895 isin 1 2Using these and a long and tedious calculation we can
show that for every 119899 ge max119871 1198701 1198702
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
exp
2
sum
119895=1
([
120582119895119899minus 1
2120582119895119899
]
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
minus
1
2
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
+ 119894120582minus12
119895119899
119899
sum
119896=1
(119890119896 11986012
119895119908)119887
(119890119896 119886)119887+ 119894(119890119899+1 119886)119887
times [
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
minus
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
]
12
)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119896 (1199020 119908)
(67)
where 119896(1199020 119908) is given by (36)
In our next theorem for119865 isin F11988611988711986011198602
we express theGFFTof 119865 as the limit of a sequence of function space integrals on1198622
119886119887[0 119879]
Theorem 14 Let 1199020and 119865 be as in Theorem 10 Let 119890
119899infin
119899=1
be a complete orthonormal set in (1198621015840119886119887[0 119879] sdot
119887) and let
(1205821119899 1205822119899)infin
119899=1be a sequence in C2
+such that 120582
119895119899rarr minus119894119902
119895
where 119902119895is a real number with |119902
119895| gt 1199020 119895 isin 1 2 Then for
119901 isin [1 2] and for s-ae (1199101 1199102) isin 1198622
119886119887[0 119879]
119879(119901)
119902(119865) (119910
1 1199102)
= lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865 (119910
1+ 1199091 1199102+ 1199092)]
(68)
where 119866119899is given by (52)
Proof ByTheorems9 and 10we know that for each119901 isin [1 2]the 119871119901analytic GFFT of 119865 119879(119901)
119902(119865) exists and is given by the
right hand side of (40) Thus it suffices to show that
119879(1)
119902(119865) (119910
1 1199102) = 119864
anf 119902119909[119865 (1199101+ 1199091 1199102+ 1199092)]
= lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times 119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092)
times 119865 (1199101+ 1199091 1199102+ 1199092)] sdot
(69)
Using (30) the Fubini theorem and (56) with 120582 and 119908replaced with 120582
119895119899and 11986012
119895119908 119895 isin 1 2 respectively we see
that
1205821198992
11198991205821198992
2119899119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865 (119910
1+ 1199091 1199102+ 1199092)]
= 1205821198992
11198991205821198992
2119899int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
times(
2
prod
119895=1
119864119909119895
[119866119899(120582minus12
119895119899 119909119895)
times exp 119894(11986012119895119908 119909119895)
sim
])119889119891 (119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
(119894(11986012
119895119908 119910119895)
sim
+ [
120582119895119899minus 1
2120582119895119899
]
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
minus
1
2
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
+ 119894120582minus12
119895119899
119899
sum
119896=1
(119890119896 119886)119887(119890119896 11986012
119895119908)119887
+ 119894(119890119899+1 119886)119887[
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
minus
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
]
12
)
119889119891 (119908)
(70)
But by Remark 13 we see that the last expression of (70) isdominated by (39) on the region Γ
1199020
given by (35) for allbut a finite number of values of 119899 Next using the dominatedconvergence theorem Parsevalrsquos relation and (40) we obtainthe desired result
Corollary 15 Let 1199020 119865 119890
119899infin
119899=1 (1205821119899 1205822119899)infin
119899=1and (119902
1 1199022)
be as in Theorem 14 Then
119864
anf119902
119909[119865 (1199091 1199092)]
= lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times 119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865 (119909
1 1199092)]
(71)
where 119866119899is given by (52)
10 Journal of Function Spaces and Applications
Corollary 16 Let 1199020 119865 and 119890
119899infin
119899=1be as in Theorem 14 and
let Γ1199020
be given by (35) Let 120582 = (1205821 1205822) isin Int (Γ
1199020
) and(1205821119899 1205822119899)infin
119899=1be a sequence in C2
+such that 120582
119895119899rarr 120582
119895
119895 isin 1 2 Then
119864
an
119909[119865 (1199091 1199092)] = lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865(1199091 1199092)]
(72)
where 119866119899is given by (52)
Our another result namely a change of scale formula forfunction space integrals now follows fromCorollary 16 above
Theorem 17 Let 119865 isin F11988611988711986011198602
and let 119890119899infin
119899=1be a complete
orthonormal set in (1198621015840119886119887[0 119879] sdot
119887) Then for any 120588
1gt 0 and
1205882gt 0
119864119909[119865 (12058811199091 12058821199092)]
= lim119899rarrinfin
120588minus119899
1120588minus119899
2
times 119864119909[119866119899(120588minus2
1 1199091)119866119899(120588minus2
2 1199092) 119865 (119909
1 1199092)]
(73)
where 119866119899is given by (52)
Proof Simply choose 120582119895= 120588minus2
119895for 119895 isin 1 2 and 120582
119895119899= 120588minus2
119895
for 119895 isin 1 2 and 119899 isin N in (72)
Remark 18 Of course if we choose 119886(119905) equiv 0 119887(119905) = 1199051198601= 119868 (identity operator) and 119860
2= 0 (zero operator) then
the function space 119862119886119887[0 119879] reduces to the classical Wiener
space 1198620[0 119879] and the generalized Fresnel type class F 119886119887
11986011198602
reduces to the Fresnel class F(1198620[0 119879]) It is known that
F(1198620[0 119879]) forms a Banach algebra over the complex field
In this case we have the relationships between the analyticFeynman integral and theWiener integral on classicalWienerspace as discussed in [14 15]
In recent paper [19] Yoo et al have studied a change ofscale formula for function space integral of the functionalsin the Banach algebra S(1198712
119886119887[0 119879]) the Banach algebra
S(1198712119886119887[0 119879]) is introduced in [12]
5 Functionals in F11988611988711986011198602
In this section we prove a theorem ensuring that variousfunctionals are inF119886119887
11986011198602
Theorem 19 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] Let (119884Y 120574) be a 120590-finite
measure space and let 120593119897 119884 rarr 119862
1015840
119886119887[0 119879] beYndashB(1198621015840
119886119887[0 119879])
measurable for 119897 isin 1 119889 Let 120579 119884 times R119889 rarr C be given by120579(120578 sdot) = ]
120578(sdot) where ]
120578isin M(R119889) for every 120578 isin 119884 and where
the family ]120578 120578 isin 119884 satisfies
(i) ]120578(119864) is a Y-measurable function of 120578 for every 119864 isin
B(R119889)(ii) ]
120578 isin 1198711
(119884Y 120574)
Under these hypothesis the functional 119865 1198622119886119887[0 119879] rarr C
given by
119865 (1199091 1199092) = int
119884
120579(120578
2
sum
119895=1
(11986012
1198951205931(120578) 119909
119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 119909
119895)
sim
)119889120574 (120578)
(74)
belongs toF11988611988711986011198602
and satisfies the inequality
119865 le int
119884
10038171003817100381710038171003817]120578
10038171003817100381710038171003817119889120574 (120578) (75)
Proof Using the techniques similar to those used in [20] wecan show that ]
120578 is measurable as a function of 120578 that 120579 is
Y-measurable and that the integrand in (74) is a measurablefunction of 120578 for every (119909
1 1199092) isin 1198622
119886119887[0 119879]
We define a measure 120591 onY timesB(R119889) by
120591 (119864) = int
119884
]120578(119864(120578)
) 119889120574 (120578) for 119864 isin Y timesB (R119889) (76)
Then by the first assertion of Theorem 31 in [17] 120591 satisfies120591 le int
119884
]120578119889120574(120578) Now let Φ 119884 times R119889 rarr 119862
1015840
119886119887[0 119879] be
defined by Φ(120578 V1 V
119889) = sum
119889
119897=1V119897120593119897(120578) Then Φ is Y times
B(R119889) ndashB(1198621015840119886119887[0 119879])-measurable on the hypothesis for 120593
119897
119897 isin 1 119889 Let 120590 = 120591 ∘Φminus1 Then clearly 120590 isinM(1198621015840119886119887[0 119879])
and satisfies 120590 le 120591From the change of variables theorem and the second
assertion of Theorem 31 in [17] it follows that for ae(1199091 1199092) isin 1198622
119886119887[0 119879] and for every 120588
1gt 0 and 120588
2gt 0
119865 (12058811199091 12058821199092)
= int
119884
]120578(
2
sum
119895=1
(11986012
1198951205931(120578) 120588
119895119909119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 120588
119895119909119895)
sim
)119889120574 (120578)
= int
119884
[
[
int
R119889exp
119894
119889
sum
119897=1
V119897
[
[
2
sum
119895=1
(11986012
119895120593119897(120578)
120588119895119909119895)
sim
]
]
119889]120578
times (V1 V
119889)]
]
119889120574 (120578)
= int
119884timesR119889exp
119894
119889
sum
119897=1
V119897
[
[
2
sum
119895=1
(11986012
119895120593119897(120578) 120588
119895119909119895)
sim
]
]
119889120591
times (120578 V1 V
119889)
Journal of Function Spaces and Applications 11
= int
119884timesR119889exp
2
sum
119895=1
119894(11986012
119895Φ(120578 V
1 V
119889) 120588119895119909119895)
sim
119889120591
times (120578 V1 V
119889)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119909119895)
sim
119889120591 ∘ Φminus1
(119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119909119895)
sim
119889120590 (119908)
(77)
Thus the functional 119865 given by (74) belongs to F11988611988711986011198602
andsatisfies the inequality
119865 = 120590 le 120591 le int
119884
10038171003817100381710038171003817]120578
10038171003817100381710038171003817119889120574 (120578) (78)
As mentioned in (2) of Remark 6 F 11988611988711986011198602
is a Banachalgebra if Ran(119860
1+ 1198602) is dense in 1198621015840
119886119887[0 119879] In this case
many analytic functionals of 119865 can be formed The followingcorollary is relevant to Feynman integration theories andquantum mechanics where exponential functions play animportant role
Corollary 20 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] such that Ran(119860
1+ 1198602)
is dense in 1198621015840119886119887[0 119879] Let 119865 be given by (74) with 120579 as in
Theorem 19 and let 120573 C rarr C be an entire function Then(120573 ∘ 119865)(119909
1 1199092) is in F119886119887
11986011198602
In particular exp119865(1199091 1199092) isin
F11988611988711986011198602
Corollary 21 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] and let 119892
1 119892
119889 be a
finite subset of 1198621015840119886119887[0 119879] Given 120573 = ] where ] isin M(R119889)
define 119865 1198622119886119887[0 119879] rarr C by
119865 (1199091 1199092) = 120573(
2
sum
119895=1
(11986012
1198951198921 119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 119909119895)
sim
)
(79)
Then 119865 is an element ofF11988611988711986011198602
Proof Let (119884Y 120574) be a probability space and for 119897 isin1 119889 let 120593
119897(120578) equiv 119892
119897 Take 120579(120578 sdot) = 120573(sdot) = ](sdot) Then for
all 1205881gt 0 and 120588
2gt 0 and for ae (119909
1 1199092) isin 1198622
119886119887[0 119879]
int
119884
120579(120578
2
sum
119895=1
(11986012
1198951205931(120578) 120588
119895119909119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 120588
119895119909119895)
sim
)119889120574 (120578)
= int
119884
120573(
2
sum
119895=1
(11986012
1198951198921 120588119895119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 120588119895119909119895)
sim
)119889120574 (120578)
= 120573(
2
sum
119895=1
(11986012
1198951198921 120588119895119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 120588119895119909119895)
sim
)
= 119865 (12058811199091 12058821199092)
(80)
Hence 119865 isin F11988611988711986011198602
Remark 22 Let 119889 = 1 and let (119884Y 120574) = ([0 119879]B([0 119879])119898119871) in Theorem 19 where 119898
119871denotes the Lebesgue measure
on [0 119879] ThenTheorems 46 47 and 49 in [18] follow fromthe results in this section by letting119860
1be the identity operator
and letting 1198602equiv 0 on 1198621015840
119886119887[0 119879] The function 120579 studied in
[18] (and mentioned in Remark 2 above) is interpreted as thepotential energy in quantum mechanics
Acknowledgments
The authors would like to express their gratitude to the refer-ees for their valuable comments and suggestions which haveimproved the original paper This research was supported bythe Basic Science Research Program through the NationalResearch Foundation of Korea (NRF) funded by theMinistryof Education (2011-0014552)
References
[1] G Kallianpur and C Bromley ldquoGeneralized Feynman integralsusing analytic continuation in several complex variablesrdquo inStochastic Analysis and Applications M A Pinsky Ed vol 7pp 217ndash267 Marcel Dekker New York NY USA 1984
[2] G Kallianpur D Kannan and R L Karandikar ldquoAnalytic andsequential Feynman integrals on abstract Wiener and Hilbertspaces and a Cameron-Martin formulardquo Annales de lrsquoInstitutHenri Poincare vol 21 no 4 pp 323ndash361 1985
[3] G W Johnson and M L Lapidus The Feynman Integral andFeynmanrsquos Operational Calculus Clarendon Press Oxford UK2000
[4] J Yeh ldquoSingularity of Gaussian measures on function spacesinduced by Brownian motion processes with non-stationaryincrementsrdquo Illinois Journal of Mathematics vol 15 pp 37ndash461971
[5] J Yeh Stochastic Processes and the Wiener Integral MarcelDekker New York NY USA 1973
[6] S J Chang J G Choi and D Skoug ldquoIntegration by partsformulas involving generalized Fourier-Feynman transformson function spacerdquo Transactions of the American MathematicalSociety vol 355 no 7 pp 2925ndash2948 2003
[7] S J Chang J G Choi and D Skoug ldquoParts formulas involvingconditional generalized Feynman integrals and conditionalgeneralized Fourier-Feynman transforms on function spacerdquo
12 Journal of Function Spaces and Applications
Integral Transforms and Special Functions vol 15 no 6 pp 491ndash512 2004
[8] S J Chang J G Choi and D Skoug ldquoEvaluation formulas forconditional function space integralsmdashIrdquo Stochastic Analysis andApplications vol 25 no 1 pp 141ndash168 2007
[9] S J Chang J G Choi and D Skoug ldquoGeneralized Fourier-Feynman transforms convolution products and first variationson function spacerdquoTheRockyMountain Journal ofMathematicsvol 40 no 3 pp 761ndash788 2010
[10] S J Chang and D M Chung ldquoConditional function spaceintegrals with applicationsrdquo The Rocky Mountain Journal ofMathematics vol 26 no 1 pp 37ndash62 1996
[11] S J Chang H S Chung and D Skoug ldquoIntegral transformsof functionals in 119871
2(119862119886119887[0 119879])rdquoThe Journal of Fourier Analysis
and Applications vol 15 no 4 pp 441ndash462 2009[12] S J Chang and D Skoug ldquoGeneralized Fourier-Feynman
transforms and a first variation on function spacerdquo IntegralTransforms and Special Functions vol 14 pp 375ndash393 2003
[13] J G Choi and S J Chang ldquoGeneralized Fourier-Feynmantransform and sequential transforms on function spacerdquo Jour-nal of the Korean Mathematical Society vol 49 pp 1065ndash10822012
[14] R H Cameron and D A Storvick ldquoRelationships between theWiener integral and the analytic Feynman integralrdquo Rendicontidel Circolo Matematico di Palermo no 17 supplement pp 117ndash133 1987
[15] RHCameron andDA Storvick ldquoChange of scale formulas forWiener integralrdquo Rendiconti del Circolo Matematico di Palermono 17 pp 105ndash115 1987
[16] G W Johnson ldquoThe equivalence of two approaches to theFeynman integralrdquo Journal of Mathematical Physics vol 23 pp2090ndash2096 1982
[17] G W Johnson and D L Skoug ldquoNotes on the FeynmanintegralmdashIII the Schroedinger equationrdquo Pacific Journal ofMathematics vol 105 pp 321ndash358 1983
[18] S J Chang J G Choi and S D Lee ldquoA Fresnel type class onfunction spacerdquo Journal of the Korean Society of MathematicalEducation Series B vol 16 no 1 pp 107ndash119 2009
[19] I Yoo B J Kim and B S Kim ldquoA change of scale formulafor a function space integral on 119862
119886119887[0 119879]rdquo Proceedings of the
American Mathematical Society vol 141 no 8 pp 2729ndash27392013
[20] K S Chang G W Johnson and D L Skoug ldquoFunctions inthe Fresnel classrdquo Proceedings of the American MathematicalSociety vol 100 no 2 pp 309ndash318 1987
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces and Applications 9
for all 1198921 1198922isin 1198621015840
119886119887[0 119879] In addition for each 119892 isin 1198621015840
119886119887[0 119879]
10038171003817100381710038171198921003817100381710038171003817
2
119887minus
119899
sum
119896=1
(119890119896 119892)2
119887=
infin
sum
119896=119899+1
(119890119896 119892)2
119887ge 0 (63)
for every 119899 isin NSince
(119892 119886)119887=
infin
sum
119899=1
(119890119899 119892)119887(119890119899 119886)119887
(64)
and for 120576 gt 1
minus12057610038171003817100381710038171198921003817100381710038171003817119887119886119887lt minus10038171003817100381710038171198921003817100381710038171003817119887119886119887le (119892 119886)
119887
le10038171003817100381710038171198921003817100381710038171003817119887119886119887lt 12057610038171003817100381710038171198921003817100381710038171003817119887119886119887
(65)
there exists a sufficiently large119870119895isin N such that for any 119899 ge 119870
119895
1003816100381610038161003816100381610038161003816100381610038161003816
119899
sum
119896=1
(119890119896 11986012
119895119908)119887
(119890119896 119886)119887
1003816100381610038161003816100381610038161003816100381610038161003816
lt 120576
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817119887119886119887
(66)
for 119895 isin 1 2Using these and a long and tedious calculation we can
show that for every 119899 ge max119871 1198701 1198702
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
exp
2
sum
119895=1
([
120582119895119899minus 1
2120582119895119899
]
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
minus
1
2
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
+ 119894120582minus12
119895119899
119899
sum
119896=1
(119890119896 11986012
119895119908)119887
(119890119896 119886)119887+ 119894(119890119899+1 119886)119887
times [
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
minus
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
]
12
)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
lt 119896 (1199020 119908)
(67)
where 119896(1199020 119908) is given by (36)
In our next theorem for119865 isin F11988611988711986011198602
we express theGFFTof 119865 as the limit of a sequence of function space integrals on1198622
119886119887[0 119879]
Theorem 14 Let 1199020and 119865 be as in Theorem 10 Let 119890
119899infin
119899=1
be a complete orthonormal set in (1198621015840119886119887[0 119879] sdot
119887) and let
(1205821119899 1205822119899)infin
119899=1be a sequence in C2
+such that 120582
119895119899rarr minus119894119902
119895
where 119902119895is a real number with |119902
119895| gt 1199020 119895 isin 1 2 Then for
119901 isin [1 2] and for s-ae (1199101 1199102) isin 1198622
119886119887[0 119879]
119879(119901)
119902(119865) (119910
1 1199102)
= lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865 (119910
1+ 1199091 1199102+ 1199092)]
(68)
where 119866119899is given by (52)
Proof ByTheorems9 and 10we know that for each119901 isin [1 2]the 119871119901analytic GFFT of 119865 119879(119901)
119902(119865) exists and is given by the
right hand side of (40) Thus it suffices to show that
119879(1)
119902(119865) (119910
1 1199102) = 119864
anf 119902119909[119865 (1199101+ 1199091 1199102+ 1199092)]
= lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times 119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092)
times 119865 (1199101+ 1199091 1199102+ 1199092)] sdot
(69)
Using (30) the Fubini theorem and (56) with 120582 and 119908replaced with 120582
119895119899and 11986012
119895119908 119895 isin 1 2 respectively we see
that
1205821198992
11198991205821198992
2119899119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865 (119910
1+ 1199091 1199102+ 1199092)]
= 1205821198992
11198991205821198992
2119899int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 119910119895)
sim
times(
2
prod
119895=1
119864119909119895
[119866119899(120582minus12
119895119899 119909119895)
times exp 119894(11986012119895119908 119909119895)
sim
])119889119891 (119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
(119894(11986012
119895119908 119910119895)
sim
+ [
120582119895119899minus 1
2120582119895119899
]
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
minus
1
2
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
+ 119894120582minus12
119895119899
119899
sum
119896=1
(119890119896 119886)119887(119890119896 11986012
119895119908)119887
+ 119894(119890119899+1 119886)119887[
1003817100381710038171003817100381711986012
119895119908
10038171003817100381710038171003817
2
119887
minus
119899
sum
119896=1
(119890119896 11986012
119895119908)
2
119887
]
12
)
119889119891 (119908)
(70)
But by Remark 13 we see that the last expression of (70) isdominated by (39) on the region Γ
1199020
given by (35) for allbut a finite number of values of 119899 Next using the dominatedconvergence theorem Parsevalrsquos relation and (40) we obtainthe desired result
Corollary 15 Let 1199020 119865 119890
119899infin
119899=1 (1205821119899 1205822119899)infin
119899=1and (119902
1 1199022)
be as in Theorem 14 Then
119864
anf119902
119909[119865 (1199091 1199092)]
= lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times 119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865 (119909
1 1199092)]
(71)
where 119866119899is given by (52)
10 Journal of Function Spaces and Applications
Corollary 16 Let 1199020 119865 and 119890
119899infin
119899=1be as in Theorem 14 and
let Γ1199020
be given by (35) Let 120582 = (1205821 1205822) isin Int (Γ
1199020
) and(1205821119899 1205822119899)infin
119899=1be a sequence in C2
+such that 120582
119895119899rarr 120582
119895
119895 isin 1 2 Then
119864
an
119909[119865 (1199091 1199092)] = lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865(1199091 1199092)]
(72)
where 119866119899is given by (52)
Our another result namely a change of scale formula forfunction space integrals now follows fromCorollary 16 above
Theorem 17 Let 119865 isin F11988611988711986011198602
and let 119890119899infin
119899=1be a complete
orthonormal set in (1198621015840119886119887[0 119879] sdot
119887) Then for any 120588
1gt 0 and
1205882gt 0
119864119909[119865 (12058811199091 12058821199092)]
= lim119899rarrinfin
120588minus119899
1120588minus119899
2
times 119864119909[119866119899(120588minus2
1 1199091)119866119899(120588minus2
2 1199092) 119865 (119909
1 1199092)]
(73)
where 119866119899is given by (52)
Proof Simply choose 120582119895= 120588minus2
119895for 119895 isin 1 2 and 120582
119895119899= 120588minus2
119895
for 119895 isin 1 2 and 119899 isin N in (72)
Remark 18 Of course if we choose 119886(119905) equiv 0 119887(119905) = 1199051198601= 119868 (identity operator) and 119860
2= 0 (zero operator) then
the function space 119862119886119887[0 119879] reduces to the classical Wiener
space 1198620[0 119879] and the generalized Fresnel type class F 119886119887
11986011198602
reduces to the Fresnel class F(1198620[0 119879]) It is known that
F(1198620[0 119879]) forms a Banach algebra over the complex field
In this case we have the relationships between the analyticFeynman integral and theWiener integral on classicalWienerspace as discussed in [14 15]
In recent paper [19] Yoo et al have studied a change ofscale formula for function space integral of the functionalsin the Banach algebra S(1198712
119886119887[0 119879]) the Banach algebra
S(1198712119886119887[0 119879]) is introduced in [12]
5 Functionals in F11988611988711986011198602
In this section we prove a theorem ensuring that variousfunctionals are inF119886119887
11986011198602
Theorem 19 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] Let (119884Y 120574) be a 120590-finite
measure space and let 120593119897 119884 rarr 119862
1015840
119886119887[0 119879] beYndashB(1198621015840
119886119887[0 119879])
measurable for 119897 isin 1 119889 Let 120579 119884 times R119889 rarr C be given by120579(120578 sdot) = ]
120578(sdot) where ]
120578isin M(R119889) for every 120578 isin 119884 and where
the family ]120578 120578 isin 119884 satisfies
(i) ]120578(119864) is a Y-measurable function of 120578 for every 119864 isin
B(R119889)(ii) ]
120578 isin 1198711
(119884Y 120574)
Under these hypothesis the functional 119865 1198622119886119887[0 119879] rarr C
given by
119865 (1199091 1199092) = int
119884
120579(120578
2
sum
119895=1
(11986012
1198951205931(120578) 119909
119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 119909
119895)
sim
)119889120574 (120578)
(74)
belongs toF11988611988711986011198602
and satisfies the inequality
119865 le int
119884
10038171003817100381710038171003817]120578
10038171003817100381710038171003817119889120574 (120578) (75)
Proof Using the techniques similar to those used in [20] wecan show that ]
120578 is measurable as a function of 120578 that 120579 is
Y-measurable and that the integrand in (74) is a measurablefunction of 120578 for every (119909
1 1199092) isin 1198622
119886119887[0 119879]
We define a measure 120591 onY timesB(R119889) by
120591 (119864) = int
119884
]120578(119864(120578)
) 119889120574 (120578) for 119864 isin Y timesB (R119889) (76)
Then by the first assertion of Theorem 31 in [17] 120591 satisfies120591 le int
119884
]120578119889120574(120578) Now let Φ 119884 times R119889 rarr 119862
1015840
119886119887[0 119879] be
defined by Φ(120578 V1 V
119889) = sum
119889
119897=1V119897120593119897(120578) Then Φ is Y times
B(R119889) ndashB(1198621015840119886119887[0 119879])-measurable on the hypothesis for 120593
119897
119897 isin 1 119889 Let 120590 = 120591 ∘Φminus1 Then clearly 120590 isinM(1198621015840119886119887[0 119879])
and satisfies 120590 le 120591From the change of variables theorem and the second
assertion of Theorem 31 in [17] it follows that for ae(1199091 1199092) isin 1198622
119886119887[0 119879] and for every 120588
1gt 0 and 120588
2gt 0
119865 (12058811199091 12058821199092)
= int
119884
]120578(
2
sum
119895=1
(11986012
1198951205931(120578) 120588
119895119909119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 120588
119895119909119895)
sim
)119889120574 (120578)
= int
119884
[
[
int
R119889exp
119894
119889
sum
119897=1
V119897
[
[
2
sum
119895=1
(11986012
119895120593119897(120578)
120588119895119909119895)
sim
]
]
119889]120578
times (V1 V
119889)]
]
119889120574 (120578)
= int
119884timesR119889exp
119894
119889
sum
119897=1
V119897
[
[
2
sum
119895=1
(11986012
119895120593119897(120578) 120588
119895119909119895)
sim
]
]
119889120591
times (120578 V1 V
119889)
Journal of Function Spaces and Applications 11
= int
119884timesR119889exp
2
sum
119895=1
119894(11986012
119895Φ(120578 V
1 V
119889) 120588119895119909119895)
sim
119889120591
times (120578 V1 V
119889)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119909119895)
sim
119889120591 ∘ Φminus1
(119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119909119895)
sim
119889120590 (119908)
(77)
Thus the functional 119865 given by (74) belongs to F11988611988711986011198602
andsatisfies the inequality
119865 = 120590 le 120591 le int
119884
10038171003817100381710038171003817]120578
10038171003817100381710038171003817119889120574 (120578) (78)
As mentioned in (2) of Remark 6 F 11988611988711986011198602
is a Banachalgebra if Ran(119860
1+ 1198602) is dense in 1198621015840
119886119887[0 119879] In this case
many analytic functionals of 119865 can be formed The followingcorollary is relevant to Feynman integration theories andquantum mechanics where exponential functions play animportant role
Corollary 20 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] such that Ran(119860
1+ 1198602)
is dense in 1198621015840119886119887[0 119879] Let 119865 be given by (74) with 120579 as in
Theorem 19 and let 120573 C rarr C be an entire function Then(120573 ∘ 119865)(119909
1 1199092) is in F119886119887
11986011198602
In particular exp119865(1199091 1199092) isin
F11988611988711986011198602
Corollary 21 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] and let 119892
1 119892
119889 be a
finite subset of 1198621015840119886119887[0 119879] Given 120573 = ] where ] isin M(R119889)
define 119865 1198622119886119887[0 119879] rarr C by
119865 (1199091 1199092) = 120573(
2
sum
119895=1
(11986012
1198951198921 119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 119909119895)
sim
)
(79)
Then 119865 is an element ofF11988611988711986011198602
Proof Let (119884Y 120574) be a probability space and for 119897 isin1 119889 let 120593
119897(120578) equiv 119892
119897 Take 120579(120578 sdot) = 120573(sdot) = ](sdot) Then for
all 1205881gt 0 and 120588
2gt 0 and for ae (119909
1 1199092) isin 1198622
119886119887[0 119879]
int
119884
120579(120578
2
sum
119895=1
(11986012
1198951205931(120578) 120588
119895119909119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 120588
119895119909119895)
sim
)119889120574 (120578)
= int
119884
120573(
2
sum
119895=1
(11986012
1198951198921 120588119895119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 120588119895119909119895)
sim
)119889120574 (120578)
= 120573(
2
sum
119895=1
(11986012
1198951198921 120588119895119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 120588119895119909119895)
sim
)
= 119865 (12058811199091 12058821199092)
(80)
Hence 119865 isin F11988611988711986011198602
Remark 22 Let 119889 = 1 and let (119884Y 120574) = ([0 119879]B([0 119879])119898119871) in Theorem 19 where 119898
119871denotes the Lebesgue measure
on [0 119879] ThenTheorems 46 47 and 49 in [18] follow fromthe results in this section by letting119860
1be the identity operator
and letting 1198602equiv 0 on 1198621015840
119886119887[0 119879] The function 120579 studied in
[18] (and mentioned in Remark 2 above) is interpreted as thepotential energy in quantum mechanics
Acknowledgments
The authors would like to express their gratitude to the refer-ees for their valuable comments and suggestions which haveimproved the original paper This research was supported bythe Basic Science Research Program through the NationalResearch Foundation of Korea (NRF) funded by theMinistryof Education (2011-0014552)
References
[1] G Kallianpur and C Bromley ldquoGeneralized Feynman integralsusing analytic continuation in several complex variablesrdquo inStochastic Analysis and Applications M A Pinsky Ed vol 7pp 217ndash267 Marcel Dekker New York NY USA 1984
[2] G Kallianpur D Kannan and R L Karandikar ldquoAnalytic andsequential Feynman integrals on abstract Wiener and Hilbertspaces and a Cameron-Martin formulardquo Annales de lrsquoInstitutHenri Poincare vol 21 no 4 pp 323ndash361 1985
[3] G W Johnson and M L Lapidus The Feynman Integral andFeynmanrsquos Operational Calculus Clarendon Press Oxford UK2000
[4] J Yeh ldquoSingularity of Gaussian measures on function spacesinduced by Brownian motion processes with non-stationaryincrementsrdquo Illinois Journal of Mathematics vol 15 pp 37ndash461971
[5] J Yeh Stochastic Processes and the Wiener Integral MarcelDekker New York NY USA 1973
[6] S J Chang J G Choi and D Skoug ldquoIntegration by partsformulas involving generalized Fourier-Feynman transformson function spacerdquo Transactions of the American MathematicalSociety vol 355 no 7 pp 2925ndash2948 2003
[7] S J Chang J G Choi and D Skoug ldquoParts formulas involvingconditional generalized Feynman integrals and conditionalgeneralized Fourier-Feynman transforms on function spacerdquo
12 Journal of Function Spaces and Applications
Integral Transforms and Special Functions vol 15 no 6 pp 491ndash512 2004
[8] S J Chang J G Choi and D Skoug ldquoEvaluation formulas forconditional function space integralsmdashIrdquo Stochastic Analysis andApplications vol 25 no 1 pp 141ndash168 2007
[9] S J Chang J G Choi and D Skoug ldquoGeneralized Fourier-Feynman transforms convolution products and first variationson function spacerdquoTheRockyMountain Journal ofMathematicsvol 40 no 3 pp 761ndash788 2010
[10] S J Chang and D M Chung ldquoConditional function spaceintegrals with applicationsrdquo The Rocky Mountain Journal ofMathematics vol 26 no 1 pp 37ndash62 1996
[11] S J Chang H S Chung and D Skoug ldquoIntegral transformsof functionals in 119871
2(119862119886119887[0 119879])rdquoThe Journal of Fourier Analysis
and Applications vol 15 no 4 pp 441ndash462 2009[12] S J Chang and D Skoug ldquoGeneralized Fourier-Feynman
transforms and a first variation on function spacerdquo IntegralTransforms and Special Functions vol 14 pp 375ndash393 2003
[13] J G Choi and S J Chang ldquoGeneralized Fourier-Feynmantransform and sequential transforms on function spacerdquo Jour-nal of the Korean Mathematical Society vol 49 pp 1065ndash10822012
[14] R H Cameron and D A Storvick ldquoRelationships between theWiener integral and the analytic Feynman integralrdquo Rendicontidel Circolo Matematico di Palermo no 17 supplement pp 117ndash133 1987
[15] RHCameron andDA Storvick ldquoChange of scale formulas forWiener integralrdquo Rendiconti del Circolo Matematico di Palermono 17 pp 105ndash115 1987
[16] G W Johnson ldquoThe equivalence of two approaches to theFeynman integralrdquo Journal of Mathematical Physics vol 23 pp2090ndash2096 1982
[17] G W Johnson and D L Skoug ldquoNotes on the FeynmanintegralmdashIII the Schroedinger equationrdquo Pacific Journal ofMathematics vol 105 pp 321ndash358 1983
[18] S J Chang J G Choi and S D Lee ldquoA Fresnel type class onfunction spacerdquo Journal of the Korean Society of MathematicalEducation Series B vol 16 no 1 pp 107ndash119 2009
[19] I Yoo B J Kim and B S Kim ldquoA change of scale formulafor a function space integral on 119862
119886119887[0 119879]rdquo Proceedings of the
American Mathematical Society vol 141 no 8 pp 2729ndash27392013
[20] K S Chang G W Johnson and D L Skoug ldquoFunctions inthe Fresnel classrdquo Proceedings of the American MathematicalSociety vol 100 no 2 pp 309ndash318 1987
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Journal of Function Spaces and Applications
Corollary 16 Let 1199020 119865 and 119890
119899infin
119899=1be as in Theorem 14 and
let Γ1199020
be given by (35) Let 120582 = (1205821 1205822) isin Int (Γ
1199020
) and(1205821119899 1205822119899)infin
119899=1be a sequence in C2
+such that 120582
119895119899rarr 120582
119895
119895 isin 1 2 Then
119864
an
119909[119865 (1199091 1199092)] = lim119899rarrinfin
1205821198992
11198991205821198992
2119899
times119864119909[119866119899(1205821119899 1199091) 119866119899(1205822119899 1199092) 119865(1199091 1199092)]
(72)
where 119866119899is given by (52)
Our another result namely a change of scale formula forfunction space integrals now follows fromCorollary 16 above
Theorem 17 Let 119865 isin F11988611988711986011198602
and let 119890119899infin
119899=1be a complete
orthonormal set in (1198621015840119886119887[0 119879] sdot
119887) Then for any 120588
1gt 0 and
1205882gt 0
119864119909[119865 (12058811199091 12058821199092)]
= lim119899rarrinfin
120588minus119899
1120588minus119899
2
times 119864119909[119866119899(120588minus2
1 1199091)119866119899(120588minus2
2 1199092) 119865 (119909
1 1199092)]
(73)
where 119866119899is given by (52)
Proof Simply choose 120582119895= 120588minus2
119895for 119895 isin 1 2 and 120582
119895119899= 120588minus2
119895
for 119895 isin 1 2 and 119899 isin N in (72)
Remark 18 Of course if we choose 119886(119905) equiv 0 119887(119905) = 1199051198601= 119868 (identity operator) and 119860
2= 0 (zero operator) then
the function space 119862119886119887[0 119879] reduces to the classical Wiener
space 1198620[0 119879] and the generalized Fresnel type class F 119886119887
11986011198602
reduces to the Fresnel class F(1198620[0 119879]) It is known that
F(1198620[0 119879]) forms a Banach algebra over the complex field
In this case we have the relationships between the analyticFeynman integral and theWiener integral on classicalWienerspace as discussed in [14 15]
In recent paper [19] Yoo et al have studied a change ofscale formula for function space integral of the functionalsin the Banach algebra S(1198712
119886119887[0 119879]) the Banach algebra
S(1198712119886119887[0 119879]) is introduced in [12]
5 Functionals in F11988611988711986011198602
In this section we prove a theorem ensuring that variousfunctionals are inF119886119887
11986011198602
Theorem 19 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] Let (119884Y 120574) be a 120590-finite
measure space and let 120593119897 119884 rarr 119862
1015840
119886119887[0 119879] beYndashB(1198621015840
119886119887[0 119879])
measurable for 119897 isin 1 119889 Let 120579 119884 times R119889 rarr C be given by120579(120578 sdot) = ]
120578(sdot) where ]
120578isin M(R119889) for every 120578 isin 119884 and where
the family ]120578 120578 isin 119884 satisfies
(i) ]120578(119864) is a Y-measurable function of 120578 for every 119864 isin
B(R119889)(ii) ]
120578 isin 1198711
(119884Y 120574)
Under these hypothesis the functional 119865 1198622119886119887[0 119879] rarr C
given by
119865 (1199091 1199092) = int
119884
120579(120578
2
sum
119895=1
(11986012
1198951205931(120578) 119909
119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 119909
119895)
sim
)119889120574 (120578)
(74)
belongs toF11988611988711986011198602
and satisfies the inequality
119865 le int
119884
10038171003817100381710038171003817]120578
10038171003817100381710038171003817119889120574 (120578) (75)
Proof Using the techniques similar to those used in [20] wecan show that ]
120578 is measurable as a function of 120578 that 120579 is
Y-measurable and that the integrand in (74) is a measurablefunction of 120578 for every (119909
1 1199092) isin 1198622
119886119887[0 119879]
We define a measure 120591 onY timesB(R119889) by
120591 (119864) = int
119884
]120578(119864(120578)
) 119889120574 (120578) for 119864 isin Y timesB (R119889) (76)
Then by the first assertion of Theorem 31 in [17] 120591 satisfies120591 le int
119884
]120578119889120574(120578) Now let Φ 119884 times R119889 rarr 119862
1015840
119886119887[0 119879] be
defined by Φ(120578 V1 V
119889) = sum
119889
119897=1V119897120593119897(120578) Then Φ is Y times
B(R119889) ndashB(1198621015840119886119887[0 119879])-measurable on the hypothesis for 120593
119897
119897 isin 1 119889 Let 120590 = 120591 ∘Φminus1 Then clearly 120590 isinM(1198621015840119886119887[0 119879])
and satisfies 120590 le 120591From the change of variables theorem and the second
assertion of Theorem 31 in [17] it follows that for ae(1199091 1199092) isin 1198622
119886119887[0 119879] and for every 120588
1gt 0 and 120588
2gt 0
119865 (12058811199091 12058821199092)
= int
119884
]120578(
2
sum
119895=1
(11986012
1198951205931(120578) 120588
119895119909119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 120588
119895119909119895)
sim
)119889120574 (120578)
= int
119884
[
[
int
R119889exp
119894
119889
sum
119897=1
V119897
[
[
2
sum
119895=1
(11986012
119895120593119897(120578)
120588119895119909119895)
sim
]
]
119889]120578
times (V1 V
119889)]
]
119889120574 (120578)
= int
119884timesR119889exp
119894
119889
sum
119897=1
V119897
[
[
2
sum
119895=1
(11986012
119895120593119897(120578) 120588
119895119909119895)
sim
]
]
119889120591
times (120578 V1 V
119889)
Journal of Function Spaces and Applications 11
= int
119884timesR119889exp
2
sum
119895=1
119894(11986012
119895Φ(120578 V
1 V
119889) 120588119895119909119895)
sim
119889120591
times (120578 V1 V
119889)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119909119895)
sim
119889120591 ∘ Φminus1
(119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119909119895)
sim
119889120590 (119908)
(77)
Thus the functional 119865 given by (74) belongs to F11988611988711986011198602
andsatisfies the inequality
119865 = 120590 le 120591 le int
119884
10038171003817100381710038171003817]120578
10038171003817100381710038171003817119889120574 (120578) (78)
As mentioned in (2) of Remark 6 F 11988611988711986011198602
is a Banachalgebra if Ran(119860
1+ 1198602) is dense in 1198621015840
119886119887[0 119879] In this case
many analytic functionals of 119865 can be formed The followingcorollary is relevant to Feynman integration theories andquantum mechanics where exponential functions play animportant role
Corollary 20 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] such that Ran(119860
1+ 1198602)
is dense in 1198621015840119886119887[0 119879] Let 119865 be given by (74) with 120579 as in
Theorem 19 and let 120573 C rarr C be an entire function Then(120573 ∘ 119865)(119909
1 1199092) is in F119886119887
11986011198602
In particular exp119865(1199091 1199092) isin
F11988611988711986011198602
Corollary 21 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] and let 119892
1 119892
119889 be a
finite subset of 1198621015840119886119887[0 119879] Given 120573 = ] where ] isin M(R119889)
define 119865 1198622119886119887[0 119879] rarr C by
119865 (1199091 1199092) = 120573(
2
sum
119895=1
(11986012
1198951198921 119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 119909119895)
sim
)
(79)
Then 119865 is an element ofF11988611988711986011198602
Proof Let (119884Y 120574) be a probability space and for 119897 isin1 119889 let 120593
119897(120578) equiv 119892
119897 Take 120579(120578 sdot) = 120573(sdot) = ](sdot) Then for
all 1205881gt 0 and 120588
2gt 0 and for ae (119909
1 1199092) isin 1198622
119886119887[0 119879]
int
119884
120579(120578
2
sum
119895=1
(11986012
1198951205931(120578) 120588
119895119909119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 120588
119895119909119895)
sim
)119889120574 (120578)
= int
119884
120573(
2
sum
119895=1
(11986012
1198951198921 120588119895119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 120588119895119909119895)
sim
)119889120574 (120578)
= 120573(
2
sum
119895=1
(11986012
1198951198921 120588119895119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 120588119895119909119895)
sim
)
= 119865 (12058811199091 12058821199092)
(80)
Hence 119865 isin F11988611988711986011198602
Remark 22 Let 119889 = 1 and let (119884Y 120574) = ([0 119879]B([0 119879])119898119871) in Theorem 19 where 119898
119871denotes the Lebesgue measure
on [0 119879] ThenTheorems 46 47 and 49 in [18] follow fromthe results in this section by letting119860
1be the identity operator
and letting 1198602equiv 0 on 1198621015840
119886119887[0 119879] The function 120579 studied in
[18] (and mentioned in Remark 2 above) is interpreted as thepotential energy in quantum mechanics
Acknowledgments
The authors would like to express their gratitude to the refer-ees for their valuable comments and suggestions which haveimproved the original paper This research was supported bythe Basic Science Research Program through the NationalResearch Foundation of Korea (NRF) funded by theMinistryof Education (2011-0014552)
References
[1] G Kallianpur and C Bromley ldquoGeneralized Feynman integralsusing analytic continuation in several complex variablesrdquo inStochastic Analysis and Applications M A Pinsky Ed vol 7pp 217ndash267 Marcel Dekker New York NY USA 1984
[2] G Kallianpur D Kannan and R L Karandikar ldquoAnalytic andsequential Feynman integrals on abstract Wiener and Hilbertspaces and a Cameron-Martin formulardquo Annales de lrsquoInstitutHenri Poincare vol 21 no 4 pp 323ndash361 1985
[3] G W Johnson and M L Lapidus The Feynman Integral andFeynmanrsquos Operational Calculus Clarendon Press Oxford UK2000
[4] J Yeh ldquoSingularity of Gaussian measures on function spacesinduced by Brownian motion processes with non-stationaryincrementsrdquo Illinois Journal of Mathematics vol 15 pp 37ndash461971
[5] J Yeh Stochastic Processes and the Wiener Integral MarcelDekker New York NY USA 1973
[6] S J Chang J G Choi and D Skoug ldquoIntegration by partsformulas involving generalized Fourier-Feynman transformson function spacerdquo Transactions of the American MathematicalSociety vol 355 no 7 pp 2925ndash2948 2003
[7] S J Chang J G Choi and D Skoug ldquoParts formulas involvingconditional generalized Feynman integrals and conditionalgeneralized Fourier-Feynman transforms on function spacerdquo
12 Journal of Function Spaces and Applications
Integral Transforms and Special Functions vol 15 no 6 pp 491ndash512 2004
[8] S J Chang J G Choi and D Skoug ldquoEvaluation formulas forconditional function space integralsmdashIrdquo Stochastic Analysis andApplications vol 25 no 1 pp 141ndash168 2007
[9] S J Chang J G Choi and D Skoug ldquoGeneralized Fourier-Feynman transforms convolution products and first variationson function spacerdquoTheRockyMountain Journal ofMathematicsvol 40 no 3 pp 761ndash788 2010
[10] S J Chang and D M Chung ldquoConditional function spaceintegrals with applicationsrdquo The Rocky Mountain Journal ofMathematics vol 26 no 1 pp 37ndash62 1996
[11] S J Chang H S Chung and D Skoug ldquoIntegral transformsof functionals in 119871
2(119862119886119887[0 119879])rdquoThe Journal of Fourier Analysis
and Applications vol 15 no 4 pp 441ndash462 2009[12] S J Chang and D Skoug ldquoGeneralized Fourier-Feynman
transforms and a first variation on function spacerdquo IntegralTransforms and Special Functions vol 14 pp 375ndash393 2003
[13] J G Choi and S J Chang ldquoGeneralized Fourier-Feynmantransform and sequential transforms on function spacerdquo Jour-nal of the Korean Mathematical Society vol 49 pp 1065ndash10822012
[14] R H Cameron and D A Storvick ldquoRelationships between theWiener integral and the analytic Feynman integralrdquo Rendicontidel Circolo Matematico di Palermo no 17 supplement pp 117ndash133 1987
[15] RHCameron andDA Storvick ldquoChange of scale formulas forWiener integralrdquo Rendiconti del Circolo Matematico di Palermono 17 pp 105ndash115 1987
[16] G W Johnson ldquoThe equivalence of two approaches to theFeynman integralrdquo Journal of Mathematical Physics vol 23 pp2090ndash2096 1982
[17] G W Johnson and D L Skoug ldquoNotes on the FeynmanintegralmdashIII the Schroedinger equationrdquo Pacific Journal ofMathematics vol 105 pp 321ndash358 1983
[18] S J Chang J G Choi and S D Lee ldquoA Fresnel type class onfunction spacerdquo Journal of the Korean Society of MathematicalEducation Series B vol 16 no 1 pp 107ndash119 2009
[19] I Yoo B J Kim and B S Kim ldquoA change of scale formulafor a function space integral on 119862
119886119887[0 119879]rdquo Proceedings of the
American Mathematical Society vol 141 no 8 pp 2729ndash27392013
[20] K S Chang G W Johnson and D L Skoug ldquoFunctions inthe Fresnel classrdquo Proceedings of the American MathematicalSociety vol 100 no 2 pp 309ndash318 1987
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces and Applications 11
= int
119884timesR119889exp
2
sum
119895=1
119894(11986012
119895Φ(120578 V
1 V
119889) 120588119895119909119895)
sim
119889120591
times (120578 V1 V
119889)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119909119895)
sim
119889120591 ∘ Φminus1
(119908)
= int
1198621015840
119886119887[0119879]
exp
2
sum
119895=1
119894(11986012
119895119908 120588119895119909119895)
sim
119889120590 (119908)
(77)
Thus the functional 119865 given by (74) belongs to F11988611988711986011198602
andsatisfies the inequality
119865 = 120590 le 120591 le int
119884
10038171003817100381710038171003817]120578
10038171003817100381710038171003817119889120574 (120578) (78)
As mentioned in (2) of Remark 6 F 11988611988711986011198602
is a Banachalgebra if Ran(119860
1+ 1198602) is dense in 1198621015840
119886119887[0 119879] In this case
many analytic functionals of 119865 can be formed The followingcorollary is relevant to Feynman integration theories andquantum mechanics where exponential functions play animportant role
Corollary 20 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] such that Ran(119860
1+ 1198602)
is dense in 1198621015840119886119887[0 119879] Let 119865 be given by (74) with 120579 as in
Theorem 19 and let 120573 C rarr C be an entire function Then(120573 ∘ 119865)(119909
1 1199092) is in F119886119887
11986011198602
In particular exp119865(1199091 1199092) isin
F11988611988711986011198602
Corollary 21 Let 1198601and 119860
2be bounded nonnegative and
self-adjoint operators on 1198621015840119886119887[0 119879] and let 119892
1 119892
119889 be a
finite subset of 1198621015840119886119887[0 119879] Given 120573 = ] where ] isin M(R119889)
define 119865 1198622119886119887[0 119879] rarr C by
119865 (1199091 1199092) = 120573(
2
sum
119895=1
(11986012
1198951198921 119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 119909119895)
sim
)
(79)
Then 119865 is an element ofF11988611988711986011198602
Proof Let (119884Y 120574) be a probability space and for 119897 isin1 119889 let 120593
119897(120578) equiv 119892
119897 Take 120579(120578 sdot) = 120573(sdot) = ](sdot) Then for
all 1205881gt 0 and 120588
2gt 0 and for ae (119909
1 1199092) isin 1198622
119886119887[0 119879]
int
119884
120579(120578
2
sum
119895=1
(11986012
1198951205931(120578) 120588
119895119909119895)
sim
2
sum
119895=1
(11986012
119895120593119889(120578) 120588
119895119909119895)
sim
)119889120574 (120578)
= int
119884
120573(
2
sum
119895=1
(11986012
1198951198921 120588119895119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 120588119895119909119895)
sim
)119889120574 (120578)
= 120573(
2
sum
119895=1
(11986012
1198951198921 120588119895119909119895)
sim
2
sum
119895=1
(11986012
119895119892119889 120588119895119909119895)
sim
)
= 119865 (12058811199091 12058821199092)
(80)
Hence 119865 isin F11988611988711986011198602
Remark 22 Let 119889 = 1 and let (119884Y 120574) = ([0 119879]B([0 119879])119898119871) in Theorem 19 where 119898
119871denotes the Lebesgue measure
on [0 119879] ThenTheorems 46 47 and 49 in [18] follow fromthe results in this section by letting119860
1be the identity operator
and letting 1198602equiv 0 on 1198621015840
119886119887[0 119879] The function 120579 studied in
[18] (and mentioned in Remark 2 above) is interpreted as thepotential energy in quantum mechanics
Acknowledgments
The authors would like to express their gratitude to the refer-ees for their valuable comments and suggestions which haveimproved the original paper This research was supported bythe Basic Science Research Program through the NationalResearch Foundation of Korea (NRF) funded by theMinistryof Education (2011-0014552)
References
[1] G Kallianpur and C Bromley ldquoGeneralized Feynman integralsusing analytic continuation in several complex variablesrdquo inStochastic Analysis and Applications M A Pinsky Ed vol 7pp 217ndash267 Marcel Dekker New York NY USA 1984
[2] G Kallianpur D Kannan and R L Karandikar ldquoAnalytic andsequential Feynman integrals on abstract Wiener and Hilbertspaces and a Cameron-Martin formulardquo Annales de lrsquoInstitutHenri Poincare vol 21 no 4 pp 323ndash361 1985
[3] G W Johnson and M L Lapidus The Feynman Integral andFeynmanrsquos Operational Calculus Clarendon Press Oxford UK2000
[4] J Yeh ldquoSingularity of Gaussian measures on function spacesinduced by Brownian motion processes with non-stationaryincrementsrdquo Illinois Journal of Mathematics vol 15 pp 37ndash461971
[5] J Yeh Stochastic Processes and the Wiener Integral MarcelDekker New York NY USA 1973
[6] S J Chang J G Choi and D Skoug ldquoIntegration by partsformulas involving generalized Fourier-Feynman transformson function spacerdquo Transactions of the American MathematicalSociety vol 355 no 7 pp 2925ndash2948 2003
[7] S J Chang J G Choi and D Skoug ldquoParts formulas involvingconditional generalized Feynman integrals and conditionalgeneralized Fourier-Feynman transforms on function spacerdquo
12 Journal of Function Spaces and Applications
Integral Transforms and Special Functions vol 15 no 6 pp 491ndash512 2004
[8] S J Chang J G Choi and D Skoug ldquoEvaluation formulas forconditional function space integralsmdashIrdquo Stochastic Analysis andApplications vol 25 no 1 pp 141ndash168 2007
[9] S J Chang J G Choi and D Skoug ldquoGeneralized Fourier-Feynman transforms convolution products and first variationson function spacerdquoTheRockyMountain Journal ofMathematicsvol 40 no 3 pp 761ndash788 2010
[10] S J Chang and D M Chung ldquoConditional function spaceintegrals with applicationsrdquo The Rocky Mountain Journal ofMathematics vol 26 no 1 pp 37ndash62 1996
[11] S J Chang H S Chung and D Skoug ldquoIntegral transformsof functionals in 119871
2(119862119886119887[0 119879])rdquoThe Journal of Fourier Analysis
and Applications vol 15 no 4 pp 441ndash462 2009[12] S J Chang and D Skoug ldquoGeneralized Fourier-Feynman
transforms and a first variation on function spacerdquo IntegralTransforms and Special Functions vol 14 pp 375ndash393 2003
[13] J G Choi and S J Chang ldquoGeneralized Fourier-Feynmantransform and sequential transforms on function spacerdquo Jour-nal of the Korean Mathematical Society vol 49 pp 1065ndash10822012
[14] R H Cameron and D A Storvick ldquoRelationships between theWiener integral and the analytic Feynman integralrdquo Rendicontidel Circolo Matematico di Palermo no 17 supplement pp 117ndash133 1987
[15] RHCameron andDA Storvick ldquoChange of scale formulas forWiener integralrdquo Rendiconti del Circolo Matematico di Palermono 17 pp 105ndash115 1987
[16] G W Johnson ldquoThe equivalence of two approaches to theFeynman integralrdquo Journal of Mathematical Physics vol 23 pp2090ndash2096 1982
[17] G W Johnson and D L Skoug ldquoNotes on the FeynmanintegralmdashIII the Schroedinger equationrdquo Pacific Journal ofMathematics vol 105 pp 321ndash358 1983
[18] S J Chang J G Choi and S D Lee ldquoA Fresnel type class onfunction spacerdquo Journal of the Korean Society of MathematicalEducation Series B vol 16 no 1 pp 107ndash119 2009
[19] I Yoo B J Kim and B S Kim ldquoA change of scale formulafor a function space integral on 119862
119886119887[0 119879]rdquo Proceedings of the
American Mathematical Society vol 141 no 8 pp 2729ndash27392013
[20] K S Chang G W Johnson and D L Skoug ldquoFunctions inthe Fresnel classrdquo Proceedings of the American MathematicalSociety vol 100 no 2 pp 309ndash318 1987
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Journal of Function Spaces and Applications
Integral Transforms and Special Functions vol 15 no 6 pp 491ndash512 2004
[8] S J Chang J G Choi and D Skoug ldquoEvaluation formulas forconditional function space integralsmdashIrdquo Stochastic Analysis andApplications vol 25 no 1 pp 141ndash168 2007
[9] S J Chang J G Choi and D Skoug ldquoGeneralized Fourier-Feynman transforms convolution products and first variationson function spacerdquoTheRockyMountain Journal ofMathematicsvol 40 no 3 pp 761ndash788 2010
[10] S J Chang and D M Chung ldquoConditional function spaceintegrals with applicationsrdquo The Rocky Mountain Journal ofMathematics vol 26 no 1 pp 37ndash62 1996
[11] S J Chang H S Chung and D Skoug ldquoIntegral transformsof functionals in 119871
2(119862119886119887[0 119879])rdquoThe Journal of Fourier Analysis
and Applications vol 15 no 4 pp 441ndash462 2009[12] S J Chang and D Skoug ldquoGeneralized Fourier-Feynman
transforms and a first variation on function spacerdquo IntegralTransforms and Special Functions vol 14 pp 375ndash393 2003
[13] J G Choi and S J Chang ldquoGeneralized Fourier-Feynmantransform and sequential transforms on function spacerdquo Jour-nal of the Korean Mathematical Society vol 49 pp 1065ndash10822012
[14] R H Cameron and D A Storvick ldquoRelationships between theWiener integral and the analytic Feynman integralrdquo Rendicontidel Circolo Matematico di Palermo no 17 supplement pp 117ndash133 1987
[15] RHCameron andDA Storvick ldquoChange of scale formulas forWiener integralrdquo Rendiconti del Circolo Matematico di Palermono 17 pp 105ndash115 1987
[16] G W Johnson ldquoThe equivalence of two approaches to theFeynman integralrdquo Journal of Mathematical Physics vol 23 pp2090ndash2096 1982
[17] G W Johnson and D L Skoug ldquoNotes on the FeynmanintegralmdashIII the Schroedinger equationrdquo Pacific Journal ofMathematics vol 105 pp 321ndash358 1983
[18] S J Chang J G Choi and S D Lee ldquoA Fresnel type class onfunction spacerdquo Journal of the Korean Society of MathematicalEducation Series B vol 16 no 1 pp 107ndash119 2009
[19] I Yoo B J Kim and B S Kim ldquoA change of scale formulafor a function space integral on 119862
119886119887[0 119879]rdquo Proceedings of the
American Mathematical Society vol 141 no 8 pp 2729ndash27392013
[20] K S Chang G W Johnson and D L Skoug ldquoFunctions inthe Fresnel classrdquo Proceedings of the American MathematicalSociety vol 100 no 2 pp 309ndash318 1987
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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