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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
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
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
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
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
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 =
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
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
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
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
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
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
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 =
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
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
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
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
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
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 =
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
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
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
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
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 =
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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