Research Article Numerical Methods for Solving Fredholm Integral … · 2019. 7. 31. · Numerical Methods for Solving Fredholm Integral Equations of Second Kind S.SahaRayandP.K.Sahu

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 426916 17 pageshttpdxdoiorg1011552013426916

Research ArticleNumerical Methods for Solving Fredholm IntegralEquations of Second Kind

S Saha Ray and P K Sahu

Department of Mathematics National Institute of Technology Rourkela 769008 India

Correspondence should be addressed to S Saha Ray santanusaharayyahoocom

Received 3 September 2013 Accepted 3 October 2013

Academic Editor Rasajit Bera

Copyright copy 2013 S S Ray and P K Sahu This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Integral equation has been one of the essential tools for various areas of applied mathematics In this paper we review differentnumerical methods for solving both linear and nonlinear Fredholm integral equations of second kindThe goal is to categorize theselectedmethods and assess their accuracy and efficiencyWe discuss challenges faced by researchers in this field andwe emphasizethe importance of interdisciplinary effort for advancing the study on numerical methods for solving integral equations

1 Introduction

Integral equations occur naturally in many fields of scienceand engineering [1] A computational approach to solveintegral equation is an essential work in scientific research

Integral equation is encountered in a variety of appli-cations in many fields including continuum mechanicspotential theory geophysics electricity and magnetismkinetic theory of gases hereditary phenomena in physicsand biology renewal theory quantum mechanics radiationoptimization optimal control systems communication the-ory mathematical economics population genetics queuingtheory medicine mathematical problems of radiative equi-librium the particle transport problems of astrophysics andreactor theory acoustics fluid mechanics steady state heatconduction fracture mechanics and radiative heat transferproblems Fredholm integral equation is one of the mostimportant integral equations

Integral equations can be viewed as equations which areresults of transformation of points in a given vector spacesof integrable functions by the use of certain specific integraloperators to points in the same space If in particular oneis concerned with function spaces spanned by polynomialsfor which the kernel of the corresponding transformingintegral operator is separable being comprised of polynomial

functions only then several approximatemethods of solutionof integral equations can be developed

A computational approach to solving integral equationis an essential work in scientific research Some methodsfor solving second kind Fredholm integral equation areavailable in the open literatureThe B-spline wavelet methodthe method of moments based on B-spline wavelets byMaleknejad and Sahlan [2] and variational iteration method(VIM) by He [3ndash5] have been applied to solve second kindFredholm linear integral equations The learned researchersMaleknejad et al proposed some numerical methods forsolving linear Fredholm integral equations system of secondkind using Rationalized Haar functions method Block-Pulsefunctions and Taylor series expansion method [6ndash8] Haarwavelet method with operational matrices of integration [9]has been applied to solve system of linear Fredholm integralequations of second kind Quadrature method [10] B-splinewavelet method [11] wavelet Galerkin method [12] andalso VIM [13] can be applied to solve nonlinear Fredholmintegral equation of second kind Some iterative methodslike Homotopy perturbation method (HPM) [14ndash16] andAdomian decomposition method (ADM) [16ndash18] have beenapplied to solve nonlinear Fredholm integral equation ofsecond kind

2 Abstract and Applied Analysis

2 Fredholm Integral Equation

The general form of linear Fredholm integral equation isdefined as follows

119892 (119909) 119910 (119909) = 119891 (119909) + 120582int

119887

119886

119870 (119909 119905) 119910 (119905) 119889119905 (1)

where 119886 and 119887 are both constants 119891(119909) 119892(119909) and 119870(119909 119905)are known functions while 119910(119909) is unknown function 120582(nonzero parameter) is called eigenvalue of the integralequation The function 119870(119909 119905) is known as kernel of theintegral equation

21 Fredholm Integral Equation of First Kind The linearintegral equation is of form (by setting 119892(119909) = 0 in (1))

119891 (119909) + 120582int

119887

119886

119870 (119909 119905) 119910 (119905) 119889119905 = 0 (2)

Equation (2) is known as Fredholm integral equation of firstkind

22 Fredholm Integral Equation of Second Kind The linearintegral equation is of form (by setting 119892(119909) = 1 in (1))

119910 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) 119910 (119905) 119889119905 (3)

Equation (3) is known as Fredholm integral equation ofsecond kind

23 System of Linear Fredholm Integral Equations The gen-eral form of system of linear Fredholm integral equations ofsecond kind is defined as follows

119899

sum

119895=1

119892119894119895119910119895 (119909) = 119891119894 (119909) +

119899

sum

119895=1

int

119887

119886

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899

(4)

where 119891119894(119909) and 119870119894119895(119909 119905) are known functions and 119910119895(119909) arethe unknown functions for 119894 119895 = 1 2 119899

24 Nonlinear Fredholm-Hammerstein Integral Equation ofSecond Kind Nonlinear Fredholm-Hammerstein integralequation of second kind is defined as follows

119910 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) 119865 (119910 (119905)) 119889119905 (5)

where 119870(119909 119905) is the kernel of the integral equation 119891(119909)and 119870(119909 119905) are known functions and 119910(119909) is the unknownfunction that is to be determined

25 System of Nonlinear Fredholm Integral Equations Systemof nonlinear Fredholm integral equations of second kind isdefined as follows119899

sum

119895=1

119892119894119895119910119895 (119909) = 119891119894 (119909) +

119899

sum

119895=1

int

119887

119886

119870119894119895 (119909 119905) 119865119894119895 (119905 119910119895 (119905)) 119889119905

119894 = 1 2 119899

(6)


3 Numerical Methods for Linear FredholmIntegral Equation of Second Kind

Consider the following Fredholm integral equation of secondkind defined in (3)

119910 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) 119910 (119905) 119889119905 119886 le 119909 le 119887 (7)

where 119870(119909 119905) and 119892(119909) are known functions and 119910(119909) isunknown function to be determined

31 B-Spline Wavelet Method

311 B-Spline Scaling and Wavelet Functions on the Interval[0 1] Semiorthogonal wavelets using B-spline are speciallyconstructed for the bounded interval and this wavelet canbe represented in a closed form This provides a compactsupport Semiorthogonal wavelets form the basis in the space1198712(119877)Using this basis an arbitrary function in 1198712(119877) can be

expressed as the wavelet series For the finite interval [0 1]the wavelet series cannot be completely presented by usingthis basisThis is because supports of some basis are truncatedat the left or right end points of the interval Hence a specialbasis has to be introduced into the wavelet expansion on thefinite intervalThese functions are referred to as the boundaryscaling functions and boundary wavelet functions

Let119898 and 119899 be two positive integers and let

119886 = 119909minus119898+1 = sdot sdot sdot = 1199090 lt 1199091

lt sdot sdot sdot lt 119909119899 = 119909119899+1

= sdot sdot sdot = 119909119899+119898minus1 = 119887

(8)

be an equally spaced knots sequence The functions

119861119898119895119883 (119909) =119909 minus 119909119895

119909119895+119898minus1 minus 119909119895

119861119898minus1119895119883 (119909)

+119909119895+119898 minus 119909

119909119895+119898 minus 119909119895+1

119861119898minus1119895+1119883 (119909)

119895 = minus119898 + 1 119899 minus 1

1198611119895119883 (119909) = 1 119909 isin [119909119895 119909119895+1)

0 otherwise

(9)

Abstract and Applied Analysis 3

are called cardinal B-spline functions of order 119898 ge 2 forthe knot sequence 119883 = 119909119894

119899+119898minus1

119894=minus119898+1and Supp119861119898119895119883(119909) =

[119909119895 119909119895+119898] cap [119886 119887]By considering the interval [119886 119887] = [0 1] at any level 119895 isin

Ζ+ the discretization step is 2minus119895 and this generates 119899 = 2119895

number of segments in [0 1] with knot sequence

119883(119895)=

119909(119895)

minus119898+1= sdot sdot sdot = 119909

(119895)

0= 0

119909(119895)

119896=119896

2119895 119896 = 1 119899 minus 1

119909(119895)

119899= sdot sdot sdot = 119909

(119895)

119899+119898minus1= 1

(10)

Let 1198950 be the level forwhich 21198950 ge 2119898minus1 for each level 119895 ge 1198950

the scaling functions of order 119898 can be defined as follows in[2]120593119898119895119894 (119909)

=

1198611198981198950119894(2119895minus1198950119909) 119894 = minus119898 + 1 minus1

11986111989811989502119895minus119898minus119894 (1 minus 2

119895minus1198950119909) 119894 = 2119895minus 119898 + 1 2

119895minus 1

11986111989811989500(2119895minus1198950119909 minus 2

minus1198950 119894) 119894 = 0 2119895minus 119898

(11)

And the two scale relations for the 119898-order semiorthogonalcompactly supported B-wavelet functions are defined asfollows

120595119898119895119894minus119898 =

2119894+2119898minus2

sum

119896=119894

119902119894119896119861119898119895119896minus119898 119894 = 1 119898 minus 1

120595119898119895119894minus119898 =

2119894+2119898minus2

sum

119896=2119894minus119898

119902119894119896119861119898119895119896minus119898 119894 = 119898 119899 minus 119898 + 1

120595119898119895119894minus119898 =

119899+119894+119898minus1

sum

119896=2119894minus119898

119902119894119896119861119898119895119896minus119898 119894 = 119899 minus 119898 + 2 119899

(12)

where 119902119894119896 = 119902119896minus2119894Hence there are 2(119898 minus 1) boundary wavelets and (119899 minus

2119898+ 2) inner wavelets in the bounded interval [119886 119887] Finallyby considering the level 119895with 119895 ge 1198950 the B-wavelet functionsin [0 1] can be expressed as follows

120595119898119895119894 (119909)

=


1205951198982119895minus2119898+1minus119894119894 (1 minus 2119895minus1198950119909) 119894 = 2

119895minus2119898+2 2

119895minus119898

12059511989811989500(2119895minus1198950119909 minus 2

minus1198950 119894) 119894 = 0 2119895minus 2119898 + 1

(13)

The scaling functions 120593119898119895119894(119909) occupy 119898 segments and thewavelet functions 120595119898119895119894(119909) occupy 2119898 minus 1 segments

When the semiorthogonal wavelets are constructed fromB-spline of order 119898 the lowest octave level 119895 = 1198950 isdetermined in [19 20] by

21198950 ge 2119898 minus 1 (14)

so as to have a minimum of one complete wavelet on theinterval [0 1]

312 Function Approximation A function 119891(119909) defined over[0 1]may be approximated by B-spline wavelets as [21 22]

119891 (119909) =

21198950minus1

sum

119896=1minus119898

1198881198950 1198961205931198950 119896

(119909)

+

infin

sum

119895=1198950

2119895minus119898

sum

119896=1minus119898

119889119895119896120595119895119896 (119909)

(15)

If the infinite series in (15) is truncated at119872 then (15) can bewritten as [2]

119891 (119909) cong

21198950minus1

sum

119896=1minus119898

1198881198950 1198961205931198950 119896

(119909)

+

119872

sum

119895=1198950

2119895minus119898

sum

119896=1minus119898

119889119895119896120595119895119896 (119909)

(16)

where 1205932119896 and 120595119895119896 are scaling and wavelets functionsrespectively and119862 andΨ are (2119872+1 +119898minus1)times1 vectors givenby

119862 = [11988811989501minus119898 1198881198950 2

1198950minus1 1198891198950 1minus119898

1198891198950 21198950minus119898 1198891198721minus119898 1198891198722119872minus119898]

119879

(17)

Ψ = [1205931198950 1minus119898 1205931198950 2

1198950minus1 12059511989501minus119898

120595119895021198950minus119898 1205951198721minus119898 1205951198722119872minus119898]

119879

(18)

with

1198881198950 119896= int

1

0

119891 (119909) 1205931198950 119896(119909) 119889119909 119896 = 1 minus 119898 2

1198950 minus 1

119889119895119896 = int

1

0

119891 (119909) 119895119896 (119909) 119889119909

119895 = 1198950 119872 119896 = 1 minus 119898 2119872minus 119898

(19)

where 1205931198950 119896(119909) and 119895119896(119909) are dual functions of 1205931198950 119896 and 120595119895119896respectivelyThese can be obtained by linear combinations of1205931198950 119896

119896 = 1 minus 119898 21198950 minus 1 and 120595119895119896 119895 = 1198950 119872 119896 =1 minus 119898 2

119872minus 119898 as follows Let

Φ = [1205931198950 1minus119898 1205931198950 2

1198950minus1]119879

(20)

Ψ = [1205951198950 1minus119898 1205951198950 2

1198950minus119898 1205951198721minus119898 1205951198722119872minus119898]119879

(21)

Using (11) (20) (12)-(13) and (21) we get

int

1

0

ΦΦ119879119889119909 = 1198751

int

1

0

ΨΨ119879

119889119909 = 1198752

(22)


Suppose that Φ and Ψ are the dual functions of Φ and Ψrespectively then

int

1

0

ΦΦ119879119889119909 = 1198681

int

1

0

ΨΨ119879

119889119909 = 1198682

(23)

Φ = 1198751minus1Φ

Ψ = 1198752

minus1Ψ

(24)

313 Application of B-SplineWavelet Method In this sectionlinear Fredholm integral equation of the second kind of form(7) has been solved by using B-spline wavelets For this weuse (16) to approximate 119910(119909) as

119910 (119909) = 119862119879Ψ (119909) (25)

where Ψ(119909) is defined in (18) and 119862 is (2119872+1 + 119898 minus 1) times 1unknown vector defined similarly as in (17) We also expand119891(119909) and 119870(119909 119905) by B-spline dual wavelets Ψ defined in (24)as

119891 (119909) = 1198621119879Ψ (119909)

119870 (119909 119905) = Ψ119879(119905) ΘΨ (119909)

(26)

where

Θ119894119895 = int

1

0

[int

1

0

119870 (119909 119905) Ψ119894 (119905) 119889119905]Ψ119895 (119909) 119889119909 (27)

From (26) and (25) we get

int

1

0

119870 (119909 119905) 119910 (119905) 119889119905 = int

1

0

119862119879Ψ (119905) Ψ

119879(119905) ΘΨ (119909) 119889119905

= 119862119879ΘΨ (119909)

(28)

since

int

1

0

Ψ (119905) Ψ119879(119905) 119889119905 = 119868 (29)

By applying (25)ndash(28) in (7) we have

119862119879Ψ (119909) minus 119862

119879ΘΨ (119909) = 119862

119879

1Ψ (119909) (30)

By multiplying both sides of (30) with Ψ119879(119909) from the rightand integrating both sides with respect to 119909 from 0 to 1 weget

119862119879119875 minus 119862

119879Θ = 119862

119879

1 (31)

since

int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = 119868 (32)

and 119875 is a (2119872+1 +119898minus1)times(2119872+1 +119898minus1) square matrix givenby

119875 = int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = (

1198751 0

0 1198752) (33)

Consequently from (31) we get 119862119879 = 1198621198791(119875 minus Θ)

minus1 Hencewe can calculate the solution for 119910(119909) = 119862119879Ψ(119909)

32 Method of Moments

321 Multiresolution Analysis (MRA) andWavelets [2] A setof subspaces 119881119895119895isin119885 is said to beMRA of 1198712(119877) if it possessesthe following properties

119881119895 sub 119881119895+1 forall119895 isin Ζ (34)

⋃

119895isinΖ

119881119895 is dense in 1198712 (119877) (35)

⋂

119895isinΖ

119881119895 = 120601 (36)

119891 (119909) isin 119881119895 lArrrArr 119891(2119909) isin 119881119895+1 forall119895 isin Ζ (37)

where119885 denotes the set of integers Properties (34)ndash(36) statethat 119881119895119895isin119885 is a nested sequence of subspaces that effectivelycovers 1198712(119877) That is every square integrable function can beapproximated as closely as desired by a function that belongsto at least one of the subspaces 119881119895 A function 120593 isin 1198712(119877) iscalled a scaling function if it generates the nested sequence ofsubspaces 119881119895 and satisfies the dilation equation namely

120593 (119909) = sum

119896

119901119896120593 (119886119909 minus 119896) (38)

with 119901119896 isin 1198972 and 119886 being any rational number

For each scale 119895 since 119881119895 sub 119881119895+1 there exists a uniqueorthogonal complementary subspace 119882119895 of 119881119895 in 119881119895+1 Thissubspace 119882119895 is called wavelet subspace and is generated by120595119895119896 = 120595(2

119895119909 minus 119896) where 120595 isin 1198712 is called the wavelet From

the above discussion these results follow easily

1198811198951cap 1198811198952

= 1198811198952 1198951 gt 1198952

1198821198951cap1198821198952

= 0 1198951 = 1198952

1198811198951cap1198821198952

= 0 1198951 le 1198952

(39)

Some of the important properties relevant to the presentanalysis are given below [2 19]

(1) Vanishing Moment A wavelet is said to have a vanishingmoment of order119898 if

int

infin

minusinfin

119909119901120595 (119909) 119889119909 = 0 119901 = 0 119898 minus 1 (40)


All wavelets must satisfy the previously mentioned conditionfor 119901 = 0

(2) SemiorthogonalityThewavelets 120595119895119896 form a semiorthogo-nal basis if

⟨120595119895119896 120595119904119894⟩ = 0 119895 = 119904 forall119895 119896 119904 119894 isin Ζ (41)

322 Method ofMoments for the Solution of Fredholm IntegralEquation In this section we solve the integral equation ofform (7) in interval [0 1] by using linear B-spline wavelets[2] The unknown function in (7) can be expanded in termsof the scaling and wavelet functions as follows

119910 (119909) asymp

21198950minus1

sum

119896=minus1

1198881198961205931198950 119896(119909)

+

119872

sum

119895=1198950

2119895minus2

sum

119896=minus1

119889119895119896120595119895119896 (119909)

= 119862119879Ψ (119909)

(42)

By substituting this expression into (7) and employing theGalerkin method the following set of linear system of order(2119872+ 1) is generated The scaling and wavelet functions are

used as testing and weighting functions

(⟨120593 120593⟩ minus ⟨119870120593 120593⟩ ⟨120595 120593⟩ minus ⟨119870120595 120593⟩

⟨120593 120595⟩ minus ⟨119870120593 120595⟩ ⟨120595 120595⟩ minus ⟨119870120595 120595⟩)(119862

119863) = (

1198651

1198652)

(43)where

119862 = [119888minus1 1198880 1198883]119879

119863 = [1198892minus1 11988922 1198893minus1 11988936

119889119872minus1 1198891198722119872minus2]119879

⟨120593 120593⟩ minus ⟨119870120593 120593⟩

= (int

1

0

1205931198950 119903(119909) 1205931198950 119894

(119909) 119889119909

minusint

1

0

1205931198950 119903(119909) int

1

0

119870(119909 119905)1205931198950 119894(119905)119889119905 119889119909)

119894119903

⟨120595 120593⟩ minus ⟨119870120595 120593⟩

= (int

1

0

1205931198950 119903(119909) 120595119896119895 (119909) 119889119909

minusint

1

0

1205931198950 119903(119909) int

1

0

119870(119909 119905)120595119896119895(119905)119889119905 119889119909)

119903119896119895

⟨120593 120595⟩ minus ⟨119870120593 120595⟩

= (int

1

0

120595119904119897 (119909) 1205931198950 119894(119909) 119889119909

minusint

1

0

120595119904119897(119909) int

1

0

119870(119909 119905)1205931198950 119894(119905)119889119905 119889119909)

119894119897119904

⟨120595 120595⟩ minus ⟨119870120595 120595⟩

= (int

1

0

120595119904119897 (119909) 120595119896119895 (119909) 119889119909

minus int

1

0

120595119904119897(119909) int

1

0

119870(119909 119905)120595119896119895(119905)119889119905 119889119909)

119897119904119896119895

1198651 = int

1

0

119891 (119909) 1205931198950 119903(119909) 119889119909

1198652 = int

1

0

119891 (119909) 120595119904119897 (119909) 119889119909

(44)

and the subscripts 119894 119903 119896 119895 119897 and 119904 assume values as givenbelow

119894 119903 = minus1 21198950 minus 1

119897 119896 = 1198950 119872

119904 119895 = minus1 2119872minus 2

(45)

In fact the entries with significant magnitude are in the⟨119870120593 120593⟩minus ⟨120593 120593⟩ and ⟨119870120595 120595⟩minus ⟨120595 120595⟩ submatrices which areof order (21198950 + 1) and (2119872+1 + 1) respectively

33 Variational Iteration Method [3ndash5] In this section Fred-holm integral equation of second kind given in (7) has beenconsidered for solving (7) by variational iteration methodFirst we have to take the partial derivative of (7) with respectto 119909 yielding

1198841015840(119909) = 119891

1015840(119909) + int

1

0

1198701015840(119909 119905) 119910 (119905) 119889119905 (46)

We apply variation iteration method for (46) According tothis method correction functional can be defined as

119910119899+1 (119909)

= 119910119899 (119909)

+ int

119909

0

120582 (120585) (1199101015840

119899(120585) minus 119891

1015840(120585) minus int

119887

119886

1198701015840(120585 119905) 119910119899 (119905) 119889119905) 119889120585

(47)

where 120582(120585) is a general Lagrange multiplier which can beidentified optimally by the variational theory the subscript119899 denotes the 119899th order approximation and 119910119899 is consideredas a restricted variation that is 120575119910119899 = 0 The successiveapproximations 119910119899(119909) 119899 ge 1 for the solution 119910(119909) can bereadily obtained after determining the Lagrange multiplierand selecting an appropriate initial function 1199100(119909) Conse-quently the approximate solution may be obtained by using

119910 (119909) = lim119899rarrinfin

119910119899 (119909) (48)


To make the above correction functional stationary we have

120575119910119899+1 (119909) = 120575119910119899 (119909)

+ 120575int

119909

0

120582 (120585) (1199101015840

119899(120585) minus 119891

1015840(120585)

minusint

119887

119886

1198701015840(120585 119905) 119910119899 (119905) 119889119905) 119889120585

= 120575119910119899 (119909) + int

119909

0

120582 (120585) 120575 (1199101015840

119899(120585)) 119889120585

= 120575119910119899 (119909) + 1205821205751199101198991003816100381610038161003816120585=119909 minus int

119909

0

1205821015840(120585) 120575119910119899 (120585) 119889120585

(49)

Under stationary condition

120575119910119899+1 = 0 (50)

implies the following Euler Lagrange equation

1205821015840(120585) = 0 (51)

with the following natural boundary condition

1 + 120582(120585)1003816100381610038161003816120585=119909 = 0 (52)

Solving (51) along with boundary condition (52) we get thegeneral Lagrange multiplier

120582 = minus1 (53)

Substituting the identified Lagrange multiplier into (47)results in the following iterative scheme

119910119899+1 (119909) = 119910119899 (119909)

minus int

119909

0

(1199101015840

119899(120585) minus 119891

1015840(120585) minus int

119887

119886

1198701015840(120585 119905) 119910119899 (119905) 119889119905) 119889120585

119899 ge 0

(54)

By starting with initial approximate function 1199100(119909) = 119891(119909)(say) we can determine the approximate solution 119910(119909) of (7)

4 Numerical Methods for System ofLinear Fredholm Integral Equations ofSecond Kind

Consider the system of linear Fredholm integral equations ofsecond kind of the following form

119899

sum

119895=1

119910119895 (119909) = 119891119894 (119909) +

119899

sum

119895=1

int

1

0

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899

(55)


41 Application of Haar Wavelet Method [9] In this sectionan efficient algorithm for solving Fredholm integral equationswith Haar wavelets is analyzed The present algorithm takesthe following essential strategy The Haar wavelet is firstused to decompose integral equations into algebraic systemsof linear equations which are then solved by collocationmethods

411 Haar Wavelets The compact set of scale functions ischosen as

ℎ0 = 1 0 le 119909 lt 1

0 others(56)

The mother wavelet function is defined as

ℎ1 (119909) =

1 0 le 119909 lt1

2

minus11

2le 119909 lt 1

0 others

(57)

The family of wavelet functions generated by translation anddilation of ℎ1(119909) are given by

ℎ119899 (119909) = ℎ1 (2119895119909 minus 119896) (58)

where 119899 = 2119895 + 119896 119895 ge 0 0 le 119896 lt 2119895Mutual orthogonalities of all Haar wavelets can be

expressed as

int

1

0

ℎ119898 (119909) ℎ119899 (119909) 119889119909 = 2minus119895120575119898119899 =

2minus119895 119898 = 119899 = 2

119895+ 119896

0 119898 = 119899

(59)

412 Function Approximation An arbitrary function 119910(119909) isin1198712[0 1) can be expanded into the following Haar series

119910 (119909) =

+infin

sum

119899=0

119888119899ℎ119899 (119909) (60)

where the coefficients 119888119899 are given by

119888119899 = 2119895int

1

0

119910 (119909) ℎ119899 (119909) 119889119909

119899 = 2119895+ 119896 119895 ge 0 0 le 119896 lt 2

119895

(61)

In particular 1198880 = int1

0119910(119909)119889119909

The previously mentioned expression in (60) can beapproximately represented with finite terms as follows

119910 (119909) asymp

119898minus1

sum

119899=0

119888119899ℎ119899 (119909) = 119862119879

(119898)ℎ(119898) (119909) (62)

where the coefficient vector119862119879(119898)

and theHaar function vectorℎ(119898)(119909) are respectively defined as

119862119879

(119898)= [1198880 1198881 119888119898minus1] 119898 = 2

119895

ℎ(119898) (119909) = [ℎ0 (119909) ℎ1 (119909) ℎ119898minus1 (119909)]119879 119898 = 2

119895

(63)


TheHaar expansion for function119870(119909 119905) of order119898 is definedas follows

119870 (119909 119905) asymp

119898minus1

sum

119906=0

119898minus1

sum

V=0

119886119906VℎV (119909) ℎ119906 (119905) (64)

where 119886119906V = 2119894+119902∬1

0119870(119909 119905)ℎV(119909)ℎ119906(119905)119889119909 119889119905 119906 = 2

119894+ 119895 V =

2119902+ 119903 119894 119902 ge 0From (62) and (64) we obtain

119870 (119909 119905) asymp ℎ119879

(119898)(119905) 119870ℎ(119898) (119909) (65)

where

119870 = (119886119906V)119879

119898times119898 (66)

413 Operational Matrices of Integration We define

119867(119898) = [ℎ(119898) (1

2119898) ℎ(119898) (

3

2119898) ℎ(119898) (

2119898 minus 1

2119898)]

(67)

where119867(1) = [1]119867(2) = [ 1 11 minus1 ]Then for 119898 = 4 the corresponding matrix can be

represented as

119867(4) = [ℎ(4) (1

8) ℎ(4) (

3

8) ℎ(4) (

7

8)]

=[[[

[

1 1 1 1

1 1 minus1 minus1

1 minus1 0 0

0 0 1 minus1

]]]

]

(68)

The integration of the Haar function vector ℎ(119898)(119905) is

int

119909

0

ℎ(119898) (119905) 119889119905 = 119875(119898)ℎ(119898) (119909)

119909 isin [0 1)

(69)

where 119875(119898) is the operational matrix of order119898 and

119875(1) = [1

2]

119875(119898) =1

2119898[

2119898119875(1198982) minus119867(1198982)

119867minus1

(1198982)0

]

(70)

By recursion of the above formula we obtain

119875(2) =1

4[2 minus1

1 0]

119875(4) =1

16

[[[

[

8 minus4 minus2 minus2

4 0 minus2 2

1 1 0 0

1 minus1 0 0

]]]

]

119875(8) =1

64

[[[[[[[[[[

[

32 minus16 minus8 minus8 minus4 minus4 minus4 minus4

16 0 minus8 8 minus4 minus4 4 4

4 4 0 0 minus4 4 0 0

4 4 0 0 0 0 minus4 4

1 1 2 0 0 0 0 0

1 1 minus2 0 0 0 0 0

1 minus1 0 2 0 0 0 0

1 minus1 0 minus2 0 0 0 0

]]]]]]]]]]

]

(71)

Therefore we get

119867minus1

(119898)= (

1

119898)119867119879

(119898)

times diag(1 1 2 2 22 22⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

22

2120572minus1 2

120572minus1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

2120572minus1

)

(72)

where119898 = 2120572 and 120572 is a positive integerThe inner product of two Haar functions can be repre-

sented as

int

1

0

ℎ(119898) (119905) ℎ119879

(119898)(119905) 119889119905 = 119863 (73)

where

119863 = diag(1 1 12 12 122 122⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

22

12120572minus1 12

120572minus1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

2120572minus1

)

(74)

414 Haar Wavelet Solution for Fredholm Integral EquationsSystem [9] Consider the following Fredholm integral equa-tions system defined in (55)

119898

sum

119895=1

119910119895 (119909) = 119891119894 (119909) +

119898

sum

119895=1

int

1

0

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119898

(75)

The Haar series of 119910119895(119909) and119870119894119895(119909 119905) 119894 = 1 2 119898 119895 =1 2 119898 are respectively expanded as

119910119895 (119909) asymp 119884119879

119895ℎ(119898) (119909) 119895 = 1 2 119898

119870119894119895 (119909 119905) asymp ℎ119879

(119898)(119905) 119870119894119895ℎ(119898) (119909)

119894 119895 = 1 2 119898

(76)


Substituting (76) into (75) we get

119898

sum

119895=1

119884119879

119895ℎ(119898) (119909)

= 119891119894 (119909) +

119898

sum

119895=1

int

1

0

119884119879

119895ℎ(119898) (119905) ℎ

119879

(119898)(119905) 119870119894119895ℎ(119898) (119909) 119889119905

119894 = 1 2 119898

(77)


119898

sum

119895=1

119884119879

119895ℎ(119898) (119909) = 119891119894 (119909) +

119898

sum

119895=1

119884119879

119895119863119870119894119895ℎ(119898) (119909)

119894 = 1 2 119898

(78)

Interpolating 119898 collocation points that is 119909119894119898

119894=1 in the

interval [0 1] leads to the following algebraic system ofequations

119898

sum

119895=1

119884119879

119895ℎ(119898) (119909119894) = 119891119894 (119909119894) +

119898

sum

119895=1

119884119879

119895119863119870119894119895ℎ(119898) (119909119894)

119894 = 1 2 119898

(79)

Hence 119884119895 119895 = 1 2 119898 can be computed by solving theabove algebraic system of equations and consequently thesolutions 119910119895(119909) asymp 119884

119879

119895ℎ(119898)(119909) 119895 = 1 2 119898

42 Taylor Series Expansion Method In this section wepresent Taylor series expansionmethod for solving Fredholmintegral equations system of second kind [7] This methodreduces the system of integral equations to a linear systemof ordinary differential equation After including boundaryconditions this system reduces to a system of equations thatcan be solved easily by any usual methods

Consider the second kind Fredholm integral equationssystem defined in (55) as follows

119910119894 (119909) = 119891119894 (119909) +

119899

sum

119895=1

int

1

0

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899 0 le 119909 le 1

(80)

A Taylor series expansion can be made for the solution of119910119895(119905) in the integral equation (80)

119910119895 (119905) = 119910119895 (119909) + 1199101015840

119895(119909) (119905 minus 119909) + sdot sdot sdot

+1

119898119910(119898)

119895(119909) (119905 minus 119909)

119898+ 119864 (119905)

(81)

where 119864(119905) denotes the error between 119910119895(119905) and its Taylorseries expansion in (81)

If we use the first 119898 term of Taylor seriesexpansion and neglect the term containing 119864(119905) that is

int1

0sum119899

119895=1119870119894119895(119909 119905)119864(119905)119889119905 then substituting (81) for 119910119895(119905) into

the integral in (80) we have

119910119894 (119909) asymp 119891119894 (119909)

+

119899

sum

119895=1

int

1

0

119870119894119895 (119909 119905)

119898

sum

119903=0

1

119903(119905 minus 119909)

119903119910(119903)

119895(119909) 119889119905

119894 = 1 2 119899

119910119894 (119909) asymp 119891119894 (119909)

+

119899

sum

119895=1

119898

sum

119903=0

1

119903119910(119903)

119895(119909) int

1

0

119870119894119895 (119909 119905) (119905 minus 119909)119903119889119905

119894 = 1 2 119899

(82)

119910119894 (119909) minus

119899

sum

119895=1

119898

sum

119903=0

1

119903119910(119903)

119895(119909) [int

1

0

119870119894119895 (119909 119905) (119905 minus 119909)119903119889119905]

asymp 119891119894 (119909) 119894 = 1 2 119899

(83)

Equation (83) becomes a linear systemof ordinary differentialequations that we have to solve For solving the linearsystem of ordinary differential equations (83) we require anappropriate number of boundary conditions

In order to construct boundary conditions we firstdifferentiate 119904 times both sides of (80) with respect to 119909 thatis

119910(119904)

119894(119909) = 119891

(119904)

119894(119909) +

119899

sum

119895=1

int

1

0

119870(119904)

119894119895(119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899 119904 = 1 2 119898

(84)

where119870(119904)119894119895(119909 119905) = 120597

(119904)119870119894119895(119909 119905)120597119909

(119904) 119904 = 1 2 119898Applying the mean value theorem for integral in (84) we

have

119910(119904)

119894(119909) minus [

[

119899

sum

119895=1

int

1

0

119870(119904)

119894119895(119909 119905) 119889119905]

]

119910119895 (119909) asymp 119891(119904)

119894(119909)

119894 = 1 2 119899 119904 = 1 2 119898

(85)

Now (83) combined with (85) becomes a linear systemof algebraic equations that can be solved analytically ornumerically

43 Block-Pulse Functions for the Solution of Fredholm IntegralEquation In this section Block-Pulse functions (BPF) havebeen utilized for the solution of system of Fredholm integralequations [6]

An119898-set of BPF is defined as follows

Φ119894 (119905) =

1 (119894 minus 1)119879

119898le 119905 lt 119894

119879

119898

0 otherwise(86)

with 119905 isin [0 119879) 119879119898 = ℎ and 119894 = 1 2 119898


431 Properties of BPF

(1) Disjointness One has

Φ119894 (119905) Φ119895 (119905) = Φ119894 (119905) 119894 = 119895

0 119894 = 119895(87)

119894 119895 = 1 2 119898 This property is obtained from definitionof BPF

(2) Orthogonality One has

int

119879

0

Φ119894 (119905) Φ119895 (119905) 119889119905 = ℎ 119894 = 119895

0 119894 = 119895(88)

119905 isin [0 119879) 119894 119895 = 1 2 119898 This property is obtained fromthe disjointness property

(3) Completeness For every119891 isin 1198712 Φ is complete if intΦ119891 =0 then 119891 = 0 almost everywhere Because of completeness ofΦ we have

int

119879

0

1198912(119905) 119889119905 =

infin

sum

119894=1

1198912

119894

1003817100381710038171003817Φ119894(119905)10038171003817100381710038172 (89)

for every real bounded function 119891(119905) which is square inte-grable in the interval 119905 isin [0 119879) and 119891119894 = (1ℎ)119891(119905)Φ119894(119905)119889119905

432 Function Approximation The orthogonality propertyof BPF is the basis of expanding functions into their Block-Pulse series For every 119891(119905) isin 1198712(119877)

119891 (119905) =

119898

sum

119894=1

119891119894Φ119894 (119905) (90)

where 119891119894 is the coefficient of Block-Pulse function withrespect to 119894th Block-Pulse functionΦ119894(119905)

The criterion of this approximation is that mean squareerror between 119891(119905) and its expansion is minimum

120576 =1

119879int

119879

0

(119891(119905) minus

119898

sum

119895=1

119891119895Φ119895(119905))

2

119889119905 (91)

so that we can evaluate Block-Pulse coefficients

Now 120597120576

120597119891119894

= minus2

119879int

119879

0

(119891 (119905) minus

119898

sum

119895=1

119891119895Φ119895 (119905))Φ119894 (119905) 119889119905 = 0

997904rArr 119891119894 =1

ℎint

119879

0

119891 (119905)Φ119894 (119905) 119889119905 (using orthogonal property)

(92)

In the matrix form we obtain the following from (90) asfollow

119891 (119905) =

119898

sum

119894=1

119891119894Φ119894 (119905) = 119865119879Φ (119905) = Φ

119879119865

where 119865 = [1198911 1198912 119891119898]119879

Φ (119905) = [Φ1 (119905) Φ2 (119905) Φ119898 (119905)]119879

(93)

Now let119870(119905 119904) be two-variable function defined on 119905 isin [0 119879)and 119904 isin [0 1) then 119870(119905 119904) can be expanded to BPF as

119870 (119905 119904) = Φ119879(119905) 119870Ψ (119904) (94)

whereΦ(119905) andΨ(119904) are1198981 and1198982 dimensional Block-Pulsefunction vectors and 119896 is a 1198981 times 1198982 Block-Pulse coefficientmatrix

There are two different cases of multiplication of two BPFThe first case is

Φ (119905)Φ119879(119905) = (

Φ1 (119905) 0 sdot sdot sdot 0

0 Φ2 (119905) sdot sdot sdot 0

d

0 0 sdot sdot sdot Φ119898 (119905)

) (95)

It is obtained from disjointness property of BPF It is adiagonal matrix with119898 Block-Pulse functions

The second case is

Φ119879(119905) Φ (119905) = 1 (96)

because sum119898119894=1(Φ119894(119905))

2= sum119898

119894=1Φ119894(119905) = 1

Operational Matrix of Integration BPF integration propertycan be expressed by an operational equation as

int

119879

0

Φ (119905) 119889119905 = 119875Φ (119905) (97)

where

Φ (119905) = [Φ1 (119905) Φ2 (119905) Φ119898 (119905)]119879 (98)

A general formula for 119875119898times119898 can be written as

119875 =1

2(

1 2 2 sdot sdot sdot 2



d


) (99)

By using this matrix we can express the integral of a function119891(119905) into its Block-Pulse series

int

119905

0

119891 (119905) 119889119905 = int

119905

0

119865119879Φ (119905) 119889119905 = 119865

119879119875Φ (119905) (100)

433 Solution for Linear Integral Equations System Considerthe integral equations system from (55) as follows

119899

sum

119895=1

119910119895 (119909) = 119891119894 (119909) +

119899

sum

119895=1

int

120573

120572

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899

(101)

Block-Pulse coefficients of119910119895(119909) 119895 = 1 2 119899 in the interval119909 isin [120572 120573) can be determined from the known functions119891119894(119909) 119894 = 1 2 119899 and the kernels 119870119894119895(119909 119905) 119894 119895 = 1 2 119899Usually we consider 120572 = 0 to facilitie the use of Block-Pulse


functions In case 120572 = 0 we set119883 = ((119909minus120572)(120573minus120572))119879 where119879 = 119898ℎ

We approximate 119891119894(119909) 119910119895(119909) 119870119894119895(119909 119905) by its BPF asfollows

119891119894 (119909) asymp 119865119879

119894Φ (119909)

119910119895 (119909) asymp 119884119879

119895Φ (119909)

119870119894119895 (119909 119905) asymp Φ119879(119905) 119870119894119895Φ (119909)

(102)

where 119865119894 119884119895 and 119870119894119895 are defined in Section 432 andsubstituting (102) into (101) we have

119899

sum

119895=1

119884119879

119895Φ (119909) = 119865

119879

119894Φ (119909)

+

119899

sum

119895=1

int

119898ℎ

0

119884119879

119895Φ (119905)Φ

119879(119905) 119870119894119895Φ (119909) 119889119905

119894 = 1 2 119899

(103)

119899

sum

119895=1

119884119879

119895Φ (119909) = 119865

119879

119894Φ (119909) +

119899

sum

119895=1

119884119879

119895ℎ119868119870119894119895Φ (119909)

119894 = 1 2 119899

(104)

since

int

119898ℎ

0

Φ (119905)Φ119879(119905) 119889119905 = ℎ119868 (105)

From (104) we get

119899

sum

119895=1

(119868 minus ℎ119870119879

119894119895) 119884119895 = 119865119894 119894 = 1 2 119899 (106)

Set 119860 119894119895 = 119868 minus ℎ119870119879

119894119895 then we have from (106)

119899

sum

119895=1

119860 119894119895119884119895 = 119865119894 119894 = 1 2 119899 (107)

which is a linear system

(

(

11986011 11986012 1198601119899

11986021 11986022 1198602119899

1198601198991 1198601198992 119860119899119899

)

)

(

(

1198841

1198842

119884119899

)

)

=(

(

1198651

1198652

119865119899

)

)

(108)

After solving the above system we can find 119884119895 119895 = 1 2 119899and hence obtain the solutions 119910119895 = Φ

119879119884119895 119895 = 1 2 119899

5 Numerical Methods for NonlinearFredholm-Hammerstein Integral Equation

We consider the second kind nonlinear Fredholm integralequation of the following form

119906 (119909) = 119891 (119909) + int

1

0

119870 (119909 119905) 119865 (119905 119906 (119905)) 119889119905

0 le 119909 le 1

(109)


51 B-Spline Wavelet Method In this section nonlinearFredholm integral equation of second kind of the form givenin (109) has been solved by using B-spline wavelets [11]

B-spline scaling and wavelet functions in the interval[0 1] and function approximation have been defined inSections 311 and 312 respectively

First we assume that

119910 (119909) = 119865 (119909 119906 (119909))

0 le 119909 le 1

(110)

Now from (16) we can approximate the functions 119906(119909) and119910(119909) as

119906 (119909) = 119860119879Ψ (119909)

119910 (119909) = 119861119879Ψ (119909)

(111)

where 119860 and 119861 are (2119872+1 +119898minus 1) times 1 column vectors similarto 119862 defined in (17)

Again by using dual of the wavelet functions we canapproximate the functions 119891(119909) and119870(119909 119905) as follows

119865 (119909) = 119863119879Ψ (119909)

119870 (119909 119905) = Ψ119879(119905) ΘΨ (119909)

(112)

where

Θ(119894119895) = int

1

0

[int

1

0

119870 (119909 119905) Ψ119894 (119905) 119889119905]Ψ119895 (119909) 119889119909 (113)

From (110)ndash(112) we get

int

1

0

119870 (119909 119905) 119865 (119905 119906 (119905)) 119889119905

= int

1

0

119861119879Ψ (119905) Ψ

119879(119905) ΘΨ (119909) 119889119905

= 119861119879[int

1

0

Ψ (119905) Ψ119879(119905) 119889119905]ΘΨ (119909)

= 119861119879ΘΨ (119909) since int

1

0

Ψ (119905) Ψ119879(119905) 119889119905 = 119868

(114)


Applying (110)ndash(114) in (109) we get

119860119879Ψ (119909) = 119863

119879Ψ (119909) + 119861

119879ΘΨ (119909) (115)

Multiplying (115) by Ψ119879(119909) both sides from the right andintegrating both sides with respect to 119909 from 0 to 1 we have

119860119879119875 = 119863

119879+ 119861119879Θ

119860119879119875 minus 119863

119879minus 119861119879Θ = 0

(116)

where 119875 is a (2119872+1 + 119898 minus 1) times (2119872+1 + 119898 minus 1) square matrixgiven by

119875 = int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = [

1198751

1198752]

int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = 119868

(117)

Equation (116) gives a system of (2119872+1 + 119898 minus 1) algebraicequations with 2(2119872+1+119898minus1) unknowns for119860 and 119861 vectorsgiven in (111)

To find the solution 119906(119909) in (111) we first utilize thefollowing equation

119865 (119909 119860119879Ψ (119909)) = 119861

119879Ψ (119909) (118)

with the collocation points 119909119894 = (119894 minus 1)(2119872+1

+119898minus2) where119894 = 1 2 2

119872+1+ 119898 minus 1

Equation (118) gives a system of (2119872+1 + 119898 minus 1) algebraicequations with 2(2119872+1+119898minus1) unknowns for119860 and119861 vectorsgiven in (111)

Combining (116) and (118) we have a total of 2(2119872+1 +119898 minus 1) system of algebraic equations with 2(2119872+1 + 119898 minus

1) unknowns for 119860 and 119861 Solving those equations for theunknown coefficients in the vectors 119860 and 119861 we can obtainthe solution 119906(119909) = 119860119879Ψ(119909)

52 Quadrature Method Applied to Fredholm Integral Equa-tion In this section Quadrature method has been appliedto solve nonlinear Fredholm-Hammerstein integral equation[10]

The quadrature methods like Simpson rule and modifiedtrapezoidmethod are applied for solving a definite integral asfollows

521 Simpsonrsquos Rule One has

int

119887

119886

119891 (119909) 119889119909 =

119899minus1

sum

119894=1

int

119909119894+1

119909119894minus1

119891 (119909) 119889119909

=ℎ

3119891 (119886) +

4ℎ

3

1198992

sum

119894=1

119891 (1199092119894minus1)

+2ℎ

3

(119899minus1)2

sum

119894=1

119891 (1199092119894)

+ℎ

3119891 (119887)

minus(119887 minus 119886)

180ℎ4119891(4)(120578)

(119)

522 Modified Trapezoid Rule One has

int

119887

119886

119891 (119909) 119889119909 =

119899

sum

119894=1

int

119909119894

119909119894minus1

119891 (119909) 119889119909

=ℎ

2119891 (119886) + ℎ

119899minus1

sum

119894=1

119891 (119909119894)

+ℎ

2119891 (119887)

+ℎ2

12[1198911015840(119886) minus 119891

1015840(119887)]

(120)

Consider the nonlinear Fredholm integral equation of secondkind defined in (109) as follows

119906 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) 119865 (119906 (119905)) 119889119905

119886 le 119909 le 119887

(121)

For solving (121) we approximate the right-hand integral of(121) with Simpsonrsquos rule and modified trapezoid rule thenwe get the following

523 Simpsonrsquos Rule One has119906 (119909) = 119891 (119909)

+ℎ

3

[

[

119870 (119909 1199050) 119865 (1199060)

+ 4

1198992

sum

119895=1

119870(119909 1199052119895minus1) 119865 (1199062119895minus1)

+ 2

(1198992)minus1

sum

119895=1

119870(119909 1199052119895) 119865 (1199062119895)

+ 119870 (119909 119905119899) 119865 (119906119899)]

]

(122)

Hence for 119909 = 1199090 1199091 119909119899 and 119905 = 1199050 1199051 119905119899 in (122) wehave119906 (119909119894) = 119891 (119909119894)

+ℎ

3

[

[

119870 (119909119894 1199050) 119865 (1199060)

+ 4

1198992

sum

119895=1

119870(119909119894 1199052119895minus1) 119865 (1199062119895minus1)


+ 2

(1198992)minus1

sum

119895=1

119870(119909119894 1199052119895) 119865 (1199062119895)

+119870 (119909119894 119905119899) 119865 (119906119899)]

]

(123)

Equation (123) is a nonlinear system of equations and bysolving (123) we obtain the unknowns119906(119909119894) for 119894 = 0 1 119899


119906 (119909) = 119891 (119909)

+ℎ

2119870 (119909 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909 119905119895) 119865 (119906119895)

+ℎ

2119870 (119909 119905119899) 119865 (119906119899)

+ℎ2

12[119869 (119909 1199050) 119865 (1199060)

+ 119870 (119909 1199050) 1199061015840

01198651015840(1199060)

minus 119869 (119909 119905119899) 119865 (119906119899)

minus119870 (119909 119905119899) 1199061015840

1198991198651015840(119906119899)]

(124)

where 119869(119909 119905) = 120597119870(119909 119905)120597119905For 119909 = 1199090 1199091 119909119899 and 119905 = 1199050 1199051 119905119899 in (124) we

have

119906 (119909119894) = 119891 (119909119894)

+ℎ

2119870 (119909119894 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909119894 119905119895) 119865 (119906119895)

+ℎ

2119870 (119909119894 119905119899) 119865 (119906119899)

+ℎ2

12[119869 (119909119894 1199050) 119865 (1199060)

+ 119870 (119909119894 1199050) 1199061015840

01198651015840(1199060)

minus 119869 (119909119894 119905119899) 119865 (119906119899)

minus119870 (119909119894 119905119899) 1199061015840

1198991198651015840(119906119899)]

(125)

for 119894 = 0 1 119899

This is a system of (119899+1) equations and (119899+3) unknownsBy taking derivative from (121) and setting 119867(119909 119905) =

120597119870(119909 119905)120597119909 we obtain

1199061015840(119909) = 119891

1015840(119909) + int

119887

119886

119867(119909 119905) 119865 (119906 (119905)) 119889119905

119886 le 119909 le 119887

(126)

If 119906 is a solution of (121) then it is also solution of (126) Byusing trapezoid rule for (126) and replacing 119909 = 119909119894 we get

1199061015840(119909119894) = 119891

1015840(119909119894)

+ℎ

2119867 (119909119894 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(119909119894 119905119895) 119865 (119906119895)

+ℎ

2119867 (119909119894 119905119899) 119865 (119906119899)

(127)

for 119894 = 0 1 119899 In case of 119894 = 0 119899 from system (127) weobtain two equations

Now (127) combined with (125) generates the nonlinearsystem of equations as follows

119906 (119909119894) = (ℎ

2119870 (119909119894 1199050) +

ℎ2

12119869 (119909119894 1199050))119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909119894 119905119895) 119865 (119906119895)

+ (ℎ

2119870 (119909119894 119905119899) minus

ℎ2

12119869 (119909119894 119905119899))119865 (119906119899)

+ℎ2

12(119870 (119909119894 1199050) 119906

1015840

01198651015840(1199060)

minus119870 (119909119894 119905119899) 1199061015840

1198991198651015840(119906119899))

1199061015840(1199090) = 119891

1015840(1199090)

+ℎ

2119867 (1199090 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(1199090 119905119895) 119865 (119906119895)

+ℎ

2119867 (1199090 119905119899) 119865 (119906119899)


1199061015840(119909119899) = 119891

1015840(119909119899)

+ℎ

2119867 (119909119899 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(119909119899 119905119895) 119865 (119906119895)

+ℎ

2119867 (119909119899 119905119899) 119865 (119906119899)

(128)

By solving this system with (119899 + 3) nonlinear equations and(119899 + 3) unknowns we can obtain the solution of (109)

53Wavelet GalerkinMethod In this section the continuousLegendre wavelets [12] constructed on the interval [0 1] areapplied to solve the nonlinear Fredholm integral equation ofthe second kindThe nonlinear part of the integral equation isapproximated by Legendre wavelets and the nonlinear inte-gral equation is reduced to a system of nonlinear equations

We have the following family of continuous wavelets withdilation parameter 119886 and the translation parameter 119887

120595119886119887 (119905) = |119886|minus12120595(

119905 minus 119887

119886)

119886 119887 isin 119877 119886 = 0

(129)

Legendre wavelets 120595119898119899(119905) = 120595(119896 119899119898 119905) have four argu-ments 119896 = 2 3 119899 = 2119899 minus 1 119899 = 1 2 2119896minus1 119898 is theorder for Legendre polynomials and 119905 is the normalized time

Legendre wavelets are defined on [0 1) by

120595119898119899 (119905)

=

(119898 +1

2)

12

21198962119871119898 (2

119896119905 minus 119899)

119899 minus 1

2119896le 119905 lt

119899 + 1

2119896

0 otherwise(130)

where 119871119898(119905) are the well-known Legendre polynomials oforder m which are orthogonal with respect to the weightfunction119908(119905) = 1 and satisfy the following recursive formula

1198710 (119905) = 1

1198711 (119905) = 119905

119871119898+1 (119905) =2119898 + 1

119898 + 1119905119871119898 (119905)

minus119898

119898 + 1119871119898minus1 (119905) 119898 = 1 2 3

(131)

The set of Legendre wavelets are an orthonormal set

531 Function Approximation A function 119891(119909) isin 1198712[0 1]can be expanded as

119891 (119909) =

infin

sum

119899=1

infin

sum

119898=0

119888119899119898120595119899119898 (119909) (132)

where

119888119899119898 = ⟨119891 (119909) 120595119899119898 (119909)⟩ (133)

If the infinite series in (132) is truncated then (132) can bewritten as

119891 (119909) asymp

2119896minus1

sum

119899=1

119872minus1

sum

119898=0

119888119899119898120595119899119898 (119909) = 119862119879Ψ (119909) (134)

where 119862 and Ψ(119909) are 2119896minus1119872times 1matrices given by

119862 = [11988810 11988811 1198881119872minus1 11988820

1198882119872minus1 1198882119896minus1 0 1198882119896minus1 119872minus1]119879

(135)

Ψ (119909) = [12059510 (119909) 1205951119872minus1 (119909)

12059520 (119909) 1205952119872minus1 (119909)

1205952119896minus1 0 (119909) 1205952119896minus1 119872minus1 (119909)]119879

(136)

Similarly a function 119896(119909 119905) isin 1198712([0 1] times [0 1]) can be

approximated as

119896 (119909 119905) asymp Ψ119879(119905) 119870Ψ (119909) (137)

where119870 is (2119896minus1119872times 2119896minus1119872)matrix with

119870119894119895 = ⟨120595119894 (119905) ⟨119896 (119909 119905) 120595119895 (119909)⟩⟩ (138)

Also the integer power of a function can be approximated as

[119910 (119909)]119901= [119884119879Ψ (119909)]

119901

= 119884lowast

119901

119879Ψ (119909) (139)

where 119884lowast119901is a column vector whose elements are nonlinear

combinations of the elements of the vector 119884 119884lowast119901is called the

operational vector of the 119901th power of the function 119910(119909)

532The Operational Matrices The integration of the vectorΨ(119909) defined in (136) can be obtained as

int

119905

0

Ψ(1199051015840) 1198891199051015840= 119875Ψ (119905) (140)

where 119875 is the (2119896minus1119872 times 2119896minus1119872) operational matrix for

integration and is given in [23] as

119875 =

[[[[[[

[

119871 119867 sdot sdot sdot 119867 119867

0 119871 sdot sdot sdot 119867 119867

d

0 0 sdot sdot sdot 119871 119867

0 0 sdot sdot sdot 0 119871

]]]]]]

]

(141)

In (141)119867 and 119871 are (119872 times119872)matrices given in [23] as


119867 =1

2119896

[[[[

[

2 0 sdot sdot sdot 0


d


]]]]

]

119871 =1

2119896

[[[[[[[[[[[[[[[[[[[[[[[

[

11

radic30 0 sdot sdot sdot 0 0

minusradic3

30

radic3

3radic50 sdot sdot sdot 0 0

0 minusradic5

5radic30

radic5

5radic7sdot sdot sdot 0 0

0 0 minusradic7


d

0 0 0 0 sdot sdot sdot 0radic2119872 minus 3

(2119872 minus 3)radic2119872 minus 1

0 0 0 0 sdot sdot sdotminusradic2119872 minus 1


]]]]]]]]]]]]]]]]]]]]]]]

]

(142)

The integration of the product of two Legendre waveletsvector functions is obtained as

int

1

0

Ψ (119905)Ψ119879(119905) 119889119905 = 119868 (143)

where 119868 is an identity matrixThe product of two Legendre wavelet vector functions is

defined as

Ψ (119905) Ψ119879(119905) 119862 = 119862

119879Ψ (119905) (144)

where 119862 is a vector given in (135) and 119862 is (2119896minus1119872 times 2119896minus1119872)

matrix which is called the product operation of Legendrewavelet vector functions [23 24]

533 Solution of Fredholm Integral Equation of Second KindConsider the nonlinear Fredholm-Hammerstein integralequation of second kind of the form

119910 (119909) = 119891 (119909) + int

1

0

119896 (119909 119905) [119910 (119905)]119901119889119905 (145)

where 119891 isin 1198712[0 1] 119896 isin 1198712([0 1] times [0 1]) 119910 is an unknownfunction and 119901 is a positive integer

We can approximate the following functions as

119891 (119909) asymp 119865119879Ψ (119909)

119910 (119909) asymp 119884119879Ψ (119909)

119896 (119909 119905) asymp Ψ119879(119905) 119870Ψ (119909)

[119910 (119909)]119901asymp 119884lowast119879Ψ (119909)

(146)

Substituting (146) into (145) we have

119884119879Ψ (119909) = 119865

119879Ψ (119909)

+ int

1

0

119884lowast119879Ψ (119905) Ψ

119879(119905) 119870Ψ (119909) 119889119905

= 119865119879Ψ (119909)

+ 119884lowast119879(int

1

0

Ψ (119905) Ψ119879(119905) 119889119905)119870Ψ (119909)

= 119865119879Ψ (119909) + 119884

lowast119879119870Ψ (119909)

= (119865119879+ 119884lowast119879119870)Ψ (119909)

997904rArr 119884119879minus 119884lowast119879119870 minus 119865

119879= 0

(147)

Equation (147) is a system of algebraic equations Solving(147) we can obtain the solution 119910(119909) asymp 119884119879Ψ(119909)

54 Homotopy Perturbation Method Consider the followingnonlinear Fredholm integral equation of second kind of theform

119906 (119909) = 119891 (119909) + int

1

0

119870 (119909 119905) 119865 (119906 (119905)) 119889119905

0 le 119909 le 1

(148)

For solving (148) by Homotopy perturbation method (HPM)[14ndash16] we consider (148) as

119871 (119906) = 119906 (119909) minus 119891 (119909) minus int

1

0

119870 (119909 119905) 119865 (119906 (119905)) 119889119905 = 0 (149)


As a possible remedy we can define119867(119906 119901) by

119867(119906 0) = 119873 (119906)

119867 (119906 1) = 119871 (119906)

(150)

where119873(119906) is an integral operator with known solution 1199060We may choose a convex homotopy by

119867(119906 119901) = (1 minus 119901)119873 (119906) + 119901119871 (119906) = 0 (151)

and continuously trace an implicitly defined curve from astarting point 119867(1199060 0) to a solution function 119867(119880 1) Theembedding parameter 119901 monotonically increases from zeroto unit as the trivial problem 119871(119906) = 0 The embeddingparameter 119901 isin (0 1] can be considered as an expandingparameter The HPM uses the homotopy parameter 119901 as anexpanding parameter that is

119906 = 1199060 + 1199011199061 + 11990121199062 + sdot sdot sdot (152a)

When 119901 rarr 1 (152a) corresponding to (151) become theapproximate solution of (149) as follows

119880 = lim119901rarr1

119906 = 1199060 + 1199061 + 1199062 + sdot sdot sdot (152b)

The series in (152b) converges in most cases and the rate ofconvergence depends on 119871(119906) [14]

Consider

119873(119906) = 119906 (119909) minus 119891 (119909) (153)

The nonlinear term 119865(119906) can be expressed in He polynomials[25] as

119865 (119906) =

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898)

= 1198670 (1199060) + 1199011198671 (1199060 1199061)

+ sdot sdot sdot + 119901119898119867119898 (1199060 1199061 119906119898) + sdot sdot sdot

(154)

where119867119898 (1199060 1199061 119906119898)

=1

119898

120597119898

120597119901119898(119865(

119898

sum

119896=0

119901119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816119901=0

119898 ge 0

(155)

Substituting (152a) (153) and (154) into (151) we have

(1 minus 119901) ((1199060 + 1199011199061 + sdot sdot sdot ) minus 119891 (119909))

+ 119901( (1199060 + 1199011199061 + sdot sdot sdot ) minus 119891 (119909)

minusint

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905) = 0

997904rArr (1199060 + 1199011199061 + sdot sdot sdot ) minus 119891 (119909)

minus 119901int

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905 = 0

(156)

Equating the termswith identical power of119901 in (156) we have

1199010 1199060 (119909) minus 119891 (119909) = 0 997904rArr 1199060 (119909) = 119891 (119909)

1199011 1199061 (119909) minus int

1

0

119870 (119909 119905)1198670119889119905 = 0 997904rArr 1199061 (119909)

= int

1

0

119870 (119909 119905)1198670119889119905

1199012 1199062 (119909) minus int

1

0

119870 (119909 119905)1198671119889119905 = 0 997904rArr 1199062 (119909)

= int

1

0

119870 (119909 119905)1198671119889119905

(157)

and in general form we have

1199060 (119909) = 119891 (119909)

119906119899+1 (119909) = int

1

0

119870 (119909 119905)119867119899119889119905 119899 = 0 1 2

(158)

Hence we can obtain the approximate solution of aforesaidequation (148) from (152b)

55 Adomian Decomposition Method Adomian decomposi-tion method (ADM) [16ndash18] has been applied to a wide classof functional equations This method gives the solution asan infinite series usually converging to an accurate solutionLet us consider the nonlinear Fredholm integral equation ofsecond kind as follows

119906 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) (119871119906 (119905) + 119873119906 (119905)) 119889119905

119886 le 119909 le 119887

(159)

where 119871(119906(119905)) and119873(119906(119905)) are the linear and nonlinear termsrespectively

The Adomian decomposition method (ADM) consists ofrepresenting 119906(119909) as a series

119906 (119909) =

infin

sum

119898=0

119906119898 (119909) (160)

In the view of ADM the nonlinear term 119873119906 can be repre-sented as

119873119906 =

infin

sum

119899=0

119860119899 (161)

where 119860119899 =1

119899(120597119899

120597120582119899119873(

infin

sum

119896=0

120582119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816120582=0

(162)


Now substituting (160) and (161) into (159) we have

infin

sum

119898=0

119906119898 (119909) = 119891 (119909)

+ int

119887

119886

119870 (119909 119905) (119871(

infin

sum

119898=0

119906119898 (119905)) +

infin

sum

119898=0

119860119898)119889119905

(163)

and then ADM uses the recursive relations

1199060 (119909) = 119891 (119909)

119906119898 (119909) = int

119887

119886

119870 (119909 119905) (119871 (119906119898minus1 (119905)) + 119860119898minus1 (119905)) 119889119905

119898 ge 1

(164)

where 119860119898 is so-called Adomian polynomialTherefore we obtain the 119899-terms approximate solution as

120593119899 = 1199060 + 1199061 + sdot sdot sdot + 119906119899 (165)

with

119906 (119909) = lim119899rarrinfin

120593119899 (166)

6 Conclusion and Discussion

In this work we have examined many numerical methodsto solve Fredholm integral equations Using these methodsexcept variational iteration method the Fredholm integralequations have been reduced to a system of algebraic equa-tions and this system can be easily solved by any usualmethods In this work we have applied compactly supportedsemiorthogonal B-spline wavelets along with their dualwavelets for solving both linear and nonlinear Fredholm inte-gral equations of second kindThe problem has been reducedto solve a system of algebraic equations In order to increasethe accuracy of the approximate solution it is necessary toapply higher-order B-spline wavelet method The method ofmoments based on compactly supported semiorthogonal B-spline wavelets via Galerkin method has been used to solveFredholm integral equation of second kind This methoddetermines a strong reduction in the computation time andmemory requirement in inverting the matrix Variationaliteration method has been successfully applied to find theapproximate solution of Fredholm integral equation of bothlinear and nonlinear types Taylor series expansion methodreduces the system of integral equations to a linear system ofordinary differential equation After including the requiredboundary conditions this system reduces to a system ofalgebraic equations that can be solved easily Block-Pulsefunctions and Haar wavelet method can be applied to thesystem of Fredholm integral equations by reducing into asystem of algebraic equations These methods give moreaccuracy if we increase their order Quadrature method canbe applied to solve the nonlinear Fredholm-Hammersteinintegral equation of second kind by reducing it to a system ofalgebraic equations Homotopy perturbationmethod (HPM)

and Adomian decomposition method (ADM) can be alsoapplied to approximate the solution of nonlinear Fredholmintegral equation of second kind The solutions obtained byHPM and ADM are applicable for not only weakly nonlinearequations but also strong onesThe approximate solutions bythese aforesaid methods highly agree with exact solutions

References

[1] A-M Wazwaz Linear and Nonlinear Integral Equations Meth-ods and Applications Springer New York NY USA 2011

[2] K Maleknejad and M N Sahlan ldquoThe method of momentsfor solution of second kind Fredholm integral equations basedon B-spline waveletsrdquo International Journal of Computer Math-ematics vol 87 no 7 pp 1602ndash1616 2010

[3] J-H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

[4] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006

[5] J-H He ldquoVariational iteration methodmdashsome recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[6] K Maleknejad M Shahrezaee and H Khatami ldquoNumericalsolution of integral equations system of the second kind byblock-pulse functionsrdquo Applied Mathematics and Computationvol 166 no 1 pp 15ndash24 2005

[7] K Maleknejad N Aghazadeh and M Rabbani ldquoNumericalsolution of second kind Fredholm integral equations system byusing a Taylor-series expansion methodrdquo Applied Mathematicsand Computation vol 175 no 2 pp 1229ndash1234 2006

[8] K Maleknejad and F Mirzaee ldquoNumerical solution of linearFredholm integral equations system by rationalized Haar func-tions methodrdquo International Journal of Computer Mathematicsvol 80 no 11 pp 1397ndash1405 2003

[9] X-Y Lin J-S Leng and Y-J Lu ldquoA Haar wavelet solution toFredholm equationsrdquo in Proceedings of the International Con-ference on Computational Intelligence and Software Engineering(CiSE rsquo09) pp 1ndash4 Wuhan China December 2009

[10] M J Emamzadeh and M T Kajani ldquoNonlinear Fredholmintegral equation of the second kind with quadrature methodsrdquoJournal of Mathematical Extension vol 4 no 2 pp 51ndash58 2010

[11] M Lakestani M Razzaghi and M Dehghan ldquoSolution ofnonlinear Fredholm-Hammerstein integral equations by usingsemiorthogonal spline waveletsrdquo Mathematical Problems inEngineering vol 2005 no 1 pp 113ndash121 2005

[12] Y Mahmoudi ldquoWavelet Galerkin method for numerical solu-tion of nonlinear integral equationrdquo Applied Mathematics andComputation vol 167 no 2 pp 1119ndash1129 2005

[13] J Biazar andH Ebrahimi ldquoIterationmethod for Fredholm inte-gral equations of second kindrdquo Iranian Journal of Optimizationvol 1 pp 13ndash23 2009

[14] D D Ganji G A Afrouzi H Hosseinzadeh and R ATalarposhti ldquoApplication of homotopy-perturbation method tothe second kind of nonlinear integral equationsrdquo Physics LettersA vol 371 no 1-2 pp 20ndash25 2007

[15] M Javidi and A Golbabai ldquoModified homotopy perturbationmethod for solving non-linear Fredholm integral equationsrdquoChaos Solitons and Fractals vol 40 no 3 pp 1408ndash1412 2009


[16] S Abbasbandy ldquoNumerical solutions of the integral equationshomotopy perturbation method and Adomianrsquos decompositionmethodrdquo Applied Mathematics and Computation vol 173 no 1pp 493ndash500 2006

[17] E Babolian J Biazar and A R Vahidi ldquoThe decompositionmethod applied to systems of Fredholm integral equations ofthe second kindrdquo Applied Mathematics and Computation vol148 no 2 pp 443ndash452 2004

[18] S Abbasbandy and E Shivanian ldquoA new analytical technique tosolve Fredholmrsquos integral equationsrdquoNumerical Algorithms vol56 no 1 pp 27ndash43 2011

[19] J C Goswami A K Chan and C K Chui ldquoOn solving first-kind integral equations using wavelets on a bounded intervalrdquoIEEE Transactions on Antennas and Propagation vol 43 no 6pp 614ndash622 1995

[20] M Lakestani M Razzaghi and M Dehghan ldquoSemiorthogonalspline wavelets approximation for fredholm integro-differentialequationsrdquo Mathematical Problems in Engineering vol 2006Article ID 96184 12 pages 2006

[21] C K Chui An Introduction to Wavelets vol 1 of WaveletAnalysis and its Applications Academic Press Boston MassUSA 1992

[22] J C Goswami and A K Chan Fundamentals of Wavelets JohnWiley amp Sons Hoboken NJ USA 2nd edition 2011

[23] M Razzaghi and S Yousefi ldquoThe Legendre wavelets operationalmatrix of integrationrdquo International Journal of Systems Sciencevol 32 no 4 pp 495ndash502 2001

[24] M Razzaghi and S Yousefi ldquoLegendre wavelets direct methodfor variational problemsrdquo Mathematics and Computers in Sim-ulation vol 53 no 3 pp 185ndash192 2000

[25] A Ghorbani ldquoBeyondAdomian polynomials He polynomialsrdquoChaos Solitons and Fractals vol 39 no 3 pp 1486ndash1492 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom

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Volume 2014

Applied MathematicsJournal of


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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


2 Fredholm Integral Equation

The general form of linear Fredholm integral equation isdefined as follows

119892 (119909) 119910 (119909) = 119891 (119909) + 120582int

119887

119886

119870 (119909 119905) 119910 (119905) 119889119905 (1)

where 119886 and 119887 are both constants 119891(119909) 119892(119909) and 119870(119909 119905)are known functions while 119910(119909) is unknown function 120582(nonzero parameter) is called eigenvalue of the integralequation The function 119870(119909 119905) is known as kernel of theintegral equation

21 Fredholm Integral Equation of First Kind The linearintegral equation is of form (by setting 119892(119909) = 0 in (1))

119891 (119909) + 120582int

119887

119886

119870 (119909 119905) 119910 (119905) 119889119905 = 0 (2)

Equation (2) is known as Fredholm integral equation of firstkind

22 Fredholm Integral Equation of Second Kind The linearintegral equation is of form (by setting 119892(119909) = 1 in (1))

119910 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) 119910 (119905) 119889119905 (3)

Equation (3) is known as Fredholm integral equation ofsecond kind

23 System of Linear Fredholm Integral Equations The gen-eral form of system of linear Fredholm integral equations ofsecond kind is defined as follows

119899

sum

119895=1

119892119894119895119910119895 (119909) = 119891119894 (119909) +

119899

sum

119895=1

int

119887

119886

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899

(4)


24 Nonlinear Fredholm-Hammerstein Integral Equation ofSecond Kind Nonlinear Fredholm-Hammerstein integralequation of second kind is defined as follows

119910 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) 119865 (119910 (119905)) 119889119905 (5)


25 System of Nonlinear Fredholm Integral Equations Systemof nonlinear Fredholm integral equations of second kind isdefined as follows119899

sum

119895=1

119892119894119895119910119895 (119909) = 119891119894 (119909) +

119899

sum

119895=1

int

119887

119886

119870119894119895 (119909 119905) 119865119894119895 (119905 119910119895 (119905)) 119889119905

119894 = 1 2 119899

(6)


3 Numerical Methods for Linear FredholmIntegral Equation of Second Kind

Consider the following Fredholm integral equation of secondkind defined in (3)

119910 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) 119910 (119905) 119889119905 119886 le 119909 le 119887 (7)

where 119870(119909 119905) and 119892(119909) are known functions and 119910(119909) isunknown function to be determined

31 B-Spline Wavelet Method

311 B-Spline Scaling and Wavelet Functions on the Interval[0 1] Semiorthogonal wavelets using B-spline are speciallyconstructed for the bounded interval and this wavelet canbe represented in a closed form This provides a compactsupport Semiorthogonal wavelets form the basis in the space1198712(119877)Using this basis an arbitrary function in 1198712(119877) can be

expressed as the wavelet series For the finite interval [0 1]the wavelet series cannot be completely presented by usingthis basisThis is because supports of some basis are truncatedat the left or right end points of the interval Hence a specialbasis has to be introduced into the wavelet expansion on thefinite intervalThese functions are referred to as the boundaryscaling functions and boundary wavelet functions

Let119898 and 119899 be two positive integers and let

119886 = 119909minus119898+1 = sdot sdot sdot = 1199090 lt 1199091

lt sdot sdot sdot lt 119909119899 = 119909119899+1

= sdot sdot sdot = 119909119899+119898minus1 = 119887

(8)

be an equally spaced knots sequence The functions

119861119898119895119883 (119909) =119909 minus 119909119895

119909119895+119898minus1 minus 119909119895

119861119898minus1119895119883 (119909)

+119909119895+119898 minus 119909

119909119895+119898 minus 119909119895+1

119861119898minus1119895+1119883 (119909)

119895 = minus119898 + 1 119899 minus 1

1198611119895119883 (119909) = 1 119909 isin [119909119895 119909119895+1)

0 otherwise

(9)



119899+119898minus1

119894=minus119898+1and Supp119861119898119895119883(119909) =




119883(119895)=

119909(119895)


(119895)

0= 0

119909(119895)

119896=119896

2119895 119896 = 1 119899 minus 1

119909(119895)

119899= sdot sdot sdot = 119909

(119895)

119899+119898minus1= 1

(10)



=



119895minus1198950119909) 119894 = 2119895minus 119898 + 1 2

119895minus 1

11986111989811989500(2119895minus1198950119909 minus 2

minus1198950 119894) 119894 = 0 2119895minus 119898

(11)


120595119898119895119894minus119898 =

2119894+2119898minus2

sum

119896=119894

119902119894119896119861119898119895119896minus119898 119894 = 1 119898 minus 1

120595119898119895119894minus119898 =

2119894+2119898minus2

sum

119896=2119894minus119898

119902119894119896119861119898119895119896minus119898 119894 = 119898 119899 minus 119898 + 1

120595119898119895119894minus119898 =

119899+119894+119898minus1

sum

119896=2119894minus119898

119902119894119896119861119898119895119896minus119898 119894 = 119899 minus 119898 + 2 119899

(12)



120595119898119895119894 (119909)

=



119895minus2119898+2 2

119895minus119898

12059511989811989500(2119895minus1198950119909 minus 2

minus1198950 119894) 119894 = 0 2119895minus 2119898 + 1

(13)



21198950 ge 2119898 minus 1 (14)



119891 (119909) =

21198950minus1

sum

119896=1minus119898

1198881198950 1198961205931198950 119896

(119909)

+

infin

sum

119895=1198950

2119895minus119898

sum

119896=1minus119898

119889119895119896120595119895119896 (119909)

(15)


119891 (119909) cong

21198950minus1

sum

119896=1minus119898

1198881198950 1198961205931198950 119896

(119909)

+

119872

sum

119895=1198950

2119895minus119898

sum

119896=1minus119898

119889119895119896120595119895119896 (119909)

(16)


119862 = [11988811989501minus119898 1198881198950 2

1198950minus1 1198891198950 1minus119898


119879

(17)

Ψ = [1205931198950 1minus119898 1205931198950 2

1198950minus1 12059511989501minus119898


119879

(18)

with

1198881198950 119896= int

1

0

119891 (119909) 1205931198950 119896(119909) 119889119909 119896 = 1 minus 119898 2

1198950 minus 1

119889119895119896 = int

1

0

119891 (119909) 119895119896 (119909) 119889119909

119895 = 1198950 119872 119896 = 1 minus 119898 2119872minus 119898

(19)




Φ = [1205931198950 1minus119898 1205931198950 2

1198950minus1]119879

(20)

Ψ = [1205951198950 1minus119898 1205951198950 2


(21)

Using (11) (20) (12)-(13) and (21) we get

int

1

0

ΦΦ119879119889119909 = 1198751

int

1

0

ΨΨ119879

119889119909 = 1198752

(22)



int

1

0

ΦΦ119879119889119909 = 1198681

int

1

0

ΨΨ119879

119889119909 = 1198682

(23)

Φ = 1198751minus1Φ

Ψ = 1198752

minus1Ψ

(24)


119910 (119909) = 119862119879Ψ (119909) (25)


119891 (119909) = 1198621119879Ψ (119909)

119870 (119909 119905) = Ψ119879(119905) ΘΨ (119909)

(26)

where

Θ119894119895 = int

1

0

[int

1

0

119870 (119909 119905) Ψ119894 (119905) 119889119905]Ψ119895 (119909) 119889119909 (27)


int

1

0

119870 (119909 119905) 119910 (119905) 119889119905 = int

1

0

119862119879Ψ (119905) Ψ

119879(119905) ΘΨ (119909) 119889119905

= 119862119879ΘΨ (119909)

(28)

since

int

1

0

Ψ (119905) Ψ119879(119905) 119889119905 = 119868 (29)


119862119879Ψ (119909) minus 119862

119879ΘΨ (119909) = 119862

119879

1Ψ (119909) (30)


119862119879119875 minus 119862

119879Θ = 119862

119879

1 (31)

since

int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = 119868 (32)


119875 = int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = (

1198751 0

0 1198752) (33)





119881119895 sub 119881119895+1 forall119895 isin Ζ (34)

⋃

119895isinΖ

119881119895 is dense in 1198712 (119877) (35)

⋂

119895isinΖ

119881119895 = 120601 (36)



120593 (119909) = sum

119896

119901119896120593 (119886119909 minus 119896) (38)





1198811198951cap 1198811198952

= 1198811198952 1198951 gt 1198952

1198821198951cap1198821198952

= 0 1198951 = 1198952

1198811198951cap1198821198952

= 0 1198951 le 1198952

(39)



int

infin

minusinfin

119909119901120595 (119909) 119889119909 = 0 119901 = 0 119898 minus 1 (40)




⟨120595119895119896 120595119904119894⟩ = 0 119895 = 119904 forall119895 119896 119904 119894 isin Ζ (41)


119910 (119909) asymp

21198950minus1

sum

119896=minus1

1198881198961205931198950 119896(119909)

+

119872

sum

119895=1198950

2119895minus2

sum

119896=minus1

119889119895119896120595119895119896 (119909)

= 119862119879Ψ (119909)

(42)



(⟨120593 120593⟩ minus ⟨119870120593 120593⟩ ⟨120595 120593⟩ minus ⟨119870120595 120593⟩

⟨120593 120595⟩ minus ⟨119870120593 120595⟩ ⟨120595 120595⟩ minus ⟨119870120595 120595⟩)(119862

119863) = (

1198651

1198652)

(43)where

119862 = [119888minus1 1198880 1198883]119879

119863 = [1198892minus1 11988922 1198893minus1 11988936

119889119872minus1 1198891198722119872minus2]119879

⟨120593 120593⟩ minus ⟨119870120593 120593⟩

= (int

1

0

1205931198950 119903(119909) 1205931198950 119894

(119909) 119889119909

minusint

1

0

1205931198950 119903(119909) int

1

0

119870(119909 119905)1205931198950 119894(119905)119889119905 119889119909)

119894119903

⟨120595 120593⟩ minus ⟨119870120595 120593⟩

= (int

1

0

1205931198950 119903(119909) 120595119896119895 (119909) 119889119909

minusint

1

0

1205931198950 119903(119909) int

1

0

119870(119909 119905)120595119896119895(119905)119889119905 119889119909)

119903119896119895

⟨120593 120595⟩ minus ⟨119870120593 120595⟩

= (int

1

0

120595119904119897 (119909) 1205931198950 119894(119909) 119889119909

minusint

1

0

120595119904119897(119909) int

1

0

119870(119909 119905)1205931198950 119894(119905)119889119905 119889119909)

119894119897119904

⟨120595 120595⟩ minus ⟨119870120595 120595⟩

= (int

1

0

120595119904119897 (119909) 120595119896119895 (119909) 119889119909

minus int

1

0

120595119904119897(119909) int

1

0

119870(119909 119905)120595119896119895(119905)119889119905 119889119909)

119897119904119896119895

1198651 = int

1

0

119891 (119909) 1205931198950 119903(119909) 119889119909

1198652 = int

1

0

119891 (119909) 120595119904119897 (119909) 119889119909

(44)


119894 119903 = minus1 21198950 minus 1

119897 119896 = 1198950 119872

119904 119895 = minus1 2119872minus 2

(45)



1198841015840(119909) = 119891

1015840(119909) + int

1

0

1198701015840(119909 119905) 119910 (119905) 119889119905 (46)


119910119899+1 (119909)

= 119910119899 (119909)

+ int

119909

0

120582 (120585) (1199101015840

119899(120585) minus 119891

1015840(120585) minus int

119887

119886

1198701015840(120585 119905) 119910119899 (119905) 119889119905) 119889120585

(47)


119910 (119909) = lim119899rarrinfin

119910119899 (119909) (48)



120575119910119899+1 (119909) = 120575119910119899 (119909)

+ 120575int

119909

0

120582 (120585) (1199101015840

119899(120585) minus 119891

1015840(120585)

minusint

119887

119886

1198701015840(120585 119905) 119910119899 (119905) 119889119905) 119889120585

= 120575119910119899 (119909) + int

119909

0

120582 (120585) 120575 (1199101015840

119899(120585)) 119889120585

= 120575119910119899 (119909) + 1205821205751199101198991003816100381610038161003816120585=119909 minus int

119909

0

1205821015840(120585) 120575119910119899 (120585) 119889120585

(49)


120575119910119899+1 = 0 (50)


1205821015840(120585) = 0 (51)


1 + 120582(120585)1003816100381610038161003816120585=119909 = 0 (52)


120582 = minus1 (53)


119910119899+1 (119909) = 119910119899 (119909)

minus int

119909

0

(1199101015840

119899(120585) minus 119891

1015840(120585) minus int

119887

119886

1198701015840(120585 119905) 119910119899 (119905) 119889119905) 119889120585

119899 ge 0

(54)




119899

sum

119895=1

119910119895 (119909) = 119891119894 (119909) +

119899

sum

119895=1

int

1

0

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899

(55)




ℎ0 = 1 0 le 119909 lt 1

0 others(56)


ℎ1 (119909) =

1 0 le 119909 lt1

2

minus11

2le 119909 lt 1

0 others

(57)


ℎ119899 (119909) = ℎ1 (2119895119909 minus 119896) (58)


expressed as

int

1

0

ℎ119898 (119909) ℎ119899 (119909) 119889119909 = 2minus119895120575119898119899 =

2minus119895 119898 = 119899 = 2

119895+ 119896

0 119898 = 119899

(59)


119910 (119909) =

+infin

sum

119899=0

119888119899ℎ119899 (119909) (60)


119888119899 = 2119895int

1

0

119910 (119909) ℎ119899 (119909) 119889119909

119899 = 2119895+ 119896 119895 ge 0 0 le 119896 lt 2

119895

(61)


0119910(119909)119889119909


119910 (119909) asymp

119898minus1

sum

119899=0

119888119899ℎ119899 (119909) = 119862119879

(119898)ℎ(119898) (119909) (62)



119862119879

(119898)= [1198880 1198881 119888119898minus1] 119898 = 2

119895

ℎ(119898) (119909) = [ℎ0 (119909) ℎ1 (119909) ℎ119898minus1 (119909)]119879 119898 = 2

119895

(63)



119870 (119909 119905) asymp

119898minus1

sum

119906=0

119898minus1

sum

V=0

119886119906VℎV (119909) ℎ119906 (119905) (64)

where 119886119906V = 2119894+119902∬1

0119870(119909 119905)ℎV(119909)ℎ119906(119905)119889119909 119889119905 119906 = 2

119894+ 119895 V =


119870 (119909 119905) asymp ℎ119879

(119898)(119905) 119870ℎ(119898) (119909) (65)

where

119870 = (119886119906V)119879

119898times119898 (66)


119867(119898) = [ℎ(119898) (1

2119898) ℎ(119898) (

3

2119898) ℎ(119898) (

2119898 minus 1

2119898)]

(67)


represented as

119867(4) = [ℎ(4) (1

8) ℎ(4) (

3

8) ℎ(4) (

7

8)]

=[[[

[

1 1 1 1

1 1 minus1 minus1

1 minus1 0 0

0 0 1 minus1

]]]

]

(68)


int

119909

0

ℎ(119898) (119905) 119889119905 = 119875(119898)ℎ(119898) (119909)

119909 isin [0 1)

(69)


119875(1) = [1

2]

119875(119898) =1

2119898[

2119898119875(1198982) minus119867(1198982)

119867minus1

(1198982)0

]

(70)


119875(2) =1

4[2 minus1

1 0]

119875(4) =1

16

[[[

[


4 0 minus2 2

1 1 0 0

1 minus1 0 0

]]]

]

119875(8) =1

64

[[[[[[[[[[

[



4 4 0 0 minus4 4 0 0

4 4 0 0 0 0 minus4 4

1 1 2 0 0 0 0 0

1 1 minus2 0 0 0 0 0

1 minus1 0 2 0 0 0 0


]]]]]]]]]]

]

(71)

Therefore we get

119867minus1

(119898)= (

1

119898)119867119879

(119898)

times diag(1 1 2 2 22 22⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

22

2120572minus1 2

120572minus1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

2120572minus1

)

(72)


sented as

int

1

0

ℎ(119898) (119905) ℎ119879

(119898)(119905) 119889119905 = 119863 (73)

where

119863 = diag(1 1 12 12 122 122⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

22

12120572minus1 12

120572minus1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

2120572minus1

)

(74)


119898

sum

119895=1

119910119895 (119909) = 119891119894 (119909) +

119898

sum

119895=1

int

1

0

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119898

(75)


119910119895 (119909) asymp 119884119879

119895ℎ(119898) (119909) 119895 = 1 2 119898

119870119894119895 (119909 119905) asymp ℎ119879

(119898)(119905) 119870119894119895ℎ(119898) (119909)

119894 119895 = 1 2 119898

(76)



119898

sum

119895=1

119884119879

119895ℎ(119898) (119909)

= 119891119894 (119909) +

119898

sum

119895=1

int

1

0

119884119879

119895ℎ(119898) (119905) ℎ

119879

(119898)(119905) 119870119894119895ℎ(119898) (119909) 119889119905

119894 = 1 2 119898

(77)


119898

sum

119895=1

119884119879

119895ℎ(119898) (119909) = 119891119894 (119909) +

119898

sum

119895=1

119884119879

119895119863119870119894119895ℎ(119898) (119909)

119894 = 1 2 119898

(78)


119894=1 in the


119898

sum

119895=1

119884119879

119895ℎ(119898) (119909119894) = 119891119894 (119909119894) +

119898

sum

119895=1

119884119879

119895119863119870119894119895ℎ(119898) (119909119894)

119894 = 1 2 119898

(79)


119879

119895ℎ(119898)(119909) 119895 = 1 2 119898



119910119894 (119909) = 119891119894 (119909) +

119899

sum

119895=1

int

1

0

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899 0 le 119909 le 1

(80)


119910119895 (119905) = 119910119895 (119909) + 1199101015840


+1

119898119910(119898)

119895(119909) (119905 minus 119909)

119898+ 119864 (119905)

(81)



int1

0sum119899



119910119894 (119909) asymp 119891119894 (119909)

+

119899

sum

119895=1

int

1

0

119870119894119895 (119909 119905)

119898

sum

119903=0

1

119903(119905 minus 119909)

119903119910(119903)

119895(119909) 119889119905

119894 = 1 2 119899

119910119894 (119909) asymp 119891119894 (119909)

+

119899

sum

119895=1

119898

sum

119903=0

1

119903119910(119903)

119895(119909) int

1

0

119870119894119895 (119909 119905) (119905 minus 119909)119903119889119905

119894 = 1 2 119899

(82)

119910119894 (119909) minus

119899

sum

119895=1

119898

sum

119903=0

1

119903119910(119903)

119895(119909) [int

1

0

119870119894119895 (119909 119905) (119905 minus 119909)119903119889119905]

asymp 119891119894 (119909) 119894 = 1 2 119899

(83)



119910(119904)

119894(119909) = 119891

(119904)

119894(119909) +

119899

sum

119895=1

int

1

0

119870(119904)

119894119895(119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899 119904 = 1 2 119898

(84)

where119870(119904)119894119895(119909 119905) = 120597

(119904)119870119894119895(119909 119905)120597119909


have

119910(119904)

119894(119909) minus [

[

119899

sum

119895=1

int

1

0

119870(119904)

119894119895(119909 119905) 119889119905]

]

119910119895 (119909) asymp 119891(119904)

119894(119909)

119894 = 1 2 119899 119904 = 1 2 119898

(85)




Φ119894 (119905) =

1 (119894 minus 1)119879

119898le 119905 lt 119894

119879

119898

0 otherwise(86)

with 119905 isin [0 119879) 119879119898 = ℎ and 119894 = 1 2 119898




Φ119894 (119905) Φ119895 (119905) = Φ119894 (119905) 119894 = 119895

0 119894 = 119895(87)



int

119879

0

Φ119894 (119905) Φ119895 (119905) 119889119905 = ℎ 119894 = 119895

0 119894 = 119895(88)



int

119879

0

1198912(119905) 119889119905 =

infin

sum

119894=1

1198912

119894

1003817100381710038171003817Φ119894(119905)10038171003817100381710038172 (89)



119891 (119905) =

119898

sum

119894=1

119891119894Φ119894 (119905) (90)



120576 =1

119879int

119879

0

(119891(119905) minus

119898

sum

119895=1

119891119895Φ119895(119905))

2

119889119905 (91)


Now 120597120576

120597119891119894

= minus2

119879int

119879

0

(119891 (119905) minus

119898

sum

119895=1

119891119895Φ119895 (119905))Φ119894 (119905) 119889119905 = 0

997904rArr 119891119894 =1

ℎint

119879

0


(92)


119891 (119905) =

119898

sum

119894=1

119891119894Φ119894 (119905) = 119865119879Φ (119905) = Φ

119879119865

where 119865 = [1198911 1198912 119891119898]119879

Φ (119905) = [Φ1 (119905) Φ2 (119905) Φ119898 (119905)]119879

(93)


119870 (119905 119904) = Φ119879(119905) 119870Ψ (119904) (94)



Φ (119905)Φ119879(119905) = (



d

0 0 sdot sdot sdot Φ119898 (119905)

) (95)


The second case is

Φ119879(119905) Φ (119905) = 1 (96)

because sum119898119894=1(Φ119894(119905))

2= sum119898

119894=1Φ119894(119905) = 1


int

119879

0

Φ (119905) 119889119905 = 119875Φ (119905) (97)

where

Φ (119905) = [Φ1 (119905) Φ2 (119905) Φ119898 (119905)]119879 (98)


119875 =1

2(




d


) (99)


int

119905

0

119891 (119905) 119889119905 = int

119905

0

119865119879Φ (119905) 119889119905 = 119865

119879119875Φ (119905) (100)


119899

sum

119895=1

119910119895 (119909) = 119891119894 (119909) +

119899

sum

119895=1

int

120573

120572

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899

(101)





119891119894 (119909) asymp 119865119879

119894Φ (119909)

119910119895 (119909) asymp 119884119879

119895Φ (119909)

119870119894119895 (119909 119905) asymp Φ119879(119905) 119870119894119895Φ (119909)

(102)


119899

sum

119895=1

119884119879

119895Φ (119909) = 119865

119879

119894Φ (119909)

+

119899

sum

119895=1

int

119898ℎ

0

119884119879

119895Φ (119905)Φ

119879(119905) 119870119894119895Φ (119909) 119889119905

119894 = 1 2 119899

(103)

119899

sum

119895=1

119884119879

119895Φ (119909) = 119865

119879

119894Φ (119909) +

119899

sum

119895=1

119884119879

119895ℎ119868119870119894119895Φ (119909)

119894 = 1 2 119899

(104)

since

int

119898ℎ

0

Φ (119905)Φ119879(119905) 119889119905 = ℎ119868 (105)

From (104) we get

119899

sum

119895=1

(119868 minus ℎ119870119879

119894119895) 119884119895 = 119865119894 119894 = 1 2 119899 (106)

Set 119860 119894119895 = 119868 minus ℎ119870119879

119894119895 then we have from (106)

119899

sum

119895=1

119860 119894119895119884119895 = 119865119894 119894 = 1 2 119899 (107)


(

(

11986011 11986012 1198601119899

11986021 11986022 1198602119899

1198601198991 1198601198992 119860119899119899

)

)

(

(

1198841

1198842

119884119899

)

)

=(

(

1198651

1198652

119865119899

)

)

(108)


119879119884119895 119895 = 1 2 119899



119906 (119909) = 119891 (119909) + int

1

0

119870 (119909 119905) 119865 (119905 119906 (119905)) 119889119905

0 le 119909 le 1

(109)





119910 (119909) = 119865 (119909 119906 (119909))

0 le 119909 le 1

(110)


119906 (119909) = 119860119879Ψ (119909)

119910 (119909) = 119861119879Ψ (119909)

(111)



119865 (119909) = 119863119879Ψ (119909)

119870 (119909 119905) = Ψ119879(119905) ΘΨ (119909)

(112)

where

Θ(119894119895) = int

1

0

[int

1

0

119870 (119909 119905) Ψ119894 (119905) 119889119905]Ψ119895 (119909) 119889119909 (113)


int

1

0

119870 (119909 119905) 119865 (119905 119906 (119905)) 119889119905

= int

1

0

119861119879Ψ (119905) Ψ

119879(119905) ΘΨ (119909) 119889119905

= 119861119879[int

1

0

Ψ (119905) Ψ119879(119905) 119889119905]ΘΨ (119909)

= 119861119879ΘΨ (119909) since int

1

0

Ψ (119905) Ψ119879(119905) 119889119905 = 119868

(114)



119860119879Ψ (119909) = 119863

119879Ψ (119909) + 119861

119879ΘΨ (119909) (115)


119860119879119875 = 119863

119879+ 119861119879Θ

119860119879119875 minus 119863

119879minus 119861119879Θ = 0

(116)


119875 = int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = [

1198751

1198752]

int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = 119868

(117)



119865 (119909 119860119879Ψ (119909)) = 119861

119879Ψ (119909) (118)


+119898minus2) where119894 = 1 2 2

119872+1+ 119898 minus 1







int

119887

119886

119891 (119909) 119889119909 =

119899minus1

sum

119894=1

int

119909119894+1

119909119894minus1

119891 (119909) 119889119909

=ℎ

3119891 (119886) +

4ℎ

3

1198992

sum

119894=1

119891 (1199092119894minus1)

+2ℎ

3

(119899minus1)2

sum

119894=1

119891 (1199092119894)

+ℎ

3119891 (119887)

minus(119887 minus 119886)

180ℎ4119891(4)(120578)

(119)


int

119887

119886

119891 (119909) 119889119909 =

119899

sum

119894=1

int

119909119894

119909119894minus1

119891 (119909) 119889119909

=ℎ

2119891 (119886) + ℎ

119899minus1

sum

119894=1

119891 (119909119894)

+ℎ

2119891 (119887)

+ℎ2

12[1198911015840(119886) minus 119891

1015840(119887)]

(120)


119906 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) 119865 (119906 (119905)) 119889119905

119886 le 119909 le 119887

(121)



+ℎ

3

[

[

119870 (119909 1199050) 119865 (1199060)

+ 4

1198992

sum

119895=1

119870(119909 1199052119895minus1) 119865 (1199062119895minus1)

+ 2

(1198992)minus1

sum

119895=1

119870(119909 1199052119895) 119865 (1199062119895)

+ 119870 (119909 119905119899) 119865 (119906119899)]

]

(122)


+ℎ

3

[

[

119870 (119909119894 1199050) 119865 (1199060)

+ 4

1198992

sum

119895=1

119870(119909119894 1199052119895minus1) 119865 (1199062119895minus1)


+ 2

(1198992)minus1

sum

119895=1

119870(119909119894 1199052119895) 119865 (1199062119895)

+119870 (119909119894 119905119899) 119865 (119906119899)]

]

(123)



119906 (119909) = 119891 (119909)

+ℎ

2119870 (119909 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909 119905119895) 119865 (119906119895)

+ℎ

2119870 (119909 119905119899) 119865 (119906119899)

+ℎ2

12[119869 (119909 1199050) 119865 (1199060)

+ 119870 (119909 1199050) 1199061015840

01198651015840(1199060)

minus 119869 (119909 119905119899) 119865 (119906119899)

minus119870 (119909 119905119899) 1199061015840

1198991198651015840(119906119899)]

(124)


have

119906 (119909119894) = 119891 (119909119894)

+ℎ

2119870 (119909119894 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909119894 119905119895) 119865 (119906119895)

+ℎ

2119870 (119909119894 119905119899) 119865 (119906119899)

+ℎ2

12[119869 (119909119894 1199050) 119865 (1199060)

+ 119870 (119909119894 1199050) 1199061015840

01198651015840(1199060)

minus 119869 (119909119894 119905119899) 119865 (119906119899)

minus119870 (119909119894 119905119899) 1199061015840

1198991198651015840(119906119899)]

(125)

for 119894 = 0 1 119899


120597119870(119909 119905)120597119909 we obtain

1199061015840(119909) = 119891

1015840(119909) + int

119887

119886

119867(119909 119905) 119865 (119906 (119905)) 119889119905

119886 le 119909 le 119887

(126)


1199061015840(119909119894) = 119891

1015840(119909119894)

+ℎ

2119867 (119909119894 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(119909119894 119905119895) 119865 (119906119895)

+ℎ

2119867 (119909119894 119905119899) 119865 (119906119899)

(127)



119906 (119909119894) = (ℎ

2119870 (119909119894 1199050) +

ℎ2

12119869 (119909119894 1199050))119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909119894 119905119895) 119865 (119906119895)

+ (ℎ

2119870 (119909119894 119905119899) minus

ℎ2

12119869 (119909119894 119905119899))119865 (119906119899)

+ℎ2

12(119870 (119909119894 1199050) 119906

1015840

01198651015840(1199060)

minus119870 (119909119894 119905119899) 1199061015840

1198991198651015840(119906119899))

1199061015840(1199090) = 119891

1015840(1199090)

+ℎ

2119867 (1199090 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(1199090 119905119895) 119865 (119906119895)

+ℎ

2119867 (1199090 119905119899) 119865 (119906119899)


1199061015840(119909119899) = 119891

1015840(119909119899)

+ℎ

2119867 (119909119899 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(119909119899 119905119895) 119865 (119906119895)

+ℎ

2119867 (119909119899 119905119899) 119865 (119906119899)

(128)




120595119886119887 (119905) = |119886|minus12120595(

119905 minus 119887

119886)

119886 119887 isin 119877 119886 = 0

(129)



120595119898119899 (119905)

=

(119898 +1

2)

12

21198962119871119898 (2

119896119905 minus 119899)

119899 minus 1

2119896le 119905 lt

119899 + 1

2119896

0 otherwise(130)


1198710 (119905) = 1

1198711 (119905) = 119905

119871119898+1 (119905) =2119898 + 1

119898 + 1119905119871119898 (119905)

minus119898

119898 + 1119871119898minus1 (119905) 119898 = 1 2 3

(131)



119891 (119909) =

infin

sum

119899=1

infin

sum

119898=0

119888119899119898120595119899119898 (119909) (132)

where

119888119899119898 = ⟨119891 (119909) 120595119899119898 (119909)⟩ (133)


119891 (119909) asymp

2119896minus1

sum

119899=1

119872minus1

sum

119898=0

119888119899119898120595119899119898 (119909) = 119862119879Ψ (119909) (134)


119862 = [11988810 11988811 1198881119872minus1 11988820


(135)

Ψ (119909) = [12059510 (119909) 1205951119872minus1 (119909)

12059520 (119909) 1205952119872minus1 (119909)


(136)


approximated as

119896 (119909 119905) asymp Ψ119879(119905) 119870Ψ (119909) (137)


119870119894119895 = ⟨120595119894 (119905) ⟨119896 (119909 119905) 120595119895 (119909)⟩⟩ (138)


[119910 (119909)]119901= [119884119879Ψ (119909)]

119901

= 119884lowast

119901

119879Ψ (119909) (139)





int

119905

0

Ψ(1199051015840) 1198891199051015840= 119875Ψ (119905) (140)



119875 =

[[[[[[

[

119871 119867 sdot sdot sdot 119867 119867

0 119871 sdot sdot sdot 119867 119867

d

0 0 sdot sdot sdot 119871 119867


]]]]]]

]

(141)



119867 =1

2119896

[[[[

[



d


]]]]

]

119871 =1

2119896

[[[[[[[[[[[[[[[[[[[[[[[

[

11


minusradic3

30

radic3


0 minusradic5

5radic30

radic5


0 0 minusradic7


d





]]]]]]]]]]]]]]]]]]]]]]]

]

(142)


int

1

0

Ψ (119905)Ψ119879(119905) 119889119905 = 119868 (143)


defined as

Ψ (119905) Ψ119879(119905) 119862 = 119862

119879Ψ (119905) (144)




119910 (119909) = 119891 (119909) + int

1

0

119896 (119909 119905) [119910 (119905)]119901119889119905 (145)



119891 (119909) asymp 119865119879Ψ (119909)

119910 (119909) asymp 119884119879Ψ (119909)

119896 (119909 119905) asymp Ψ119879(119905) 119870Ψ (119909)

[119910 (119909)]119901asymp 119884lowast119879Ψ (119909)

(146)


119884119879Ψ (119909) = 119865

119879Ψ (119909)

+ int

1

0

119884lowast119879Ψ (119905) Ψ

119879(119905) 119870Ψ (119909) 119889119905

= 119865119879Ψ (119909)

+ 119884lowast119879(int

1

0

Ψ (119905) Ψ119879(119905) 119889119905)119870Ψ (119909)

= 119865119879Ψ (119909) + 119884

lowast119879119870Ψ (119909)

= (119865119879+ 119884lowast119879119870)Ψ (119909)


119879= 0

(147)



119906 (119909) = 119891 (119909) + int

1

0

119870 (119909 119905) 119865 (119906 (119905)) 119889119905

0 le 119909 le 1

(148)


119871 (119906) = 119906 (119909) minus 119891 (119909) minus int

1

0

119870 (119909 119905) 119865 (119906 (119905)) 119889119905 = 0 (149)



119867(119906 0) = 119873 (119906)

119867 (119906 1) = 119871 (119906)

(150)


119867(119906 119901) = (1 minus 119901)119873 (119906) + 119901119871 (119906) = 0 (151)


119906 = 1199060 + 1199011199061 + 11990121199062 + sdot sdot sdot (152a)


119880 = lim119901rarr1

119906 = 1199060 + 1199061 + 1199062 + sdot sdot sdot (152b)


Consider

119873(119906) = 119906 (119909) minus 119891 (119909) (153)


119865 (119906) =

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898)

= 1198670 (1199060) + 1199011198671 (1199060 1199061)


(154)

where119867119898 (1199060 1199061 119906119898)

=1

119898

120597119898

120597119901119898(119865(

119898

sum

119896=0

119901119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816119901=0

119898 ge 0

(155)




minusint

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905) = 0


minus 119901int

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905 = 0

(156)


1199010 1199060 (119909) minus 119891 (119909) = 0 997904rArr 1199060 (119909) = 119891 (119909)

1199011 1199061 (119909) minus int

1

0

119870 (119909 119905)1198670119889119905 = 0 997904rArr 1199061 (119909)

= int

1

0

119870 (119909 119905)1198670119889119905

1199012 1199062 (119909) minus int

1

0

119870 (119909 119905)1198671119889119905 = 0 997904rArr 1199062 (119909)

= int

1

0

119870 (119909 119905)1198671119889119905

(157)


1199060 (119909) = 119891 (119909)

119906119899+1 (119909) = int

1

0

119870 (119909 119905)119867119899119889119905 119899 = 0 1 2

(158)



119906 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) (119871119906 (119905) + 119873119906 (119905)) 119889119905

119886 le 119909 le 119887

(159)



119906 (119909) =

infin

sum

119898=0

119906119898 (119909) (160)


119873119906 =

infin

sum

119899=0

119860119899 (161)

where 119860119899 =1

119899(120597119899

120597120582119899119873(

infin

sum

119896=0

120582119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816120582=0

(162)



infin

sum

119898=0

119906119898 (119909) = 119891 (119909)

+ int

119887

119886

119870 (119909 119905) (119871(

infin

sum

119898=0

119906119898 (119905)) +

infin

sum

119898=0

119860119898)119889119905

(163)


1199060 (119909) = 119891 (119909)

119906119898 (119909) = int

119887

119886

119870 (119909 119905) (119871 (119906119898minus1 (119905)) + 119860119898minus1 (119905)) 119889119905

119898 ge 1

(164)


120593119899 = 1199060 + 1199061 + sdot sdot sdot + 119906119899 (165)

with

119906 (119909) = lim119899rarrinfin

120593119899 (166)




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119899+119898minus1

119894=minus119898+1and Supp119861119898119895119883(119909) =




119883(119895)=

119909(119895)


(119895)

0= 0

119909(119895)

119896=119896

2119895 119896 = 1 119899 minus 1

119909(119895)

119899= sdot sdot sdot = 119909

(119895)

119899+119898minus1= 1

(10)



=



119895minus1198950119909) 119894 = 2119895minus 119898 + 1 2

119895minus 1

11986111989811989500(2119895minus1198950119909 minus 2

minus1198950 119894) 119894 = 0 2119895minus 119898

(11)


120595119898119895119894minus119898 =

2119894+2119898minus2

sum

119896=119894

119902119894119896119861119898119895119896minus119898 119894 = 1 119898 minus 1

120595119898119895119894minus119898 =

2119894+2119898minus2

sum

119896=2119894minus119898

119902119894119896119861119898119895119896minus119898 119894 = 119898 119899 minus 119898 + 1

120595119898119895119894minus119898 =

119899+119894+119898minus1

sum

119896=2119894minus119898

119902119894119896119861119898119895119896minus119898 119894 = 119899 minus 119898 + 2 119899

(12)



120595119898119895119894 (119909)

=



119895minus2119898+2 2

119895minus119898

12059511989811989500(2119895minus1198950119909 minus 2

minus1198950 119894) 119894 = 0 2119895minus 2119898 + 1

(13)



21198950 ge 2119898 minus 1 (14)



119891 (119909) =

21198950minus1

sum

119896=1minus119898

1198881198950 1198961205931198950 119896

(119909)

+

infin

sum

119895=1198950

2119895minus119898

sum

119896=1minus119898

119889119895119896120595119895119896 (119909)

(15)


119891 (119909) cong

21198950minus1

sum

119896=1minus119898

1198881198950 1198961205931198950 119896

(119909)

+

119872

sum

119895=1198950

2119895minus119898

sum

119896=1minus119898

119889119895119896120595119895119896 (119909)

(16)


119862 = [11988811989501minus119898 1198881198950 2

1198950minus1 1198891198950 1minus119898


119879

(17)

Ψ = [1205931198950 1minus119898 1205931198950 2

1198950minus1 12059511989501minus119898


119879

(18)

with

1198881198950 119896= int

1

0

119891 (119909) 1205931198950 119896(119909) 119889119909 119896 = 1 minus 119898 2

1198950 minus 1

119889119895119896 = int

1

0

119891 (119909) 119895119896 (119909) 119889119909

119895 = 1198950 119872 119896 = 1 minus 119898 2119872minus 119898

(19)




Φ = [1205931198950 1minus119898 1205931198950 2

1198950minus1]119879

(20)

Ψ = [1205951198950 1minus119898 1205951198950 2


(21)

Using (11) (20) (12)-(13) and (21) we get

int

1

0

ΦΦ119879119889119909 = 1198751

int

1

0

ΨΨ119879

119889119909 = 1198752

(22)



int

1

0

ΦΦ119879119889119909 = 1198681

int

1

0

ΨΨ119879

119889119909 = 1198682

(23)

Φ = 1198751minus1Φ

Ψ = 1198752

minus1Ψ

(24)


119910 (119909) = 119862119879Ψ (119909) (25)


119891 (119909) = 1198621119879Ψ (119909)

119870 (119909 119905) = Ψ119879(119905) ΘΨ (119909)

(26)

where

Θ119894119895 = int

1

0

[int

1

0

119870 (119909 119905) Ψ119894 (119905) 119889119905]Ψ119895 (119909) 119889119909 (27)


int

1

0

119870 (119909 119905) 119910 (119905) 119889119905 = int

1

0

119862119879Ψ (119905) Ψ

119879(119905) ΘΨ (119909) 119889119905

= 119862119879ΘΨ (119909)

(28)

since

int

1

0

Ψ (119905) Ψ119879(119905) 119889119905 = 119868 (29)


119862119879Ψ (119909) minus 119862

119879ΘΨ (119909) = 119862

119879

1Ψ (119909) (30)


119862119879119875 minus 119862

119879Θ = 119862

119879

1 (31)

since

int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = 119868 (32)


119875 = int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = (

1198751 0

0 1198752) (33)





119881119895 sub 119881119895+1 forall119895 isin Ζ (34)

⋃

119895isinΖ

119881119895 is dense in 1198712 (119877) (35)

⋂

119895isinΖ

119881119895 = 120601 (36)



120593 (119909) = sum

119896

119901119896120593 (119886119909 minus 119896) (38)





1198811198951cap 1198811198952

= 1198811198952 1198951 gt 1198952

1198821198951cap1198821198952

= 0 1198951 = 1198952

1198811198951cap1198821198952

= 0 1198951 le 1198952

(39)



int

infin

minusinfin

119909119901120595 (119909) 119889119909 = 0 119901 = 0 119898 minus 1 (40)




⟨120595119895119896 120595119904119894⟩ = 0 119895 = 119904 forall119895 119896 119904 119894 isin Ζ (41)


119910 (119909) asymp

21198950minus1

sum

119896=minus1

1198881198961205931198950 119896(119909)

+

119872

sum

119895=1198950

2119895minus2

sum

119896=minus1

119889119895119896120595119895119896 (119909)

= 119862119879Ψ (119909)

(42)



(⟨120593 120593⟩ minus ⟨119870120593 120593⟩ ⟨120595 120593⟩ minus ⟨119870120595 120593⟩

⟨120593 120595⟩ minus ⟨119870120593 120595⟩ ⟨120595 120595⟩ minus ⟨119870120595 120595⟩)(119862

119863) = (

1198651

1198652)

(43)where

119862 = [119888minus1 1198880 1198883]119879

119863 = [1198892minus1 11988922 1198893minus1 11988936

119889119872minus1 1198891198722119872minus2]119879

⟨120593 120593⟩ minus ⟨119870120593 120593⟩

= (int

1

0

1205931198950 119903(119909) 1205931198950 119894

(119909) 119889119909

minusint

1

0

1205931198950 119903(119909) int

1

0

119870(119909 119905)1205931198950 119894(119905)119889119905 119889119909)

119894119903

⟨120595 120593⟩ minus ⟨119870120595 120593⟩

= (int

1

0

1205931198950 119903(119909) 120595119896119895 (119909) 119889119909

minusint

1

0

1205931198950 119903(119909) int

1

0

119870(119909 119905)120595119896119895(119905)119889119905 119889119909)

119903119896119895

⟨120593 120595⟩ minus ⟨119870120593 120595⟩

= (int

1

0

120595119904119897 (119909) 1205931198950 119894(119909) 119889119909

minusint

1

0

120595119904119897(119909) int

1

0

119870(119909 119905)1205931198950 119894(119905)119889119905 119889119909)

119894119897119904

⟨120595 120595⟩ minus ⟨119870120595 120595⟩

= (int

1

0

120595119904119897 (119909) 120595119896119895 (119909) 119889119909

minus int

1

0

120595119904119897(119909) int

1

0

119870(119909 119905)120595119896119895(119905)119889119905 119889119909)

119897119904119896119895

1198651 = int

1

0

119891 (119909) 1205931198950 119903(119909) 119889119909

1198652 = int

1

0

119891 (119909) 120595119904119897 (119909) 119889119909

(44)


119894 119903 = minus1 21198950 minus 1

119897 119896 = 1198950 119872

119904 119895 = minus1 2119872minus 2

(45)



1198841015840(119909) = 119891

1015840(119909) + int

1

0

1198701015840(119909 119905) 119910 (119905) 119889119905 (46)


119910119899+1 (119909)

= 119910119899 (119909)

+ int

119909

0

120582 (120585) (1199101015840

119899(120585) minus 119891

1015840(120585) minus int

119887

119886

1198701015840(120585 119905) 119910119899 (119905) 119889119905) 119889120585

(47)


119910 (119909) = lim119899rarrinfin

119910119899 (119909) (48)



120575119910119899+1 (119909) = 120575119910119899 (119909)

+ 120575int

119909

0

120582 (120585) (1199101015840

119899(120585) minus 119891

1015840(120585)

minusint

119887

119886

1198701015840(120585 119905) 119910119899 (119905) 119889119905) 119889120585

= 120575119910119899 (119909) + int

119909

0

120582 (120585) 120575 (1199101015840

119899(120585)) 119889120585

= 120575119910119899 (119909) + 1205821205751199101198991003816100381610038161003816120585=119909 minus int

119909

0

1205821015840(120585) 120575119910119899 (120585) 119889120585

(49)


120575119910119899+1 = 0 (50)


1205821015840(120585) = 0 (51)


1 + 120582(120585)1003816100381610038161003816120585=119909 = 0 (52)


120582 = minus1 (53)


119910119899+1 (119909) = 119910119899 (119909)

minus int

119909

0

(1199101015840

119899(120585) minus 119891

1015840(120585) minus int

119887

119886

1198701015840(120585 119905) 119910119899 (119905) 119889119905) 119889120585

119899 ge 0

(54)




119899

sum

119895=1

119910119895 (119909) = 119891119894 (119909) +

119899

sum

119895=1

int

1

0

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899

(55)




ℎ0 = 1 0 le 119909 lt 1

0 others(56)


ℎ1 (119909) =

1 0 le 119909 lt1

2

minus11

2le 119909 lt 1

0 others

(57)


ℎ119899 (119909) = ℎ1 (2119895119909 minus 119896) (58)


expressed as

int

1

0

ℎ119898 (119909) ℎ119899 (119909) 119889119909 = 2minus119895120575119898119899 =

2minus119895 119898 = 119899 = 2

119895+ 119896

0 119898 = 119899

(59)


119910 (119909) =

+infin

sum

119899=0

119888119899ℎ119899 (119909) (60)


119888119899 = 2119895int

1

0

119910 (119909) ℎ119899 (119909) 119889119909

119899 = 2119895+ 119896 119895 ge 0 0 le 119896 lt 2

119895

(61)


0119910(119909)119889119909


119910 (119909) asymp

119898minus1

sum

119899=0

119888119899ℎ119899 (119909) = 119862119879

(119898)ℎ(119898) (119909) (62)



119862119879

(119898)= [1198880 1198881 119888119898minus1] 119898 = 2

119895

ℎ(119898) (119909) = [ℎ0 (119909) ℎ1 (119909) ℎ119898minus1 (119909)]119879 119898 = 2

119895

(63)



119870 (119909 119905) asymp

119898minus1

sum

119906=0

119898minus1

sum

V=0

119886119906VℎV (119909) ℎ119906 (119905) (64)

where 119886119906V = 2119894+119902∬1

0119870(119909 119905)ℎV(119909)ℎ119906(119905)119889119909 119889119905 119906 = 2

119894+ 119895 V =


119870 (119909 119905) asymp ℎ119879

(119898)(119905) 119870ℎ(119898) (119909) (65)

where

119870 = (119886119906V)119879

119898times119898 (66)


119867(119898) = [ℎ(119898) (1

2119898) ℎ(119898) (

3

2119898) ℎ(119898) (

2119898 minus 1

2119898)]

(67)


represented as

119867(4) = [ℎ(4) (1

8) ℎ(4) (

3

8) ℎ(4) (

7

8)]

=[[[

[

1 1 1 1

1 1 minus1 minus1

1 minus1 0 0

0 0 1 minus1

]]]

]

(68)


int

119909

0

ℎ(119898) (119905) 119889119905 = 119875(119898)ℎ(119898) (119909)

119909 isin [0 1)

(69)


119875(1) = [1

2]

119875(119898) =1

2119898[

2119898119875(1198982) minus119867(1198982)

119867minus1

(1198982)0

]

(70)


119875(2) =1

4[2 minus1

1 0]

119875(4) =1

16

[[[

[


4 0 minus2 2

1 1 0 0

1 minus1 0 0

]]]

]

119875(8) =1

64

[[[[[[[[[[

[



4 4 0 0 minus4 4 0 0

4 4 0 0 0 0 minus4 4

1 1 2 0 0 0 0 0

1 1 minus2 0 0 0 0 0

1 minus1 0 2 0 0 0 0


]]]]]]]]]]

]

(71)

Therefore we get

119867minus1

(119898)= (

1

119898)119867119879

(119898)

times diag(1 1 2 2 22 22⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

22

2120572minus1 2

120572minus1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

2120572minus1

)

(72)


sented as

int

1

0

ℎ(119898) (119905) ℎ119879

(119898)(119905) 119889119905 = 119863 (73)

where

119863 = diag(1 1 12 12 122 122⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

22

12120572minus1 12

120572minus1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

2120572minus1

)

(74)


119898

sum

119895=1

119910119895 (119909) = 119891119894 (119909) +

119898

sum

119895=1

int

1

0

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119898

(75)


119910119895 (119909) asymp 119884119879

119895ℎ(119898) (119909) 119895 = 1 2 119898

119870119894119895 (119909 119905) asymp ℎ119879

(119898)(119905) 119870119894119895ℎ(119898) (119909)

119894 119895 = 1 2 119898

(76)



119898

sum

119895=1

119884119879

119895ℎ(119898) (119909)

= 119891119894 (119909) +

119898

sum

119895=1

int

1

0

119884119879

119895ℎ(119898) (119905) ℎ

119879

(119898)(119905) 119870119894119895ℎ(119898) (119909) 119889119905

119894 = 1 2 119898

(77)


119898

sum

119895=1

119884119879

119895ℎ(119898) (119909) = 119891119894 (119909) +

119898

sum

119895=1

119884119879

119895119863119870119894119895ℎ(119898) (119909)

119894 = 1 2 119898

(78)


119894=1 in the


119898

sum

119895=1

119884119879

119895ℎ(119898) (119909119894) = 119891119894 (119909119894) +

119898

sum

119895=1

119884119879

119895119863119870119894119895ℎ(119898) (119909119894)

119894 = 1 2 119898

(79)


119879

119895ℎ(119898)(119909) 119895 = 1 2 119898



119910119894 (119909) = 119891119894 (119909) +

119899

sum

119895=1

int

1

0

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899 0 le 119909 le 1

(80)


119910119895 (119905) = 119910119895 (119909) + 1199101015840


+1

119898119910(119898)

119895(119909) (119905 minus 119909)

119898+ 119864 (119905)

(81)



int1

0sum119899



119910119894 (119909) asymp 119891119894 (119909)

+

119899

sum

119895=1

int

1

0

119870119894119895 (119909 119905)

119898

sum

119903=0

1

119903(119905 minus 119909)

119903119910(119903)

119895(119909) 119889119905

119894 = 1 2 119899

119910119894 (119909) asymp 119891119894 (119909)

+

119899

sum

119895=1

119898

sum

119903=0

1

119903119910(119903)

119895(119909) int

1

0

119870119894119895 (119909 119905) (119905 minus 119909)119903119889119905

119894 = 1 2 119899

(82)

119910119894 (119909) minus

119899

sum

119895=1

119898

sum

119903=0

1

119903119910(119903)

119895(119909) [int

1

0

119870119894119895 (119909 119905) (119905 minus 119909)119903119889119905]

asymp 119891119894 (119909) 119894 = 1 2 119899

(83)



119910(119904)

119894(119909) = 119891

(119904)

119894(119909) +

119899

sum

119895=1

int

1

0

119870(119904)

119894119895(119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899 119904 = 1 2 119898

(84)

where119870(119904)119894119895(119909 119905) = 120597

(119904)119870119894119895(119909 119905)120597119909


have

119910(119904)

119894(119909) minus [

[

119899

sum

119895=1

int

1

0

119870(119904)

119894119895(119909 119905) 119889119905]

]

119910119895 (119909) asymp 119891(119904)

119894(119909)

119894 = 1 2 119899 119904 = 1 2 119898

(85)




Φ119894 (119905) =

1 (119894 minus 1)119879

119898le 119905 lt 119894

119879

119898

0 otherwise(86)

with 119905 isin [0 119879) 119879119898 = ℎ and 119894 = 1 2 119898




Φ119894 (119905) Φ119895 (119905) = Φ119894 (119905) 119894 = 119895

0 119894 = 119895(87)



int

119879

0

Φ119894 (119905) Φ119895 (119905) 119889119905 = ℎ 119894 = 119895

0 119894 = 119895(88)



int

119879

0

1198912(119905) 119889119905 =

infin

sum

119894=1

1198912

119894

1003817100381710038171003817Φ119894(119905)10038171003817100381710038172 (89)



119891 (119905) =

119898

sum

119894=1

119891119894Φ119894 (119905) (90)



120576 =1

119879int

119879

0

(119891(119905) minus

119898

sum

119895=1

119891119895Φ119895(119905))

2

119889119905 (91)


Now 120597120576

120597119891119894

= minus2

119879int

119879

0

(119891 (119905) minus

119898

sum

119895=1

119891119895Φ119895 (119905))Φ119894 (119905) 119889119905 = 0

997904rArr 119891119894 =1

ℎint

119879

0


(92)


119891 (119905) =

119898

sum

119894=1

119891119894Φ119894 (119905) = 119865119879Φ (119905) = Φ

119879119865

where 119865 = [1198911 1198912 119891119898]119879

Φ (119905) = [Φ1 (119905) Φ2 (119905) Φ119898 (119905)]119879

(93)


119870 (119905 119904) = Φ119879(119905) 119870Ψ (119904) (94)



Φ (119905)Φ119879(119905) = (



d

0 0 sdot sdot sdot Φ119898 (119905)

) (95)


The second case is

Φ119879(119905) Φ (119905) = 1 (96)

because sum119898119894=1(Φ119894(119905))

2= sum119898

119894=1Φ119894(119905) = 1


int

119879

0

Φ (119905) 119889119905 = 119875Φ (119905) (97)

where

Φ (119905) = [Φ1 (119905) Φ2 (119905) Φ119898 (119905)]119879 (98)


119875 =1

2(




d


) (99)


int

119905

0

119891 (119905) 119889119905 = int

119905

0

119865119879Φ (119905) 119889119905 = 119865

119879119875Φ (119905) (100)


119899

sum

119895=1

119910119895 (119909) = 119891119894 (119909) +

119899

sum

119895=1

int

120573

120572

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899

(101)





119891119894 (119909) asymp 119865119879

119894Φ (119909)

119910119895 (119909) asymp 119884119879

119895Φ (119909)

119870119894119895 (119909 119905) asymp Φ119879(119905) 119870119894119895Φ (119909)

(102)


119899

sum

119895=1

119884119879

119895Φ (119909) = 119865

119879

119894Φ (119909)

+

119899

sum

119895=1

int

119898ℎ

0

119884119879

119895Φ (119905)Φ

119879(119905) 119870119894119895Φ (119909) 119889119905

119894 = 1 2 119899

(103)

119899

sum

119895=1

119884119879

119895Φ (119909) = 119865

119879

119894Φ (119909) +

119899

sum

119895=1

119884119879

119895ℎ119868119870119894119895Φ (119909)

119894 = 1 2 119899

(104)

since

int

119898ℎ

0

Φ (119905)Φ119879(119905) 119889119905 = ℎ119868 (105)

From (104) we get

119899

sum

119895=1

(119868 minus ℎ119870119879

119894119895) 119884119895 = 119865119894 119894 = 1 2 119899 (106)

Set 119860 119894119895 = 119868 minus ℎ119870119879

119894119895 then we have from (106)

119899

sum

119895=1

119860 119894119895119884119895 = 119865119894 119894 = 1 2 119899 (107)


(

(

11986011 11986012 1198601119899

11986021 11986022 1198602119899

1198601198991 1198601198992 119860119899119899

)

)

(

(

1198841

1198842

119884119899

)

)

=(

(

1198651

1198652

119865119899

)

)

(108)


119879119884119895 119895 = 1 2 119899



119906 (119909) = 119891 (119909) + int

1

0

119870 (119909 119905) 119865 (119905 119906 (119905)) 119889119905

0 le 119909 le 1

(109)





119910 (119909) = 119865 (119909 119906 (119909))

0 le 119909 le 1

(110)


119906 (119909) = 119860119879Ψ (119909)

119910 (119909) = 119861119879Ψ (119909)

(111)



119865 (119909) = 119863119879Ψ (119909)

119870 (119909 119905) = Ψ119879(119905) ΘΨ (119909)

(112)

where

Θ(119894119895) = int

1

0

[int

1

0

119870 (119909 119905) Ψ119894 (119905) 119889119905]Ψ119895 (119909) 119889119909 (113)


int

1

0

119870 (119909 119905) 119865 (119905 119906 (119905)) 119889119905

= int

1

0

119861119879Ψ (119905) Ψ

119879(119905) ΘΨ (119909) 119889119905

= 119861119879[int

1

0

Ψ (119905) Ψ119879(119905) 119889119905]ΘΨ (119909)

= 119861119879ΘΨ (119909) since int

1

0

Ψ (119905) Ψ119879(119905) 119889119905 = 119868

(114)



119860119879Ψ (119909) = 119863

119879Ψ (119909) + 119861

119879ΘΨ (119909) (115)


119860119879119875 = 119863

119879+ 119861119879Θ

119860119879119875 minus 119863

119879minus 119861119879Θ = 0

(116)


119875 = int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = [

1198751

1198752]

int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = 119868

(117)



119865 (119909 119860119879Ψ (119909)) = 119861

119879Ψ (119909) (118)


+119898minus2) where119894 = 1 2 2

119872+1+ 119898 minus 1







int

119887

119886

119891 (119909) 119889119909 =

119899minus1

sum

119894=1

int

119909119894+1

119909119894minus1

119891 (119909) 119889119909

=ℎ

3119891 (119886) +

4ℎ

3

1198992

sum

119894=1

119891 (1199092119894minus1)

+2ℎ

3

(119899minus1)2

sum

119894=1

119891 (1199092119894)

+ℎ

3119891 (119887)

minus(119887 minus 119886)

180ℎ4119891(4)(120578)

(119)


int

119887

119886

119891 (119909) 119889119909 =

119899

sum

119894=1

int

119909119894

119909119894minus1

119891 (119909) 119889119909

=ℎ

2119891 (119886) + ℎ

119899minus1

sum

119894=1

119891 (119909119894)

+ℎ

2119891 (119887)

+ℎ2

12[1198911015840(119886) minus 119891

1015840(119887)]

(120)


119906 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) 119865 (119906 (119905)) 119889119905

119886 le 119909 le 119887

(121)



+ℎ

3

[

[

119870 (119909 1199050) 119865 (1199060)

+ 4

1198992

sum

119895=1

119870(119909 1199052119895minus1) 119865 (1199062119895minus1)

+ 2

(1198992)minus1

sum

119895=1

119870(119909 1199052119895) 119865 (1199062119895)

+ 119870 (119909 119905119899) 119865 (119906119899)]

]

(122)


+ℎ

3

[

[

119870 (119909119894 1199050) 119865 (1199060)

+ 4

1198992

sum

119895=1

119870(119909119894 1199052119895minus1) 119865 (1199062119895minus1)


+ 2

(1198992)minus1

sum

119895=1

119870(119909119894 1199052119895) 119865 (1199062119895)

+119870 (119909119894 119905119899) 119865 (119906119899)]

]

(123)



119906 (119909) = 119891 (119909)

+ℎ

2119870 (119909 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909 119905119895) 119865 (119906119895)

+ℎ

2119870 (119909 119905119899) 119865 (119906119899)

+ℎ2

12[119869 (119909 1199050) 119865 (1199060)

+ 119870 (119909 1199050) 1199061015840

01198651015840(1199060)

minus 119869 (119909 119905119899) 119865 (119906119899)

minus119870 (119909 119905119899) 1199061015840

1198991198651015840(119906119899)]

(124)


have

119906 (119909119894) = 119891 (119909119894)

+ℎ

2119870 (119909119894 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909119894 119905119895) 119865 (119906119895)

+ℎ

2119870 (119909119894 119905119899) 119865 (119906119899)

+ℎ2

12[119869 (119909119894 1199050) 119865 (1199060)

+ 119870 (119909119894 1199050) 1199061015840

01198651015840(1199060)

minus 119869 (119909119894 119905119899) 119865 (119906119899)

minus119870 (119909119894 119905119899) 1199061015840

1198991198651015840(119906119899)]

(125)

for 119894 = 0 1 119899


120597119870(119909 119905)120597119909 we obtain

1199061015840(119909) = 119891

1015840(119909) + int

119887

119886

119867(119909 119905) 119865 (119906 (119905)) 119889119905

119886 le 119909 le 119887

(126)


1199061015840(119909119894) = 119891

1015840(119909119894)

+ℎ

2119867 (119909119894 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(119909119894 119905119895) 119865 (119906119895)

+ℎ

2119867 (119909119894 119905119899) 119865 (119906119899)

(127)



119906 (119909119894) = (ℎ

2119870 (119909119894 1199050) +

ℎ2

12119869 (119909119894 1199050))119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909119894 119905119895) 119865 (119906119895)

+ (ℎ

2119870 (119909119894 119905119899) minus

ℎ2

12119869 (119909119894 119905119899))119865 (119906119899)

+ℎ2

12(119870 (119909119894 1199050) 119906

1015840

01198651015840(1199060)

minus119870 (119909119894 119905119899) 1199061015840

1198991198651015840(119906119899))

1199061015840(1199090) = 119891

1015840(1199090)

+ℎ

2119867 (1199090 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(1199090 119905119895) 119865 (119906119895)

+ℎ

2119867 (1199090 119905119899) 119865 (119906119899)


1199061015840(119909119899) = 119891

1015840(119909119899)

+ℎ

2119867 (119909119899 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(119909119899 119905119895) 119865 (119906119895)

+ℎ

2119867 (119909119899 119905119899) 119865 (119906119899)

(128)




120595119886119887 (119905) = |119886|minus12120595(

119905 minus 119887

119886)

119886 119887 isin 119877 119886 = 0

(129)



120595119898119899 (119905)

=

(119898 +1

2)

12

21198962119871119898 (2

119896119905 minus 119899)

119899 minus 1

2119896le 119905 lt

119899 + 1

2119896

0 otherwise(130)


1198710 (119905) = 1

1198711 (119905) = 119905

119871119898+1 (119905) =2119898 + 1

119898 + 1119905119871119898 (119905)

minus119898

119898 + 1119871119898minus1 (119905) 119898 = 1 2 3

(131)



119891 (119909) =

infin

sum

119899=1

infin

sum

119898=0

119888119899119898120595119899119898 (119909) (132)

where

119888119899119898 = ⟨119891 (119909) 120595119899119898 (119909)⟩ (133)


119891 (119909) asymp

2119896minus1

sum

119899=1

119872minus1

sum

119898=0

119888119899119898120595119899119898 (119909) = 119862119879Ψ (119909) (134)


119862 = [11988810 11988811 1198881119872minus1 11988820


(135)

Ψ (119909) = [12059510 (119909) 1205951119872minus1 (119909)

12059520 (119909) 1205952119872minus1 (119909)


(136)


approximated as

119896 (119909 119905) asymp Ψ119879(119905) 119870Ψ (119909) (137)


119870119894119895 = ⟨120595119894 (119905) ⟨119896 (119909 119905) 120595119895 (119909)⟩⟩ (138)


[119910 (119909)]119901= [119884119879Ψ (119909)]

119901

= 119884lowast

119901

119879Ψ (119909) (139)





int

119905

0

Ψ(1199051015840) 1198891199051015840= 119875Ψ (119905) (140)



119875 =

[[[[[[

[

119871 119867 sdot sdot sdot 119867 119867

0 119871 sdot sdot sdot 119867 119867

d

0 0 sdot sdot sdot 119871 119867


]]]]]]

]

(141)



119867 =1

2119896

[[[[

[



d


]]]]

]

119871 =1

2119896

[[[[[[[[[[[[[[[[[[[[[[[

[

11


minusradic3

30

radic3


0 minusradic5

5radic30

radic5


0 0 minusradic7


d





]]]]]]]]]]]]]]]]]]]]]]]

]

(142)


int

1

0

Ψ (119905)Ψ119879(119905) 119889119905 = 119868 (143)


defined as

Ψ (119905) Ψ119879(119905) 119862 = 119862

119879Ψ (119905) (144)




119910 (119909) = 119891 (119909) + int

1

0

119896 (119909 119905) [119910 (119905)]119901119889119905 (145)



119891 (119909) asymp 119865119879Ψ (119909)

119910 (119909) asymp 119884119879Ψ (119909)

119896 (119909 119905) asymp Ψ119879(119905) 119870Ψ (119909)

[119910 (119909)]119901asymp 119884lowast119879Ψ (119909)

(146)


119884119879Ψ (119909) = 119865

119879Ψ (119909)

+ int

1

0

119884lowast119879Ψ (119905) Ψ

119879(119905) 119870Ψ (119909) 119889119905

= 119865119879Ψ (119909)

+ 119884lowast119879(int

1

0

Ψ (119905) Ψ119879(119905) 119889119905)119870Ψ (119909)

= 119865119879Ψ (119909) + 119884

lowast119879119870Ψ (119909)

= (119865119879+ 119884lowast119879119870)Ψ (119909)


119879= 0

(147)



119906 (119909) = 119891 (119909) + int

1

0

119870 (119909 119905) 119865 (119906 (119905)) 119889119905

0 le 119909 le 1

(148)


119871 (119906) = 119906 (119909) minus 119891 (119909) minus int

1

0

119870 (119909 119905) 119865 (119906 (119905)) 119889119905 = 0 (149)



119867(119906 0) = 119873 (119906)

119867 (119906 1) = 119871 (119906)

(150)


119867(119906 119901) = (1 minus 119901)119873 (119906) + 119901119871 (119906) = 0 (151)


119906 = 1199060 + 1199011199061 + 11990121199062 + sdot sdot sdot (152a)


119880 = lim119901rarr1

119906 = 1199060 + 1199061 + 1199062 + sdot sdot sdot (152b)


Consider

119873(119906) = 119906 (119909) minus 119891 (119909) (153)


119865 (119906) =

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898)

= 1198670 (1199060) + 1199011198671 (1199060 1199061)


(154)

where119867119898 (1199060 1199061 119906119898)

=1

119898

120597119898

120597119901119898(119865(

119898

sum

119896=0

119901119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816119901=0

119898 ge 0

(155)




minusint

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905) = 0


minus 119901int

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905 = 0

(156)


1199010 1199060 (119909) minus 119891 (119909) = 0 997904rArr 1199060 (119909) = 119891 (119909)

1199011 1199061 (119909) minus int

1

0

119870 (119909 119905)1198670119889119905 = 0 997904rArr 1199061 (119909)

= int

1

0

119870 (119909 119905)1198670119889119905

1199012 1199062 (119909) minus int

1

0

119870 (119909 119905)1198671119889119905 = 0 997904rArr 1199062 (119909)

= int

1

0

119870 (119909 119905)1198671119889119905

(157)


1199060 (119909) = 119891 (119909)

119906119899+1 (119909) = int

1

0

119870 (119909 119905)119867119899119889119905 119899 = 0 1 2

(158)



119906 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) (119871119906 (119905) + 119873119906 (119905)) 119889119905

119886 le 119909 le 119887

(159)



119906 (119909) =

infin

sum

119898=0

119906119898 (119909) (160)


119873119906 =

infin

sum

119899=0

119860119899 (161)

where 119860119899 =1

119899(120597119899

120597120582119899119873(

infin

sum

119896=0

120582119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816120582=0

(162)



infin

sum

119898=0

119906119898 (119909) = 119891 (119909)

+ int

119887

119886

119870 (119909 119905) (119871(

infin

sum

119898=0

119906119898 (119905)) +

infin

sum

119898=0

119860119898)119889119905

(163)


1199060 (119909) = 119891 (119909)

119906119898 (119909) = int

119887

119886

119870 (119909 119905) (119871 (119906119898minus1 (119905)) + 119860119898minus1 (119905)) 119889119905

119898 ge 1

(164)


120593119899 = 1199060 + 1199061 + sdot sdot sdot + 119906119899 (165)

with

119906 (119909) = lim119899rarrinfin

120593119899 (166)




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











int

1

0

ΦΦ119879119889119909 = 1198681

int

1

0

ΨΨ119879

119889119909 = 1198682

(23)

Φ = 1198751minus1Φ

Ψ = 1198752

minus1Ψ

(24)


119910 (119909) = 119862119879Ψ (119909) (25)


119891 (119909) = 1198621119879Ψ (119909)

119870 (119909 119905) = Ψ119879(119905) ΘΨ (119909)

(26)

where

Θ119894119895 = int

1

0

[int

1

0

119870 (119909 119905) Ψ119894 (119905) 119889119905]Ψ119895 (119909) 119889119909 (27)


int

1

0

119870 (119909 119905) 119910 (119905) 119889119905 = int

1

0

119862119879Ψ (119905) Ψ

119879(119905) ΘΨ (119909) 119889119905

= 119862119879ΘΨ (119909)

(28)

since

int

1

0

Ψ (119905) Ψ119879(119905) 119889119905 = 119868 (29)


119862119879Ψ (119909) minus 119862

119879ΘΨ (119909) = 119862

119879

1Ψ (119909) (30)


119862119879119875 minus 119862

119879Θ = 119862

119879

1 (31)

since

int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = 119868 (32)


119875 = int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = (

1198751 0

0 1198752) (33)





119881119895 sub 119881119895+1 forall119895 isin Ζ (34)

⋃

119895isinΖ

119881119895 is dense in 1198712 (119877) (35)

⋂

119895isinΖ

119881119895 = 120601 (36)



120593 (119909) = sum

119896

119901119896120593 (119886119909 minus 119896) (38)





1198811198951cap 1198811198952

= 1198811198952 1198951 gt 1198952

1198821198951cap1198821198952

= 0 1198951 = 1198952

1198811198951cap1198821198952

= 0 1198951 le 1198952

(39)



int

infin

minusinfin

119909119901120595 (119909) 119889119909 = 0 119901 = 0 119898 minus 1 (40)




⟨120595119895119896 120595119904119894⟩ = 0 119895 = 119904 forall119895 119896 119904 119894 isin Ζ (41)


119910 (119909) asymp

21198950minus1

sum

119896=minus1

1198881198961205931198950 119896(119909)

+

119872

sum

119895=1198950

2119895minus2

sum

119896=minus1

119889119895119896120595119895119896 (119909)

= 119862119879Ψ (119909)

(42)



(⟨120593 120593⟩ minus ⟨119870120593 120593⟩ ⟨120595 120593⟩ minus ⟨119870120595 120593⟩

⟨120593 120595⟩ minus ⟨119870120593 120595⟩ ⟨120595 120595⟩ minus ⟨119870120595 120595⟩)(119862

119863) = (

1198651

1198652)

(43)where

119862 = [119888minus1 1198880 1198883]119879

119863 = [1198892minus1 11988922 1198893minus1 11988936

119889119872minus1 1198891198722119872minus2]119879

⟨120593 120593⟩ minus ⟨119870120593 120593⟩

= (int

1

0

1205931198950 119903(119909) 1205931198950 119894

(119909) 119889119909

minusint

1

0

1205931198950 119903(119909) int

1

0

119870(119909 119905)1205931198950 119894(119905)119889119905 119889119909)

119894119903

⟨120595 120593⟩ minus ⟨119870120595 120593⟩

= (int

1

0

1205931198950 119903(119909) 120595119896119895 (119909) 119889119909

minusint

1

0

1205931198950 119903(119909) int

1

0

119870(119909 119905)120595119896119895(119905)119889119905 119889119909)

119903119896119895

⟨120593 120595⟩ minus ⟨119870120593 120595⟩

= (int

1

0

120595119904119897 (119909) 1205931198950 119894(119909) 119889119909

minusint

1

0

120595119904119897(119909) int

1

0

119870(119909 119905)1205931198950 119894(119905)119889119905 119889119909)

119894119897119904

⟨120595 120595⟩ minus ⟨119870120595 120595⟩

= (int

1

0

120595119904119897 (119909) 120595119896119895 (119909) 119889119909

minus int

1

0

120595119904119897(119909) int

1

0

119870(119909 119905)120595119896119895(119905)119889119905 119889119909)

119897119904119896119895

1198651 = int

1

0

119891 (119909) 1205931198950 119903(119909) 119889119909

1198652 = int

1

0

119891 (119909) 120595119904119897 (119909) 119889119909

(44)


119894 119903 = minus1 21198950 minus 1

119897 119896 = 1198950 119872

119904 119895 = minus1 2119872minus 2

(45)



1198841015840(119909) = 119891

1015840(119909) + int

1

0

1198701015840(119909 119905) 119910 (119905) 119889119905 (46)


119910119899+1 (119909)

= 119910119899 (119909)

+ int

119909

0

120582 (120585) (1199101015840

119899(120585) minus 119891

1015840(120585) minus int

119887

119886

1198701015840(120585 119905) 119910119899 (119905) 119889119905) 119889120585

(47)


119910 (119909) = lim119899rarrinfin

119910119899 (119909) (48)



120575119910119899+1 (119909) = 120575119910119899 (119909)

+ 120575int

119909

0

120582 (120585) (1199101015840

119899(120585) minus 119891

1015840(120585)

minusint

119887

119886

1198701015840(120585 119905) 119910119899 (119905) 119889119905) 119889120585

= 120575119910119899 (119909) + int

119909

0

120582 (120585) 120575 (1199101015840

119899(120585)) 119889120585

= 120575119910119899 (119909) + 1205821205751199101198991003816100381610038161003816120585=119909 minus int

119909

0

1205821015840(120585) 120575119910119899 (120585) 119889120585

(49)


120575119910119899+1 = 0 (50)


1205821015840(120585) = 0 (51)


1 + 120582(120585)1003816100381610038161003816120585=119909 = 0 (52)


120582 = minus1 (53)


119910119899+1 (119909) = 119910119899 (119909)

minus int

119909

0

(1199101015840

119899(120585) minus 119891

1015840(120585) minus int

119887

119886

1198701015840(120585 119905) 119910119899 (119905) 119889119905) 119889120585

119899 ge 0

(54)




119899

sum

119895=1

119910119895 (119909) = 119891119894 (119909) +

119899

sum

119895=1

int

1

0

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899

(55)




ℎ0 = 1 0 le 119909 lt 1

0 others(56)


ℎ1 (119909) =

1 0 le 119909 lt1

2

minus11

2le 119909 lt 1

0 others

(57)


ℎ119899 (119909) = ℎ1 (2119895119909 minus 119896) (58)


expressed as

int

1

0

ℎ119898 (119909) ℎ119899 (119909) 119889119909 = 2minus119895120575119898119899 =

2minus119895 119898 = 119899 = 2

119895+ 119896

0 119898 = 119899

(59)


119910 (119909) =

+infin

sum

119899=0

119888119899ℎ119899 (119909) (60)


119888119899 = 2119895int

1

0

119910 (119909) ℎ119899 (119909) 119889119909

119899 = 2119895+ 119896 119895 ge 0 0 le 119896 lt 2

119895

(61)


0119910(119909)119889119909


119910 (119909) asymp

119898minus1

sum

119899=0

119888119899ℎ119899 (119909) = 119862119879

(119898)ℎ(119898) (119909) (62)



119862119879

(119898)= [1198880 1198881 119888119898minus1] 119898 = 2

119895

ℎ(119898) (119909) = [ℎ0 (119909) ℎ1 (119909) ℎ119898minus1 (119909)]119879 119898 = 2

119895

(63)



119870 (119909 119905) asymp

119898minus1

sum

119906=0

119898minus1

sum

V=0

119886119906VℎV (119909) ℎ119906 (119905) (64)

where 119886119906V = 2119894+119902∬1

0119870(119909 119905)ℎV(119909)ℎ119906(119905)119889119909 119889119905 119906 = 2

119894+ 119895 V =


119870 (119909 119905) asymp ℎ119879

(119898)(119905) 119870ℎ(119898) (119909) (65)

where

119870 = (119886119906V)119879

119898times119898 (66)


119867(119898) = [ℎ(119898) (1

2119898) ℎ(119898) (

3

2119898) ℎ(119898) (

2119898 minus 1

2119898)]

(67)


represented as

119867(4) = [ℎ(4) (1

8) ℎ(4) (

3

8) ℎ(4) (

7

8)]

=[[[

[

1 1 1 1

1 1 minus1 minus1

1 minus1 0 0

0 0 1 minus1

]]]

]

(68)


int

119909

0

ℎ(119898) (119905) 119889119905 = 119875(119898)ℎ(119898) (119909)

119909 isin [0 1)

(69)


119875(1) = [1

2]

119875(119898) =1

2119898[

2119898119875(1198982) minus119867(1198982)

119867minus1

(1198982)0

]

(70)


119875(2) =1

4[2 minus1

1 0]

119875(4) =1

16

[[[

[


4 0 minus2 2

1 1 0 0

1 minus1 0 0

]]]

]

119875(8) =1

64

[[[[[[[[[[

[



4 4 0 0 minus4 4 0 0

4 4 0 0 0 0 minus4 4

1 1 2 0 0 0 0 0

1 1 minus2 0 0 0 0 0

1 minus1 0 2 0 0 0 0


]]]]]]]]]]

]

(71)

Therefore we get

119867minus1

(119898)= (

1

119898)119867119879

(119898)

times diag(1 1 2 2 22 22⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

22

2120572minus1 2

120572minus1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

2120572minus1

)

(72)


sented as

int

1

0

ℎ(119898) (119905) ℎ119879

(119898)(119905) 119889119905 = 119863 (73)

where

119863 = diag(1 1 12 12 122 122⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

22

12120572minus1 12

120572minus1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

2120572minus1

)

(74)


119898

sum

119895=1

119910119895 (119909) = 119891119894 (119909) +

119898

sum

119895=1

int

1

0

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119898

(75)


119910119895 (119909) asymp 119884119879

119895ℎ(119898) (119909) 119895 = 1 2 119898

119870119894119895 (119909 119905) asymp ℎ119879

(119898)(119905) 119870119894119895ℎ(119898) (119909)

119894 119895 = 1 2 119898

(76)



119898

sum

119895=1

119884119879

119895ℎ(119898) (119909)

= 119891119894 (119909) +

119898

sum

119895=1

int

1

0

119884119879

119895ℎ(119898) (119905) ℎ

119879

(119898)(119905) 119870119894119895ℎ(119898) (119909) 119889119905

119894 = 1 2 119898

(77)


119898

sum

119895=1

119884119879

119895ℎ(119898) (119909) = 119891119894 (119909) +

119898

sum

119895=1

119884119879

119895119863119870119894119895ℎ(119898) (119909)

119894 = 1 2 119898

(78)


119894=1 in the


119898

sum

119895=1

119884119879

119895ℎ(119898) (119909119894) = 119891119894 (119909119894) +

119898

sum

119895=1

119884119879

119895119863119870119894119895ℎ(119898) (119909119894)

119894 = 1 2 119898

(79)


119879

119895ℎ(119898)(119909) 119895 = 1 2 119898



119910119894 (119909) = 119891119894 (119909) +

119899

sum

119895=1

int

1

0

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899 0 le 119909 le 1

(80)


119910119895 (119905) = 119910119895 (119909) + 1199101015840


+1

119898119910(119898)

119895(119909) (119905 minus 119909)

119898+ 119864 (119905)

(81)



int1

0sum119899



119910119894 (119909) asymp 119891119894 (119909)

+

119899

sum

119895=1

int

1

0

119870119894119895 (119909 119905)

119898

sum

119903=0

1

119903(119905 minus 119909)

119903119910(119903)

119895(119909) 119889119905

119894 = 1 2 119899

119910119894 (119909) asymp 119891119894 (119909)

+

119899

sum

119895=1

119898

sum

119903=0

1

119903119910(119903)

119895(119909) int

1

0

119870119894119895 (119909 119905) (119905 minus 119909)119903119889119905

119894 = 1 2 119899

(82)

119910119894 (119909) minus

119899

sum

119895=1

119898

sum

119903=0

1

119903119910(119903)

119895(119909) [int

1

0

119870119894119895 (119909 119905) (119905 minus 119909)119903119889119905]

asymp 119891119894 (119909) 119894 = 1 2 119899

(83)



119910(119904)

119894(119909) = 119891

(119904)

119894(119909) +

119899

sum

119895=1

int

1

0

119870(119904)

119894119895(119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899 119904 = 1 2 119898

(84)

where119870(119904)119894119895(119909 119905) = 120597

(119904)119870119894119895(119909 119905)120597119909


have

119910(119904)

119894(119909) minus [

[

119899

sum

119895=1

int

1

0

119870(119904)

119894119895(119909 119905) 119889119905]

]

119910119895 (119909) asymp 119891(119904)

119894(119909)

119894 = 1 2 119899 119904 = 1 2 119898

(85)




Φ119894 (119905) =

1 (119894 minus 1)119879

119898le 119905 lt 119894

119879

119898

0 otherwise(86)

with 119905 isin [0 119879) 119879119898 = ℎ and 119894 = 1 2 119898




Φ119894 (119905) Φ119895 (119905) = Φ119894 (119905) 119894 = 119895

0 119894 = 119895(87)



int

119879

0

Φ119894 (119905) Φ119895 (119905) 119889119905 = ℎ 119894 = 119895

0 119894 = 119895(88)



int

119879

0

1198912(119905) 119889119905 =

infin

sum

119894=1

1198912

119894

1003817100381710038171003817Φ119894(119905)10038171003817100381710038172 (89)



119891 (119905) =

119898

sum

119894=1

119891119894Φ119894 (119905) (90)



120576 =1

119879int

119879

0

(119891(119905) minus

119898

sum

119895=1

119891119895Φ119895(119905))

2

119889119905 (91)


Now 120597120576

120597119891119894

= minus2

119879int

119879

0

(119891 (119905) minus

119898

sum

119895=1

119891119895Φ119895 (119905))Φ119894 (119905) 119889119905 = 0

997904rArr 119891119894 =1

ℎint

119879

0


(92)


119891 (119905) =

119898

sum

119894=1

119891119894Φ119894 (119905) = 119865119879Φ (119905) = Φ

119879119865

where 119865 = [1198911 1198912 119891119898]119879

Φ (119905) = [Φ1 (119905) Φ2 (119905) Φ119898 (119905)]119879

(93)


119870 (119905 119904) = Φ119879(119905) 119870Ψ (119904) (94)



Φ (119905)Φ119879(119905) = (



d

0 0 sdot sdot sdot Φ119898 (119905)

) (95)


The second case is

Φ119879(119905) Φ (119905) = 1 (96)

because sum119898119894=1(Φ119894(119905))

2= sum119898

119894=1Φ119894(119905) = 1


int

119879

0

Φ (119905) 119889119905 = 119875Φ (119905) (97)

where

Φ (119905) = [Φ1 (119905) Φ2 (119905) Φ119898 (119905)]119879 (98)


119875 =1

2(




d


) (99)


int

119905

0

119891 (119905) 119889119905 = int

119905

0

119865119879Φ (119905) 119889119905 = 119865

119879119875Φ (119905) (100)


119899

sum

119895=1

119910119895 (119909) = 119891119894 (119909) +

119899

sum

119895=1

int

120573

120572

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899

(101)





119891119894 (119909) asymp 119865119879

119894Φ (119909)

119910119895 (119909) asymp 119884119879

119895Φ (119909)

119870119894119895 (119909 119905) asymp Φ119879(119905) 119870119894119895Φ (119909)

(102)


119899

sum

119895=1

119884119879

119895Φ (119909) = 119865

119879

119894Φ (119909)

+

119899

sum

119895=1

int

119898ℎ

0

119884119879

119895Φ (119905)Φ

119879(119905) 119870119894119895Φ (119909) 119889119905

119894 = 1 2 119899

(103)

119899

sum

119895=1

119884119879

119895Φ (119909) = 119865

119879

119894Φ (119909) +

119899

sum

119895=1

119884119879

119895ℎ119868119870119894119895Φ (119909)

119894 = 1 2 119899

(104)

since

int

119898ℎ

0

Φ (119905)Φ119879(119905) 119889119905 = ℎ119868 (105)

From (104) we get

119899

sum

119895=1

(119868 minus ℎ119870119879

119894119895) 119884119895 = 119865119894 119894 = 1 2 119899 (106)

Set 119860 119894119895 = 119868 minus ℎ119870119879

119894119895 then we have from (106)

119899

sum

119895=1

119860 119894119895119884119895 = 119865119894 119894 = 1 2 119899 (107)


(

(

11986011 11986012 1198601119899

11986021 11986022 1198602119899

1198601198991 1198601198992 119860119899119899

)

)

(

(

1198841

1198842

119884119899

)

)

=(

(

1198651

1198652

119865119899

)

)

(108)


119879119884119895 119895 = 1 2 119899



119906 (119909) = 119891 (119909) + int

1

0

119870 (119909 119905) 119865 (119905 119906 (119905)) 119889119905

0 le 119909 le 1

(109)





119910 (119909) = 119865 (119909 119906 (119909))

0 le 119909 le 1

(110)


119906 (119909) = 119860119879Ψ (119909)

119910 (119909) = 119861119879Ψ (119909)

(111)



119865 (119909) = 119863119879Ψ (119909)

119870 (119909 119905) = Ψ119879(119905) ΘΨ (119909)

(112)

where

Θ(119894119895) = int

1

0

[int

1

0

119870 (119909 119905) Ψ119894 (119905) 119889119905]Ψ119895 (119909) 119889119909 (113)


int

1

0

119870 (119909 119905) 119865 (119905 119906 (119905)) 119889119905

= int

1

0

119861119879Ψ (119905) Ψ

119879(119905) ΘΨ (119909) 119889119905

= 119861119879[int

1

0

Ψ (119905) Ψ119879(119905) 119889119905]ΘΨ (119909)

= 119861119879ΘΨ (119909) since int

1

0

Ψ (119905) Ψ119879(119905) 119889119905 = 119868

(114)



119860119879Ψ (119909) = 119863

119879Ψ (119909) + 119861

119879ΘΨ (119909) (115)


119860119879119875 = 119863

119879+ 119861119879Θ

119860119879119875 minus 119863

119879minus 119861119879Θ = 0

(116)


119875 = int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = [

1198751

1198752]

int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = 119868

(117)



119865 (119909 119860119879Ψ (119909)) = 119861

119879Ψ (119909) (118)


+119898minus2) where119894 = 1 2 2

119872+1+ 119898 minus 1







int

119887

119886

119891 (119909) 119889119909 =

119899minus1

sum

119894=1

int

119909119894+1

119909119894minus1

119891 (119909) 119889119909

=ℎ

3119891 (119886) +

4ℎ

3

1198992

sum

119894=1

119891 (1199092119894minus1)

+2ℎ

3

(119899minus1)2

sum

119894=1

119891 (1199092119894)

+ℎ

3119891 (119887)

minus(119887 minus 119886)

180ℎ4119891(4)(120578)

(119)


int

119887

119886

119891 (119909) 119889119909 =

119899

sum

119894=1

int

119909119894

119909119894minus1

119891 (119909) 119889119909

=ℎ

2119891 (119886) + ℎ

119899minus1

sum

119894=1

119891 (119909119894)

+ℎ

2119891 (119887)

+ℎ2

12[1198911015840(119886) minus 119891

1015840(119887)]

(120)


119906 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) 119865 (119906 (119905)) 119889119905

119886 le 119909 le 119887

(121)



+ℎ

3

[

[

119870 (119909 1199050) 119865 (1199060)

+ 4

1198992

sum

119895=1

119870(119909 1199052119895minus1) 119865 (1199062119895minus1)

+ 2

(1198992)minus1

sum

119895=1

119870(119909 1199052119895) 119865 (1199062119895)

+ 119870 (119909 119905119899) 119865 (119906119899)]

]

(122)


+ℎ

3

[

[

119870 (119909119894 1199050) 119865 (1199060)

+ 4

1198992

sum

119895=1

119870(119909119894 1199052119895minus1) 119865 (1199062119895minus1)


+ 2

(1198992)minus1

sum

119895=1

119870(119909119894 1199052119895) 119865 (1199062119895)

+119870 (119909119894 119905119899) 119865 (119906119899)]

]

(123)



119906 (119909) = 119891 (119909)

+ℎ

2119870 (119909 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909 119905119895) 119865 (119906119895)

+ℎ

2119870 (119909 119905119899) 119865 (119906119899)

+ℎ2

12[119869 (119909 1199050) 119865 (1199060)

+ 119870 (119909 1199050) 1199061015840

01198651015840(1199060)

minus 119869 (119909 119905119899) 119865 (119906119899)

minus119870 (119909 119905119899) 1199061015840

1198991198651015840(119906119899)]

(124)


have

119906 (119909119894) = 119891 (119909119894)

+ℎ

2119870 (119909119894 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909119894 119905119895) 119865 (119906119895)

+ℎ

2119870 (119909119894 119905119899) 119865 (119906119899)

+ℎ2

12[119869 (119909119894 1199050) 119865 (1199060)

+ 119870 (119909119894 1199050) 1199061015840

01198651015840(1199060)

minus 119869 (119909119894 119905119899) 119865 (119906119899)

minus119870 (119909119894 119905119899) 1199061015840

1198991198651015840(119906119899)]

(125)

for 119894 = 0 1 119899


120597119870(119909 119905)120597119909 we obtain

1199061015840(119909) = 119891

1015840(119909) + int

119887

119886

119867(119909 119905) 119865 (119906 (119905)) 119889119905

119886 le 119909 le 119887

(126)


1199061015840(119909119894) = 119891

1015840(119909119894)

+ℎ

2119867 (119909119894 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(119909119894 119905119895) 119865 (119906119895)

+ℎ

2119867 (119909119894 119905119899) 119865 (119906119899)

(127)



119906 (119909119894) = (ℎ

2119870 (119909119894 1199050) +

ℎ2

12119869 (119909119894 1199050))119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909119894 119905119895) 119865 (119906119895)

+ (ℎ

2119870 (119909119894 119905119899) minus

ℎ2

12119869 (119909119894 119905119899))119865 (119906119899)

+ℎ2

12(119870 (119909119894 1199050) 119906

1015840

01198651015840(1199060)

minus119870 (119909119894 119905119899) 1199061015840

1198991198651015840(119906119899))

1199061015840(1199090) = 119891

1015840(1199090)

+ℎ

2119867 (1199090 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(1199090 119905119895) 119865 (119906119895)

+ℎ

2119867 (1199090 119905119899) 119865 (119906119899)


1199061015840(119909119899) = 119891

1015840(119909119899)

+ℎ

2119867 (119909119899 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(119909119899 119905119895) 119865 (119906119895)

+ℎ

2119867 (119909119899 119905119899) 119865 (119906119899)

(128)




120595119886119887 (119905) = |119886|minus12120595(

119905 minus 119887

119886)

119886 119887 isin 119877 119886 = 0

(129)



120595119898119899 (119905)

=

(119898 +1

2)

12

21198962119871119898 (2

119896119905 minus 119899)

119899 minus 1

2119896le 119905 lt

119899 + 1

2119896

0 otherwise(130)


1198710 (119905) = 1

1198711 (119905) = 119905

119871119898+1 (119905) =2119898 + 1

119898 + 1119905119871119898 (119905)

minus119898

119898 + 1119871119898minus1 (119905) 119898 = 1 2 3

(131)



119891 (119909) =

infin

sum

119899=1

infin

sum

119898=0

119888119899119898120595119899119898 (119909) (132)

where

119888119899119898 = ⟨119891 (119909) 120595119899119898 (119909)⟩ (133)


119891 (119909) asymp

2119896minus1

sum

119899=1

119872minus1

sum

119898=0

119888119899119898120595119899119898 (119909) = 119862119879Ψ (119909) (134)


119862 = [11988810 11988811 1198881119872minus1 11988820


(135)

Ψ (119909) = [12059510 (119909) 1205951119872minus1 (119909)

12059520 (119909) 1205952119872minus1 (119909)


(136)


approximated as

119896 (119909 119905) asymp Ψ119879(119905) 119870Ψ (119909) (137)


119870119894119895 = ⟨120595119894 (119905) ⟨119896 (119909 119905) 120595119895 (119909)⟩⟩ (138)


[119910 (119909)]119901= [119884119879Ψ (119909)]

119901

= 119884lowast

119901

119879Ψ (119909) (139)





int

119905

0

Ψ(1199051015840) 1198891199051015840= 119875Ψ (119905) (140)



119875 =

[[[[[[

[

119871 119867 sdot sdot sdot 119867 119867

0 119871 sdot sdot sdot 119867 119867

d

0 0 sdot sdot sdot 119871 119867


]]]]]]

]

(141)



119867 =1

2119896

[[[[

[



d


]]]]

]

119871 =1

2119896

[[[[[[[[[[[[[[[[[[[[[[[

[

11


minusradic3

30

radic3


0 minusradic5

5radic30

radic5


0 0 minusradic7


d





]]]]]]]]]]]]]]]]]]]]]]]

]

(142)


int

1

0

Ψ (119905)Ψ119879(119905) 119889119905 = 119868 (143)


defined as

Ψ (119905) Ψ119879(119905) 119862 = 119862

119879Ψ (119905) (144)




119910 (119909) = 119891 (119909) + int

1

0

119896 (119909 119905) [119910 (119905)]119901119889119905 (145)



119891 (119909) asymp 119865119879Ψ (119909)

119910 (119909) asymp 119884119879Ψ (119909)

119896 (119909 119905) asymp Ψ119879(119905) 119870Ψ (119909)

[119910 (119909)]119901asymp 119884lowast119879Ψ (119909)

(146)


119884119879Ψ (119909) = 119865

119879Ψ (119909)

+ int

1

0

119884lowast119879Ψ (119905) Ψ

119879(119905) 119870Ψ (119909) 119889119905

= 119865119879Ψ (119909)

+ 119884lowast119879(int

1

0

Ψ (119905) Ψ119879(119905) 119889119905)119870Ψ (119909)

= 119865119879Ψ (119909) + 119884

lowast119879119870Ψ (119909)

= (119865119879+ 119884lowast119879119870)Ψ (119909)


119879= 0

(147)



119906 (119909) = 119891 (119909) + int

1

0

119870 (119909 119905) 119865 (119906 (119905)) 119889119905

0 le 119909 le 1

(148)


119871 (119906) = 119906 (119909) minus 119891 (119909) minus int

1

0

119870 (119909 119905) 119865 (119906 (119905)) 119889119905 = 0 (149)



119867(119906 0) = 119873 (119906)

119867 (119906 1) = 119871 (119906)

(150)


119867(119906 119901) = (1 minus 119901)119873 (119906) + 119901119871 (119906) = 0 (151)


119906 = 1199060 + 1199011199061 + 11990121199062 + sdot sdot sdot (152a)


119880 = lim119901rarr1

119906 = 1199060 + 1199061 + 1199062 + sdot sdot sdot (152b)


Consider

119873(119906) = 119906 (119909) minus 119891 (119909) (153)


119865 (119906) =

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898)

= 1198670 (1199060) + 1199011198671 (1199060 1199061)


(154)

where119867119898 (1199060 1199061 119906119898)

=1

119898

120597119898

120597119901119898(119865(

119898

sum

119896=0

119901119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816119901=0

119898 ge 0

(155)




minusint

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905) = 0


minus 119901int

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905 = 0

(156)


1199010 1199060 (119909) minus 119891 (119909) = 0 997904rArr 1199060 (119909) = 119891 (119909)

1199011 1199061 (119909) minus int

1

0

119870 (119909 119905)1198670119889119905 = 0 997904rArr 1199061 (119909)

= int

1

0

119870 (119909 119905)1198670119889119905

1199012 1199062 (119909) minus int

1

0

119870 (119909 119905)1198671119889119905 = 0 997904rArr 1199062 (119909)

= int

1

0

119870 (119909 119905)1198671119889119905

(157)


1199060 (119909) = 119891 (119909)

119906119899+1 (119909) = int

1

0

119870 (119909 119905)119867119899119889119905 119899 = 0 1 2

(158)



119906 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) (119871119906 (119905) + 119873119906 (119905)) 119889119905

119886 le 119909 le 119887

(159)



119906 (119909) =

infin

sum

119898=0

119906119898 (119909) (160)


119873119906 =

infin

sum

119899=0

119860119899 (161)

where 119860119899 =1

119899(120597119899

120597120582119899119873(

infin

sum

119896=0

120582119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816120582=0

(162)



infin

sum

119898=0

119906119898 (119909) = 119891 (119909)

+ int

119887

119886

119870 (119909 119905) (119871(

infin

sum

119898=0

119906119898 (119905)) +

infin

sum

119898=0

119860119898)119889119905

(163)


1199060 (119909) = 119891 (119909)

119906119898 (119909) = int

119887

119886

119870 (119909 119905) (119871 (119906119898minus1 (119905)) + 119860119898minus1 (119905)) 119889119905

119898 ge 1

(164)


120593119899 = 1199060 + 1199061 + sdot sdot sdot + 119906119899 (165)

with

119906 (119909) = lim119899rarrinfin

120593119899 (166)




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












⟨120595119895119896 120595119904119894⟩ = 0 119895 = 119904 forall119895 119896 119904 119894 isin Ζ (41)


119910 (119909) asymp

21198950minus1

sum

119896=minus1

1198881198961205931198950 119896(119909)

+

119872

sum

119895=1198950

2119895minus2

sum

119896=minus1

119889119895119896120595119895119896 (119909)

= 119862119879Ψ (119909)

(42)



(⟨120593 120593⟩ minus ⟨119870120593 120593⟩ ⟨120595 120593⟩ minus ⟨119870120595 120593⟩

⟨120593 120595⟩ minus ⟨119870120593 120595⟩ ⟨120595 120595⟩ minus ⟨119870120595 120595⟩)(119862

119863) = (

1198651

1198652)

(43)where

119862 = [119888minus1 1198880 1198883]119879

119863 = [1198892minus1 11988922 1198893minus1 11988936

119889119872minus1 1198891198722119872minus2]119879

⟨120593 120593⟩ minus ⟨119870120593 120593⟩

= (int

1

0

1205931198950 119903(119909) 1205931198950 119894

(119909) 119889119909

minusint

1

0

1205931198950 119903(119909) int

1

0

119870(119909 119905)1205931198950 119894(119905)119889119905 119889119909)

119894119903

⟨120595 120593⟩ minus ⟨119870120595 120593⟩

= (int

1

0

1205931198950 119903(119909) 120595119896119895 (119909) 119889119909

minusint

1

0

1205931198950 119903(119909) int

1

0

119870(119909 119905)120595119896119895(119905)119889119905 119889119909)

119903119896119895

⟨120593 120595⟩ minus ⟨119870120593 120595⟩

= (int

1

0

120595119904119897 (119909) 1205931198950 119894(119909) 119889119909

minusint

1

0

120595119904119897(119909) int

1

0

119870(119909 119905)1205931198950 119894(119905)119889119905 119889119909)

119894119897119904

⟨120595 120595⟩ minus ⟨119870120595 120595⟩

= (int

1

0

120595119904119897 (119909) 120595119896119895 (119909) 119889119909

minus int

1

0

120595119904119897(119909) int

1

0

119870(119909 119905)120595119896119895(119905)119889119905 119889119909)

119897119904119896119895

1198651 = int

1

0

119891 (119909) 1205931198950 119903(119909) 119889119909

1198652 = int

1

0

119891 (119909) 120595119904119897 (119909) 119889119909

(44)


119894 119903 = minus1 21198950 minus 1

119897 119896 = 1198950 119872

119904 119895 = minus1 2119872minus 2

(45)



1198841015840(119909) = 119891

1015840(119909) + int

1

0

1198701015840(119909 119905) 119910 (119905) 119889119905 (46)


119910119899+1 (119909)

= 119910119899 (119909)

+ int

119909

0

120582 (120585) (1199101015840

119899(120585) minus 119891

1015840(120585) minus int

119887

119886

1198701015840(120585 119905) 119910119899 (119905) 119889119905) 119889120585

(47)


119910 (119909) = lim119899rarrinfin

119910119899 (119909) (48)



120575119910119899+1 (119909) = 120575119910119899 (119909)

+ 120575int

119909

0

120582 (120585) (1199101015840

119899(120585) minus 119891

1015840(120585)

minusint

119887

119886

1198701015840(120585 119905) 119910119899 (119905) 119889119905) 119889120585

= 120575119910119899 (119909) + int

119909

0

120582 (120585) 120575 (1199101015840

119899(120585)) 119889120585

= 120575119910119899 (119909) + 1205821205751199101198991003816100381610038161003816120585=119909 minus int

119909

0

1205821015840(120585) 120575119910119899 (120585) 119889120585

(49)


120575119910119899+1 = 0 (50)


1205821015840(120585) = 0 (51)


1 + 120582(120585)1003816100381610038161003816120585=119909 = 0 (52)


120582 = minus1 (53)


119910119899+1 (119909) = 119910119899 (119909)

minus int

119909

0

(1199101015840

119899(120585) minus 119891

1015840(120585) minus int

119887

119886

1198701015840(120585 119905) 119910119899 (119905) 119889119905) 119889120585

119899 ge 0

(54)




119899

sum

119895=1

119910119895 (119909) = 119891119894 (119909) +

119899

sum

119895=1

int

1

0

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899

(55)




ℎ0 = 1 0 le 119909 lt 1

0 others(56)


ℎ1 (119909) =

1 0 le 119909 lt1

2

minus11

2le 119909 lt 1

0 others

(57)


ℎ119899 (119909) = ℎ1 (2119895119909 minus 119896) (58)


expressed as

int

1

0

ℎ119898 (119909) ℎ119899 (119909) 119889119909 = 2minus119895120575119898119899 =

2minus119895 119898 = 119899 = 2

119895+ 119896

0 119898 = 119899

(59)


119910 (119909) =

+infin

sum

119899=0

119888119899ℎ119899 (119909) (60)


119888119899 = 2119895int

1

0

119910 (119909) ℎ119899 (119909) 119889119909

119899 = 2119895+ 119896 119895 ge 0 0 le 119896 lt 2

119895

(61)


0119910(119909)119889119909


119910 (119909) asymp

119898minus1

sum

119899=0

119888119899ℎ119899 (119909) = 119862119879

(119898)ℎ(119898) (119909) (62)



119862119879

(119898)= [1198880 1198881 119888119898minus1] 119898 = 2

119895

ℎ(119898) (119909) = [ℎ0 (119909) ℎ1 (119909) ℎ119898minus1 (119909)]119879 119898 = 2

119895

(63)



119870 (119909 119905) asymp

119898minus1

sum

119906=0

119898minus1

sum

V=0

119886119906VℎV (119909) ℎ119906 (119905) (64)

where 119886119906V = 2119894+119902∬1

0119870(119909 119905)ℎV(119909)ℎ119906(119905)119889119909 119889119905 119906 = 2

119894+ 119895 V =


119870 (119909 119905) asymp ℎ119879

(119898)(119905) 119870ℎ(119898) (119909) (65)

where

119870 = (119886119906V)119879

119898times119898 (66)


119867(119898) = [ℎ(119898) (1

2119898) ℎ(119898) (

3

2119898) ℎ(119898) (

2119898 minus 1

2119898)]

(67)


represented as

119867(4) = [ℎ(4) (1

8) ℎ(4) (

3

8) ℎ(4) (

7

8)]

=[[[

[

1 1 1 1

1 1 minus1 minus1

1 minus1 0 0

0 0 1 minus1

]]]

]

(68)


int

119909

0

ℎ(119898) (119905) 119889119905 = 119875(119898)ℎ(119898) (119909)

119909 isin [0 1)

(69)


119875(1) = [1

2]

119875(119898) =1

2119898[

2119898119875(1198982) minus119867(1198982)

119867minus1

(1198982)0

]

(70)


119875(2) =1

4[2 minus1

1 0]

119875(4) =1

16

[[[

[


4 0 minus2 2

1 1 0 0

1 minus1 0 0

]]]

]

119875(8) =1

64

[[[[[[[[[[

[



4 4 0 0 minus4 4 0 0

4 4 0 0 0 0 minus4 4

1 1 2 0 0 0 0 0

1 1 minus2 0 0 0 0 0

1 minus1 0 2 0 0 0 0


]]]]]]]]]]

]

(71)

Therefore we get

119867minus1

(119898)= (

1

119898)119867119879

(119898)

times diag(1 1 2 2 22 22⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

22

2120572minus1 2

120572minus1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

2120572minus1

)

(72)


sented as

int

1

0

ℎ(119898) (119905) ℎ119879

(119898)(119905) 119889119905 = 119863 (73)

where

119863 = diag(1 1 12 12 122 122⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

22

12120572minus1 12

120572minus1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

2120572minus1

)

(74)


119898

sum

119895=1

119910119895 (119909) = 119891119894 (119909) +

119898

sum

119895=1

int

1

0

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119898

(75)


119910119895 (119909) asymp 119884119879

119895ℎ(119898) (119909) 119895 = 1 2 119898

119870119894119895 (119909 119905) asymp ℎ119879

(119898)(119905) 119870119894119895ℎ(119898) (119909)

119894 119895 = 1 2 119898

(76)



119898

sum

119895=1

119884119879

119895ℎ(119898) (119909)

= 119891119894 (119909) +

119898

sum

119895=1

int

1

0

119884119879

119895ℎ(119898) (119905) ℎ

119879

(119898)(119905) 119870119894119895ℎ(119898) (119909) 119889119905

119894 = 1 2 119898

(77)


119898

sum

119895=1

119884119879

119895ℎ(119898) (119909) = 119891119894 (119909) +

119898

sum

119895=1

119884119879

119895119863119870119894119895ℎ(119898) (119909)

119894 = 1 2 119898

(78)


119894=1 in the


119898

sum

119895=1

119884119879

119895ℎ(119898) (119909119894) = 119891119894 (119909119894) +

119898

sum

119895=1

119884119879

119895119863119870119894119895ℎ(119898) (119909119894)

119894 = 1 2 119898

(79)


119879

119895ℎ(119898)(119909) 119895 = 1 2 119898



119910119894 (119909) = 119891119894 (119909) +

119899

sum

119895=1

int

1

0

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899 0 le 119909 le 1

(80)


119910119895 (119905) = 119910119895 (119909) + 1199101015840


+1

119898119910(119898)

119895(119909) (119905 minus 119909)

119898+ 119864 (119905)

(81)



int1

0sum119899



119910119894 (119909) asymp 119891119894 (119909)

+

119899

sum

119895=1

int

1

0

119870119894119895 (119909 119905)

119898

sum

119903=0

1

119903(119905 minus 119909)

119903119910(119903)

119895(119909) 119889119905

119894 = 1 2 119899

119910119894 (119909) asymp 119891119894 (119909)

+

119899

sum

119895=1

119898

sum

119903=0

1

119903119910(119903)

119895(119909) int

1

0

119870119894119895 (119909 119905) (119905 minus 119909)119903119889119905

119894 = 1 2 119899

(82)

119910119894 (119909) minus

119899

sum

119895=1

119898

sum

119903=0

1

119903119910(119903)

119895(119909) [int

1

0

119870119894119895 (119909 119905) (119905 minus 119909)119903119889119905]

asymp 119891119894 (119909) 119894 = 1 2 119899

(83)



119910(119904)

119894(119909) = 119891

(119904)

119894(119909) +

119899

sum

119895=1

int

1

0

119870(119904)

119894119895(119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899 119904 = 1 2 119898

(84)

where119870(119904)119894119895(119909 119905) = 120597

(119904)119870119894119895(119909 119905)120597119909


have

119910(119904)

119894(119909) minus [

[

119899

sum

119895=1

int

1

0

119870(119904)

119894119895(119909 119905) 119889119905]

]

119910119895 (119909) asymp 119891(119904)

119894(119909)

119894 = 1 2 119899 119904 = 1 2 119898

(85)




Φ119894 (119905) =

1 (119894 minus 1)119879

119898le 119905 lt 119894

119879

119898

0 otherwise(86)

with 119905 isin [0 119879) 119879119898 = ℎ and 119894 = 1 2 119898




Φ119894 (119905) Φ119895 (119905) = Φ119894 (119905) 119894 = 119895

0 119894 = 119895(87)



int

119879

0

Φ119894 (119905) Φ119895 (119905) 119889119905 = ℎ 119894 = 119895

0 119894 = 119895(88)



int

119879

0

1198912(119905) 119889119905 =

infin

sum

119894=1

1198912

119894

1003817100381710038171003817Φ119894(119905)10038171003817100381710038172 (89)



119891 (119905) =

119898

sum

119894=1

119891119894Φ119894 (119905) (90)



120576 =1

119879int

119879

0

(119891(119905) minus

119898

sum

119895=1

119891119895Φ119895(119905))

2

119889119905 (91)


Now 120597120576

120597119891119894

= minus2

119879int

119879

0

(119891 (119905) minus

119898

sum

119895=1

119891119895Φ119895 (119905))Φ119894 (119905) 119889119905 = 0

997904rArr 119891119894 =1

ℎint

119879

0


(92)


119891 (119905) =

119898

sum

119894=1

119891119894Φ119894 (119905) = 119865119879Φ (119905) = Φ

119879119865

where 119865 = [1198911 1198912 119891119898]119879

Φ (119905) = [Φ1 (119905) Φ2 (119905) Φ119898 (119905)]119879

(93)


119870 (119905 119904) = Φ119879(119905) 119870Ψ (119904) (94)



Φ (119905)Φ119879(119905) = (



d

0 0 sdot sdot sdot Φ119898 (119905)

) (95)


The second case is

Φ119879(119905) Φ (119905) = 1 (96)

because sum119898119894=1(Φ119894(119905))

2= sum119898

119894=1Φ119894(119905) = 1


int

119879

0

Φ (119905) 119889119905 = 119875Φ (119905) (97)

where

Φ (119905) = [Φ1 (119905) Φ2 (119905) Φ119898 (119905)]119879 (98)


119875 =1

2(




d


) (99)


int

119905

0

119891 (119905) 119889119905 = int

119905

0

119865119879Φ (119905) 119889119905 = 119865

119879119875Φ (119905) (100)


119899

sum

119895=1

119910119895 (119909) = 119891119894 (119909) +

119899

sum

119895=1

int

120573

120572

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899

(101)





119891119894 (119909) asymp 119865119879

119894Φ (119909)

119910119895 (119909) asymp 119884119879

119895Φ (119909)

119870119894119895 (119909 119905) asymp Φ119879(119905) 119870119894119895Φ (119909)

(102)


119899

sum

119895=1

119884119879

119895Φ (119909) = 119865

119879

119894Φ (119909)

+

119899

sum

119895=1

int

119898ℎ

0

119884119879

119895Φ (119905)Φ

119879(119905) 119870119894119895Φ (119909) 119889119905

119894 = 1 2 119899

(103)

119899

sum

119895=1

119884119879

119895Φ (119909) = 119865

119879

119894Φ (119909) +

119899

sum

119895=1

119884119879

119895ℎ119868119870119894119895Φ (119909)

119894 = 1 2 119899

(104)

since

int

119898ℎ

0

Φ (119905)Φ119879(119905) 119889119905 = ℎ119868 (105)

From (104) we get

119899

sum

119895=1

(119868 minus ℎ119870119879

119894119895) 119884119895 = 119865119894 119894 = 1 2 119899 (106)

Set 119860 119894119895 = 119868 minus ℎ119870119879

119894119895 then we have from (106)

119899

sum

119895=1

119860 119894119895119884119895 = 119865119894 119894 = 1 2 119899 (107)


(

(

11986011 11986012 1198601119899

11986021 11986022 1198602119899

1198601198991 1198601198992 119860119899119899

)

)

(

(

1198841

1198842

119884119899

)

)

=(

(

1198651

1198652

119865119899

)

)

(108)


119879119884119895 119895 = 1 2 119899



119906 (119909) = 119891 (119909) + int

1

0

119870 (119909 119905) 119865 (119905 119906 (119905)) 119889119905

0 le 119909 le 1

(109)





119910 (119909) = 119865 (119909 119906 (119909))

0 le 119909 le 1

(110)


119906 (119909) = 119860119879Ψ (119909)

119910 (119909) = 119861119879Ψ (119909)

(111)



119865 (119909) = 119863119879Ψ (119909)

119870 (119909 119905) = Ψ119879(119905) ΘΨ (119909)

(112)

where

Θ(119894119895) = int

1

0

[int

1

0

119870 (119909 119905) Ψ119894 (119905) 119889119905]Ψ119895 (119909) 119889119909 (113)


int

1

0

119870 (119909 119905) 119865 (119905 119906 (119905)) 119889119905

= int

1

0

119861119879Ψ (119905) Ψ

119879(119905) ΘΨ (119909) 119889119905

= 119861119879[int

1

0

Ψ (119905) Ψ119879(119905) 119889119905]ΘΨ (119909)

= 119861119879ΘΨ (119909) since int

1

0

Ψ (119905) Ψ119879(119905) 119889119905 = 119868

(114)



119860119879Ψ (119909) = 119863

119879Ψ (119909) + 119861

119879ΘΨ (119909) (115)


119860119879119875 = 119863

119879+ 119861119879Θ

119860119879119875 minus 119863

119879minus 119861119879Θ = 0

(116)


119875 = int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = [

1198751

1198752]

int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = 119868

(117)



119865 (119909 119860119879Ψ (119909)) = 119861

119879Ψ (119909) (118)


+119898minus2) where119894 = 1 2 2

119872+1+ 119898 minus 1







int

119887

119886

119891 (119909) 119889119909 =

119899minus1

sum

119894=1

int

119909119894+1

119909119894minus1

119891 (119909) 119889119909

=ℎ

3119891 (119886) +

4ℎ

3

1198992

sum

119894=1

119891 (1199092119894minus1)

+2ℎ

3

(119899minus1)2

sum

119894=1

119891 (1199092119894)

+ℎ

3119891 (119887)

minus(119887 minus 119886)

180ℎ4119891(4)(120578)

(119)


int

119887

119886

119891 (119909) 119889119909 =

119899

sum

119894=1

int

119909119894

119909119894minus1

119891 (119909) 119889119909

=ℎ

2119891 (119886) + ℎ

119899minus1

sum

119894=1

119891 (119909119894)

+ℎ

2119891 (119887)

+ℎ2

12[1198911015840(119886) minus 119891

1015840(119887)]

(120)


119906 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) 119865 (119906 (119905)) 119889119905

119886 le 119909 le 119887

(121)



+ℎ

3

[

[

119870 (119909 1199050) 119865 (1199060)

+ 4

1198992

sum

119895=1

119870(119909 1199052119895minus1) 119865 (1199062119895minus1)

+ 2

(1198992)minus1

sum

119895=1

119870(119909 1199052119895) 119865 (1199062119895)

+ 119870 (119909 119905119899) 119865 (119906119899)]

]

(122)


+ℎ

3

[

[

119870 (119909119894 1199050) 119865 (1199060)

+ 4

1198992

sum

119895=1

119870(119909119894 1199052119895minus1) 119865 (1199062119895minus1)


+ 2

(1198992)minus1

sum

119895=1

119870(119909119894 1199052119895) 119865 (1199062119895)

+119870 (119909119894 119905119899) 119865 (119906119899)]

]

(123)



119906 (119909) = 119891 (119909)

+ℎ

2119870 (119909 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909 119905119895) 119865 (119906119895)

+ℎ

2119870 (119909 119905119899) 119865 (119906119899)

+ℎ2

12[119869 (119909 1199050) 119865 (1199060)

+ 119870 (119909 1199050) 1199061015840

01198651015840(1199060)

minus 119869 (119909 119905119899) 119865 (119906119899)

minus119870 (119909 119905119899) 1199061015840

1198991198651015840(119906119899)]

(124)


have

119906 (119909119894) = 119891 (119909119894)

+ℎ

2119870 (119909119894 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909119894 119905119895) 119865 (119906119895)

+ℎ

2119870 (119909119894 119905119899) 119865 (119906119899)

+ℎ2

12[119869 (119909119894 1199050) 119865 (1199060)

+ 119870 (119909119894 1199050) 1199061015840

01198651015840(1199060)

minus 119869 (119909119894 119905119899) 119865 (119906119899)

minus119870 (119909119894 119905119899) 1199061015840

1198991198651015840(119906119899)]

(125)

for 119894 = 0 1 119899


120597119870(119909 119905)120597119909 we obtain

1199061015840(119909) = 119891

1015840(119909) + int

119887

119886

119867(119909 119905) 119865 (119906 (119905)) 119889119905

119886 le 119909 le 119887

(126)


1199061015840(119909119894) = 119891

1015840(119909119894)

+ℎ

2119867 (119909119894 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(119909119894 119905119895) 119865 (119906119895)

+ℎ

2119867 (119909119894 119905119899) 119865 (119906119899)

(127)



119906 (119909119894) = (ℎ

2119870 (119909119894 1199050) +

ℎ2

12119869 (119909119894 1199050))119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909119894 119905119895) 119865 (119906119895)

+ (ℎ

2119870 (119909119894 119905119899) minus

ℎ2

12119869 (119909119894 119905119899))119865 (119906119899)

+ℎ2

12(119870 (119909119894 1199050) 119906

1015840

01198651015840(1199060)

minus119870 (119909119894 119905119899) 1199061015840

1198991198651015840(119906119899))

1199061015840(1199090) = 119891

1015840(1199090)

+ℎ

2119867 (1199090 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(1199090 119905119895) 119865 (119906119895)

+ℎ

2119867 (1199090 119905119899) 119865 (119906119899)


1199061015840(119909119899) = 119891

1015840(119909119899)

+ℎ

2119867 (119909119899 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(119909119899 119905119895) 119865 (119906119895)

+ℎ

2119867 (119909119899 119905119899) 119865 (119906119899)

(128)




120595119886119887 (119905) = |119886|minus12120595(

119905 minus 119887

119886)

119886 119887 isin 119877 119886 = 0

(129)



120595119898119899 (119905)

=

(119898 +1

2)

12

21198962119871119898 (2

119896119905 minus 119899)

119899 minus 1

2119896le 119905 lt

119899 + 1

2119896

0 otherwise(130)


1198710 (119905) = 1

1198711 (119905) = 119905

119871119898+1 (119905) =2119898 + 1

119898 + 1119905119871119898 (119905)

minus119898

119898 + 1119871119898minus1 (119905) 119898 = 1 2 3

(131)



119891 (119909) =

infin

sum

119899=1

infin

sum

119898=0

119888119899119898120595119899119898 (119909) (132)

where

119888119899119898 = ⟨119891 (119909) 120595119899119898 (119909)⟩ (133)


119891 (119909) asymp

2119896minus1

sum

119899=1

119872minus1

sum

119898=0

119888119899119898120595119899119898 (119909) = 119862119879Ψ (119909) (134)


119862 = [11988810 11988811 1198881119872minus1 11988820


(135)

Ψ (119909) = [12059510 (119909) 1205951119872minus1 (119909)

12059520 (119909) 1205952119872minus1 (119909)


(136)


approximated as

119896 (119909 119905) asymp Ψ119879(119905) 119870Ψ (119909) (137)


119870119894119895 = ⟨120595119894 (119905) ⟨119896 (119909 119905) 120595119895 (119909)⟩⟩ (138)


[119910 (119909)]119901= [119884119879Ψ (119909)]

119901

= 119884lowast

119901

119879Ψ (119909) (139)





int

119905

0

Ψ(1199051015840) 1198891199051015840= 119875Ψ (119905) (140)



119875 =

[[[[[[

[

119871 119867 sdot sdot sdot 119867 119867

0 119871 sdot sdot sdot 119867 119867

d

0 0 sdot sdot sdot 119871 119867


]]]]]]

]

(141)



119867 =1

2119896

[[[[

[



d


]]]]

]

119871 =1

2119896

[[[[[[[[[[[[[[[[[[[[[[[

[

11


minusradic3

30

radic3


0 minusradic5

5radic30

radic5


0 0 minusradic7


d





]]]]]]]]]]]]]]]]]]]]]]]

]

(142)


int

1

0

Ψ (119905)Ψ119879(119905) 119889119905 = 119868 (143)


defined as

Ψ (119905) Ψ119879(119905) 119862 = 119862

119879Ψ (119905) (144)




119910 (119909) = 119891 (119909) + int

1

0

119896 (119909 119905) [119910 (119905)]119901119889119905 (145)



119891 (119909) asymp 119865119879Ψ (119909)

119910 (119909) asymp 119884119879Ψ (119909)

119896 (119909 119905) asymp Ψ119879(119905) 119870Ψ (119909)

[119910 (119909)]119901asymp 119884lowast119879Ψ (119909)

(146)


119884119879Ψ (119909) = 119865

119879Ψ (119909)

+ int

1

0

119884lowast119879Ψ (119905) Ψ

119879(119905) 119870Ψ (119909) 119889119905

= 119865119879Ψ (119909)

+ 119884lowast119879(int

1

0

Ψ (119905) Ψ119879(119905) 119889119905)119870Ψ (119909)

= 119865119879Ψ (119909) + 119884

lowast119879119870Ψ (119909)

= (119865119879+ 119884lowast119879119870)Ψ (119909)


119879= 0

(147)



119906 (119909) = 119891 (119909) + int

1

0

119870 (119909 119905) 119865 (119906 (119905)) 119889119905

0 le 119909 le 1

(148)


119871 (119906) = 119906 (119909) minus 119891 (119909) minus int

1

0

119870 (119909 119905) 119865 (119906 (119905)) 119889119905 = 0 (149)



119867(119906 0) = 119873 (119906)

119867 (119906 1) = 119871 (119906)

(150)


119867(119906 119901) = (1 minus 119901)119873 (119906) + 119901119871 (119906) = 0 (151)


119906 = 1199060 + 1199011199061 + 11990121199062 + sdot sdot sdot (152a)


119880 = lim119901rarr1

119906 = 1199060 + 1199061 + 1199062 + sdot sdot sdot (152b)


Consider

119873(119906) = 119906 (119909) minus 119891 (119909) (153)


119865 (119906) =

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898)

= 1198670 (1199060) + 1199011198671 (1199060 1199061)


(154)

where119867119898 (1199060 1199061 119906119898)

=1

119898

120597119898

120597119901119898(119865(

119898

sum

119896=0

119901119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816119901=0

119898 ge 0

(155)




minusint

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905) = 0


minus 119901int

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905 = 0

(156)


1199010 1199060 (119909) minus 119891 (119909) = 0 997904rArr 1199060 (119909) = 119891 (119909)

1199011 1199061 (119909) minus int

1

0

119870 (119909 119905)1198670119889119905 = 0 997904rArr 1199061 (119909)

= int

1

0

119870 (119909 119905)1198670119889119905

1199012 1199062 (119909) minus int

1

0

119870 (119909 119905)1198671119889119905 = 0 997904rArr 1199062 (119909)

= int

1

0

119870 (119909 119905)1198671119889119905

(157)


1199060 (119909) = 119891 (119909)

119906119899+1 (119909) = int

1

0

119870 (119909 119905)119867119899119889119905 119899 = 0 1 2

(158)



119906 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) (119871119906 (119905) + 119873119906 (119905)) 119889119905

119886 le 119909 le 119887

(159)



119906 (119909) =

infin

sum

119898=0

119906119898 (119909) (160)


119873119906 =

infin

sum

119899=0

119860119899 (161)

where 119860119899 =1

119899(120597119899

120597120582119899119873(

infin

sum

119896=0

120582119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816120582=0

(162)



infin

sum

119898=0

119906119898 (119909) = 119891 (119909)

+ int

119887

119886

119870 (119909 119905) (119871(

infin

sum

119898=0

119906119898 (119905)) +

infin

sum

119898=0

119860119898)119889119905

(163)


1199060 (119909) = 119891 (119909)

119906119898 (119909) = int

119887

119886

119870 (119909 119905) (119871 (119906119898minus1 (119905)) + 119860119898minus1 (119905)) 119889119905

119898 ge 1

(164)


120593119899 = 1199060 + 1199061 + sdot sdot sdot + 119906119899 (165)

with

119906 (119909) = lim119899rarrinfin

120593119899 (166)




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120575119910119899+1 (119909) = 120575119910119899 (119909)

+ 120575int

119909

0

120582 (120585) (1199101015840

119899(120585) minus 119891

1015840(120585)

minusint

119887

119886

1198701015840(120585 119905) 119910119899 (119905) 119889119905) 119889120585

= 120575119910119899 (119909) + int

119909

0

120582 (120585) 120575 (1199101015840

119899(120585)) 119889120585

= 120575119910119899 (119909) + 1205821205751199101198991003816100381610038161003816120585=119909 minus int

119909

0

1205821015840(120585) 120575119910119899 (120585) 119889120585

(49)


120575119910119899+1 = 0 (50)


1205821015840(120585) = 0 (51)


1 + 120582(120585)1003816100381610038161003816120585=119909 = 0 (52)


120582 = minus1 (53)


119910119899+1 (119909) = 119910119899 (119909)

minus int

119909

0

(1199101015840

119899(120585) minus 119891

1015840(120585) minus int

119887

119886

1198701015840(120585 119905) 119910119899 (119905) 119889119905) 119889120585

119899 ge 0

(54)




119899

sum

119895=1

119910119895 (119909) = 119891119894 (119909) +

119899

sum

119895=1

int

1

0

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899

(55)




ℎ0 = 1 0 le 119909 lt 1

0 others(56)


ℎ1 (119909) =

1 0 le 119909 lt1

2

minus11

2le 119909 lt 1

0 others

(57)


ℎ119899 (119909) = ℎ1 (2119895119909 minus 119896) (58)


expressed as

int

1

0

ℎ119898 (119909) ℎ119899 (119909) 119889119909 = 2minus119895120575119898119899 =

2minus119895 119898 = 119899 = 2

119895+ 119896

0 119898 = 119899

(59)


119910 (119909) =

+infin

sum

119899=0

119888119899ℎ119899 (119909) (60)


119888119899 = 2119895int

1

0

119910 (119909) ℎ119899 (119909) 119889119909

119899 = 2119895+ 119896 119895 ge 0 0 le 119896 lt 2

119895

(61)


0119910(119909)119889119909


119910 (119909) asymp

119898minus1

sum

119899=0

119888119899ℎ119899 (119909) = 119862119879

(119898)ℎ(119898) (119909) (62)



119862119879

(119898)= [1198880 1198881 119888119898minus1] 119898 = 2

119895

ℎ(119898) (119909) = [ℎ0 (119909) ℎ1 (119909) ℎ119898minus1 (119909)]119879 119898 = 2

119895

(63)



119870 (119909 119905) asymp

119898minus1

sum

119906=0

119898minus1

sum

V=0

119886119906VℎV (119909) ℎ119906 (119905) (64)

where 119886119906V = 2119894+119902∬1

0119870(119909 119905)ℎV(119909)ℎ119906(119905)119889119909 119889119905 119906 = 2

119894+ 119895 V =


119870 (119909 119905) asymp ℎ119879

(119898)(119905) 119870ℎ(119898) (119909) (65)

where

119870 = (119886119906V)119879

119898times119898 (66)


119867(119898) = [ℎ(119898) (1

2119898) ℎ(119898) (

3

2119898) ℎ(119898) (

2119898 minus 1

2119898)]

(67)


represented as

119867(4) = [ℎ(4) (1

8) ℎ(4) (

3

8) ℎ(4) (

7

8)]

=[[[

[

1 1 1 1

1 1 minus1 minus1

1 minus1 0 0

0 0 1 minus1

]]]

]

(68)


int

119909

0

ℎ(119898) (119905) 119889119905 = 119875(119898)ℎ(119898) (119909)

119909 isin [0 1)

(69)


119875(1) = [1

2]

119875(119898) =1

2119898[

2119898119875(1198982) minus119867(1198982)

119867minus1

(1198982)0

]

(70)


119875(2) =1

4[2 minus1

1 0]

119875(4) =1

16

[[[

[


4 0 minus2 2

1 1 0 0

1 minus1 0 0

]]]

]

119875(8) =1

64

[[[[[[[[[[

[



4 4 0 0 minus4 4 0 0

4 4 0 0 0 0 minus4 4

1 1 2 0 0 0 0 0

1 1 minus2 0 0 0 0 0

1 minus1 0 2 0 0 0 0


]]]]]]]]]]

]

(71)

Therefore we get

119867minus1

(119898)= (

1

119898)119867119879

(119898)

times diag(1 1 2 2 22 22⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

22

2120572minus1 2

120572minus1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

2120572minus1

)

(72)


sented as

int

1

0

ℎ(119898) (119905) ℎ119879

(119898)(119905) 119889119905 = 119863 (73)

where

119863 = diag(1 1 12 12 122 122⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

22

12120572minus1 12

120572minus1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

2120572minus1

)

(74)


119898

sum

119895=1

119910119895 (119909) = 119891119894 (119909) +

119898

sum

119895=1

int

1

0

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119898

(75)


119910119895 (119909) asymp 119884119879

119895ℎ(119898) (119909) 119895 = 1 2 119898

119870119894119895 (119909 119905) asymp ℎ119879

(119898)(119905) 119870119894119895ℎ(119898) (119909)

119894 119895 = 1 2 119898

(76)



119898

sum

119895=1

119884119879

119895ℎ(119898) (119909)

= 119891119894 (119909) +

119898

sum

119895=1

int

1

0

119884119879

119895ℎ(119898) (119905) ℎ

119879

(119898)(119905) 119870119894119895ℎ(119898) (119909) 119889119905

119894 = 1 2 119898

(77)


119898

sum

119895=1

119884119879

119895ℎ(119898) (119909) = 119891119894 (119909) +

119898

sum

119895=1

119884119879

119895119863119870119894119895ℎ(119898) (119909)

119894 = 1 2 119898

(78)


119894=1 in the


119898

sum

119895=1

119884119879

119895ℎ(119898) (119909119894) = 119891119894 (119909119894) +

119898

sum

119895=1

119884119879

119895119863119870119894119895ℎ(119898) (119909119894)

119894 = 1 2 119898

(79)


119879

119895ℎ(119898)(119909) 119895 = 1 2 119898



119910119894 (119909) = 119891119894 (119909) +

119899

sum

119895=1

int

1

0

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899 0 le 119909 le 1

(80)


119910119895 (119905) = 119910119895 (119909) + 1199101015840


+1

119898119910(119898)

119895(119909) (119905 minus 119909)

119898+ 119864 (119905)

(81)



int1

0sum119899



119910119894 (119909) asymp 119891119894 (119909)

+

119899

sum

119895=1

int

1

0

119870119894119895 (119909 119905)

119898

sum

119903=0

1

119903(119905 minus 119909)

119903119910(119903)

119895(119909) 119889119905

119894 = 1 2 119899

119910119894 (119909) asymp 119891119894 (119909)

+

119899

sum

119895=1

119898

sum

119903=0

1

119903119910(119903)

119895(119909) int

1

0

119870119894119895 (119909 119905) (119905 minus 119909)119903119889119905

119894 = 1 2 119899

(82)

119910119894 (119909) minus

119899

sum

119895=1

119898

sum

119903=0

1

119903119910(119903)

119895(119909) [int

1

0

119870119894119895 (119909 119905) (119905 minus 119909)119903119889119905]

asymp 119891119894 (119909) 119894 = 1 2 119899

(83)



119910(119904)

119894(119909) = 119891

(119904)

119894(119909) +

119899

sum

119895=1

int

1

0

119870(119904)

119894119895(119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899 119904 = 1 2 119898

(84)

where119870(119904)119894119895(119909 119905) = 120597

(119904)119870119894119895(119909 119905)120597119909


have

119910(119904)

119894(119909) minus [

[

119899

sum

119895=1

int

1

0

119870(119904)

119894119895(119909 119905) 119889119905]

]

119910119895 (119909) asymp 119891(119904)

119894(119909)

119894 = 1 2 119899 119904 = 1 2 119898

(85)




Φ119894 (119905) =

1 (119894 minus 1)119879

119898le 119905 lt 119894

119879

119898

0 otherwise(86)

with 119905 isin [0 119879) 119879119898 = ℎ and 119894 = 1 2 119898




Φ119894 (119905) Φ119895 (119905) = Φ119894 (119905) 119894 = 119895

0 119894 = 119895(87)



int

119879

0

Φ119894 (119905) Φ119895 (119905) 119889119905 = ℎ 119894 = 119895

0 119894 = 119895(88)



int

119879

0

1198912(119905) 119889119905 =

infin

sum

119894=1

1198912

119894

1003817100381710038171003817Φ119894(119905)10038171003817100381710038172 (89)



119891 (119905) =

119898

sum

119894=1

119891119894Φ119894 (119905) (90)



120576 =1

119879int

119879

0

(119891(119905) minus

119898

sum

119895=1

119891119895Φ119895(119905))

2

119889119905 (91)


Now 120597120576

120597119891119894

= minus2

119879int

119879

0

(119891 (119905) minus

119898

sum

119895=1

119891119895Φ119895 (119905))Φ119894 (119905) 119889119905 = 0

997904rArr 119891119894 =1

ℎint

119879

0


(92)


119891 (119905) =

119898

sum

119894=1

119891119894Φ119894 (119905) = 119865119879Φ (119905) = Φ

119879119865

where 119865 = [1198911 1198912 119891119898]119879

Φ (119905) = [Φ1 (119905) Φ2 (119905) Φ119898 (119905)]119879

(93)


119870 (119905 119904) = Φ119879(119905) 119870Ψ (119904) (94)



Φ (119905)Φ119879(119905) = (



d

0 0 sdot sdot sdot Φ119898 (119905)

) (95)


The second case is

Φ119879(119905) Φ (119905) = 1 (96)

because sum119898119894=1(Φ119894(119905))

2= sum119898

119894=1Φ119894(119905) = 1


int

119879

0

Φ (119905) 119889119905 = 119875Φ (119905) (97)

where

Φ (119905) = [Φ1 (119905) Φ2 (119905) Φ119898 (119905)]119879 (98)


119875 =1

2(




d


) (99)


int

119905

0

119891 (119905) 119889119905 = int

119905

0

119865119879Φ (119905) 119889119905 = 119865

119879119875Φ (119905) (100)


119899

sum

119895=1

119910119895 (119909) = 119891119894 (119909) +

119899

sum

119895=1

int

120573

120572

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899

(101)





119891119894 (119909) asymp 119865119879

119894Φ (119909)

119910119895 (119909) asymp 119884119879

119895Φ (119909)

119870119894119895 (119909 119905) asymp Φ119879(119905) 119870119894119895Φ (119909)

(102)


119899

sum

119895=1

119884119879

119895Φ (119909) = 119865

119879

119894Φ (119909)

+

119899

sum

119895=1

int

119898ℎ

0

119884119879

119895Φ (119905)Φ

119879(119905) 119870119894119895Φ (119909) 119889119905

119894 = 1 2 119899

(103)

119899

sum

119895=1

119884119879

119895Φ (119909) = 119865

119879

119894Φ (119909) +

119899

sum

119895=1

119884119879

119895ℎ119868119870119894119895Φ (119909)

119894 = 1 2 119899

(104)

since

int

119898ℎ

0

Φ (119905)Φ119879(119905) 119889119905 = ℎ119868 (105)

From (104) we get

119899

sum

119895=1

(119868 minus ℎ119870119879

119894119895) 119884119895 = 119865119894 119894 = 1 2 119899 (106)

Set 119860 119894119895 = 119868 minus ℎ119870119879

119894119895 then we have from (106)

119899

sum

119895=1

119860 119894119895119884119895 = 119865119894 119894 = 1 2 119899 (107)


(

(

11986011 11986012 1198601119899

11986021 11986022 1198602119899

1198601198991 1198601198992 119860119899119899

)

)

(

(

1198841

1198842

119884119899

)

)

=(

(

1198651

1198652

119865119899

)

)

(108)


119879119884119895 119895 = 1 2 119899



119906 (119909) = 119891 (119909) + int

1

0

119870 (119909 119905) 119865 (119905 119906 (119905)) 119889119905

0 le 119909 le 1

(109)





119910 (119909) = 119865 (119909 119906 (119909))

0 le 119909 le 1

(110)


119906 (119909) = 119860119879Ψ (119909)

119910 (119909) = 119861119879Ψ (119909)

(111)



119865 (119909) = 119863119879Ψ (119909)

119870 (119909 119905) = Ψ119879(119905) ΘΨ (119909)

(112)

where

Θ(119894119895) = int

1

0

[int

1

0

119870 (119909 119905) Ψ119894 (119905) 119889119905]Ψ119895 (119909) 119889119909 (113)


int

1

0

119870 (119909 119905) 119865 (119905 119906 (119905)) 119889119905

= int

1

0

119861119879Ψ (119905) Ψ

119879(119905) ΘΨ (119909) 119889119905

= 119861119879[int

1

0

Ψ (119905) Ψ119879(119905) 119889119905]ΘΨ (119909)

= 119861119879ΘΨ (119909) since int

1

0

Ψ (119905) Ψ119879(119905) 119889119905 = 119868

(114)



119860119879Ψ (119909) = 119863

119879Ψ (119909) + 119861

119879ΘΨ (119909) (115)


119860119879119875 = 119863

119879+ 119861119879Θ

119860119879119875 minus 119863

119879minus 119861119879Θ = 0

(116)


119875 = int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = [

1198751

1198752]

int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = 119868

(117)



119865 (119909 119860119879Ψ (119909)) = 119861

119879Ψ (119909) (118)


+119898minus2) where119894 = 1 2 2

119872+1+ 119898 minus 1







int

119887

119886

119891 (119909) 119889119909 =

119899minus1

sum

119894=1

int

119909119894+1

119909119894minus1

119891 (119909) 119889119909

=ℎ

3119891 (119886) +

4ℎ

3

1198992

sum

119894=1

119891 (1199092119894minus1)

+2ℎ

3

(119899minus1)2

sum

119894=1

119891 (1199092119894)

+ℎ

3119891 (119887)

minus(119887 minus 119886)

180ℎ4119891(4)(120578)

(119)


int

119887

119886

119891 (119909) 119889119909 =

119899

sum

119894=1

int

119909119894

119909119894minus1

119891 (119909) 119889119909

=ℎ

2119891 (119886) + ℎ

119899minus1

sum

119894=1

119891 (119909119894)

+ℎ

2119891 (119887)

+ℎ2

12[1198911015840(119886) minus 119891

1015840(119887)]

(120)


119906 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) 119865 (119906 (119905)) 119889119905

119886 le 119909 le 119887

(121)



+ℎ

3

[

[

119870 (119909 1199050) 119865 (1199060)

+ 4

1198992

sum

119895=1

119870(119909 1199052119895minus1) 119865 (1199062119895minus1)

+ 2

(1198992)minus1

sum

119895=1

119870(119909 1199052119895) 119865 (1199062119895)

+ 119870 (119909 119905119899) 119865 (119906119899)]

]

(122)


+ℎ

3

[

[

119870 (119909119894 1199050) 119865 (1199060)

+ 4

1198992

sum

119895=1

119870(119909119894 1199052119895minus1) 119865 (1199062119895minus1)


+ 2

(1198992)minus1

sum

119895=1

119870(119909119894 1199052119895) 119865 (1199062119895)

+119870 (119909119894 119905119899) 119865 (119906119899)]

]

(123)



119906 (119909) = 119891 (119909)

+ℎ

2119870 (119909 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909 119905119895) 119865 (119906119895)

+ℎ

2119870 (119909 119905119899) 119865 (119906119899)

+ℎ2

12[119869 (119909 1199050) 119865 (1199060)

+ 119870 (119909 1199050) 1199061015840

01198651015840(1199060)

minus 119869 (119909 119905119899) 119865 (119906119899)

minus119870 (119909 119905119899) 1199061015840

1198991198651015840(119906119899)]

(124)


have

119906 (119909119894) = 119891 (119909119894)

+ℎ

2119870 (119909119894 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909119894 119905119895) 119865 (119906119895)

+ℎ

2119870 (119909119894 119905119899) 119865 (119906119899)

+ℎ2

12[119869 (119909119894 1199050) 119865 (1199060)

+ 119870 (119909119894 1199050) 1199061015840

01198651015840(1199060)

minus 119869 (119909119894 119905119899) 119865 (119906119899)

minus119870 (119909119894 119905119899) 1199061015840

1198991198651015840(119906119899)]

(125)

for 119894 = 0 1 119899


120597119870(119909 119905)120597119909 we obtain

1199061015840(119909) = 119891

1015840(119909) + int

119887

119886

119867(119909 119905) 119865 (119906 (119905)) 119889119905

119886 le 119909 le 119887

(126)


1199061015840(119909119894) = 119891

1015840(119909119894)

+ℎ

2119867 (119909119894 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(119909119894 119905119895) 119865 (119906119895)

+ℎ

2119867 (119909119894 119905119899) 119865 (119906119899)

(127)



119906 (119909119894) = (ℎ

2119870 (119909119894 1199050) +

ℎ2

12119869 (119909119894 1199050))119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909119894 119905119895) 119865 (119906119895)

+ (ℎ

2119870 (119909119894 119905119899) minus

ℎ2

12119869 (119909119894 119905119899))119865 (119906119899)

+ℎ2

12(119870 (119909119894 1199050) 119906

1015840

01198651015840(1199060)

minus119870 (119909119894 119905119899) 1199061015840

1198991198651015840(119906119899))

1199061015840(1199090) = 119891

1015840(1199090)

+ℎ

2119867 (1199090 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(1199090 119905119895) 119865 (119906119895)

+ℎ

2119867 (1199090 119905119899) 119865 (119906119899)


1199061015840(119909119899) = 119891

1015840(119909119899)

+ℎ

2119867 (119909119899 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(119909119899 119905119895) 119865 (119906119895)

+ℎ

2119867 (119909119899 119905119899) 119865 (119906119899)

(128)




120595119886119887 (119905) = |119886|minus12120595(

119905 minus 119887

119886)

119886 119887 isin 119877 119886 = 0

(129)



120595119898119899 (119905)

=

(119898 +1

2)

12

21198962119871119898 (2

119896119905 minus 119899)

119899 minus 1

2119896le 119905 lt

119899 + 1

2119896

0 otherwise(130)


1198710 (119905) = 1

1198711 (119905) = 119905

119871119898+1 (119905) =2119898 + 1

119898 + 1119905119871119898 (119905)

minus119898

119898 + 1119871119898minus1 (119905) 119898 = 1 2 3

(131)



119891 (119909) =

infin

sum

119899=1

infin

sum

119898=0

119888119899119898120595119899119898 (119909) (132)

where

119888119899119898 = ⟨119891 (119909) 120595119899119898 (119909)⟩ (133)


119891 (119909) asymp

2119896minus1

sum

119899=1

119872minus1

sum

119898=0

119888119899119898120595119899119898 (119909) = 119862119879Ψ (119909) (134)


119862 = [11988810 11988811 1198881119872minus1 11988820


(135)

Ψ (119909) = [12059510 (119909) 1205951119872minus1 (119909)

12059520 (119909) 1205952119872minus1 (119909)


(136)


approximated as

119896 (119909 119905) asymp Ψ119879(119905) 119870Ψ (119909) (137)


119870119894119895 = ⟨120595119894 (119905) ⟨119896 (119909 119905) 120595119895 (119909)⟩⟩ (138)


[119910 (119909)]119901= [119884119879Ψ (119909)]

119901

= 119884lowast

119901

119879Ψ (119909) (139)





int

119905

0

Ψ(1199051015840) 1198891199051015840= 119875Ψ (119905) (140)



119875 =

[[[[[[

[

119871 119867 sdot sdot sdot 119867 119867

0 119871 sdot sdot sdot 119867 119867

d

0 0 sdot sdot sdot 119871 119867


]]]]]]

]

(141)



119867 =1

2119896

[[[[

[



d


]]]]

]

119871 =1

2119896

[[[[[[[[[[[[[[[[[[[[[[[

[

11


minusradic3

30

radic3


0 minusradic5

5radic30

radic5


0 0 minusradic7


d





]]]]]]]]]]]]]]]]]]]]]]]

]

(142)


int

1

0

Ψ (119905)Ψ119879(119905) 119889119905 = 119868 (143)


defined as

Ψ (119905) Ψ119879(119905) 119862 = 119862

119879Ψ (119905) (144)




119910 (119909) = 119891 (119909) + int

1

0

119896 (119909 119905) [119910 (119905)]119901119889119905 (145)



119891 (119909) asymp 119865119879Ψ (119909)

119910 (119909) asymp 119884119879Ψ (119909)

119896 (119909 119905) asymp Ψ119879(119905) 119870Ψ (119909)

[119910 (119909)]119901asymp 119884lowast119879Ψ (119909)

(146)


119884119879Ψ (119909) = 119865

119879Ψ (119909)

+ int

1

0

119884lowast119879Ψ (119905) Ψ

119879(119905) 119870Ψ (119909) 119889119905

= 119865119879Ψ (119909)

+ 119884lowast119879(int

1

0

Ψ (119905) Ψ119879(119905) 119889119905)119870Ψ (119909)

= 119865119879Ψ (119909) + 119884

lowast119879119870Ψ (119909)

= (119865119879+ 119884lowast119879119870)Ψ (119909)


119879= 0

(147)



119906 (119909) = 119891 (119909) + int

1

0

119870 (119909 119905) 119865 (119906 (119905)) 119889119905

0 le 119909 le 1

(148)


119871 (119906) = 119906 (119909) minus 119891 (119909) minus int

1

0

119870 (119909 119905) 119865 (119906 (119905)) 119889119905 = 0 (149)



119867(119906 0) = 119873 (119906)

119867 (119906 1) = 119871 (119906)

(150)


119867(119906 119901) = (1 minus 119901)119873 (119906) + 119901119871 (119906) = 0 (151)


119906 = 1199060 + 1199011199061 + 11990121199062 + sdot sdot sdot (152a)


119880 = lim119901rarr1

119906 = 1199060 + 1199061 + 1199062 + sdot sdot sdot (152b)


Consider

119873(119906) = 119906 (119909) minus 119891 (119909) (153)


119865 (119906) =

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898)

= 1198670 (1199060) + 1199011198671 (1199060 1199061)


(154)

where119867119898 (1199060 1199061 119906119898)

=1

119898

120597119898

120597119901119898(119865(

119898

sum

119896=0

119901119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816119901=0

119898 ge 0

(155)




minusint

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905) = 0


minus 119901int

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905 = 0

(156)


1199010 1199060 (119909) minus 119891 (119909) = 0 997904rArr 1199060 (119909) = 119891 (119909)

1199011 1199061 (119909) minus int

1

0

119870 (119909 119905)1198670119889119905 = 0 997904rArr 1199061 (119909)

= int

1

0

119870 (119909 119905)1198670119889119905

1199012 1199062 (119909) minus int

1

0

119870 (119909 119905)1198671119889119905 = 0 997904rArr 1199062 (119909)

= int

1

0

119870 (119909 119905)1198671119889119905

(157)


1199060 (119909) = 119891 (119909)

119906119899+1 (119909) = int

1

0

119870 (119909 119905)119867119899119889119905 119899 = 0 1 2

(158)



119906 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) (119871119906 (119905) + 119873119906 (119905)) 119889119905

119886 le 119909 le 119887

(159)



119906 (119909) =

infin

sum

119898=0

119906119898 (119909) (160)


119873119906 =

infin

sum

119899=0

119860119899 (161)

where 119860119899 =1

119899(120597119899

120597120582119899119873(

infin

sum

119896=0

120582119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816120582=0

(162)



infin

sum

119898=0

119906119898 (119909) = 119891 (119909)

+ int

119887

119886

119870 (119909 119905) (119871(

infin

sum

119898=0

119906119898 (119905)) +

infin

sum

119898=0

119860119898)119889119905

(163)


1199060 (119909) = 119891 (119909)

119906119898 (119909) = int

119887

119886

119870 (119909 119905) (119871 (119906119898minus1 (119905)) + 119860119898minus1 (119905)) 119889119905

119898 ge 1

(164)


120593119899 = 1199060 + 1199061 + sdot sdot sdot + 119906119899 (165)

with

119906 (119909) = lim119899rarrinfin

120593119899 (166)




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119870 (119909 119905) asymp

119898minus1

sum

119906=0

119898minus1

sum

V=0

119886119906VℎV (119909) ℎ119906 (119905) (64)

where 119886119906V = 2119894+119902∬1

0119870(119909 119905)ℎV(119909)ℎ119906(119905)119889119909 119889119905 119906 = 2

119894+ 119895 V =


119870 (119909 119905) asymp ℎ119879

(119898)(119905) 119870ℎ(119898) (119909) (65)

where

119870 = (119886119906V)119879

119898times119898 (66)


119867(119898) = [ℎ(119898) (1

2119898) ℎ(119898) (

3

2119898) ℎ(119898) (

2119898 minus 1

2119898)]

(67)


represented as

119867(4) = [ℎ(4) (1

8) ℎ(4) (

3

8) ℎ(4) (

7

8)]

=[[[

[

1 1 1 1

1 1 minus1 minus1

1 minus1 0 0

0 0 1 minus1

]]]

]

(68)


int

119909

0

ℎ(119898) (119905) 119889119905 = 119875(119898)ℎ(119898) (119909)

119909 isin [0 1)

(69)


119875(1) = [1

2]

119875(119898) =1

2119898[

2119898119875(1198982) minus119867(1198982)

119867minus1

(1198982)0

]

(70)


119875(2) =1

4[2 minus1

1 0]

119875(4) =1

16

[[[

[


4 0 minus2 2

1 1 0 0

1 minus1 0 0

]]]

]

119875(8) =1

64

[[[[[[[[[[

[



4 4 0 0 minus4 4 0 0

4 4 0 0 0 0 minus4 4

1 1 2 0 0 0 0 0

1 1 minus2 0 0 0 0 0

1 minus1 0 2 0 0 0 0


]]]]]]]]]]

]

(71)

Therefore we get

119867minus1

(119898)= (

1

119898)119867119879

(119898)

times diag(1 1 2 2 22 22⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

22

2120572minus1 2

120572minus1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

2120572minus1

)

(72)


sented as

int

1

0

ℎ(119898) (119905) ℎ119879

(119898)(119905) 119889119905 = 119863 (73)

where

119863 = diag(1 1 12 12 122 122⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

22

12120572minus1 12

120572minus1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

2120572minus1

)

(74)


119898

sum

119895=1

119910119895 (119909) = 119891119894 (119909) +

119898

sum

119895=1

int

1

0

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119898

(75)


119910119895 (119909) asymp 119884119879

119895ℎ(119898) (119909) 119895 = 1 2 119898

119870119894119895 (119909 119905) asymp ℎ119879

(119898)(119905) 119870119894119895ℎ(119898) (119909)

119894 119895 = 1 2 119898

(76)



119898

sum

119895=1

119884119879

119895ℎ(119898) (119909)

= 119891119894 (119909) +

119898

sum

119895=1

int

1

0

119884119879

119895ℎ(119898) (119905) ℎ

119879

(119898)(119905) 119870119894119895ℎ(119898) (119909) 119889119905

119894 = 1 2 119898

(77)


119898

sum

119895=1

119884119879

119895ℎ(119898) (119909) = 119891119894 (119909) +

119898

sum

119895=1

119884119879

119895119863119870119894119895ℎ(119898) (119909)

119894 = 1 2 119898

(78)


119894=1 in the


119898

sum

119895=1

119884119879

119895ℎ(119898) (119909119894) = 119891119894 (119909119894) +

119898

sum

119895=1

119884119879

119895119863119870119894119895ℎ(119898) (119909119894)

119894 = 1 2 119898

(79)


119879

119895ℎ(119898)(119909) 119895 = 1 2 119898



119910119894 (119909) = 119891119894 (119909) +

119899

sum

119895=1

int

1

0

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899 0 le 119909 le 1

(80)


119910119895 (119905) = 119910119895 (119909) + 1199101015840


+1

119898119910(119898)

119895(119909) (119905 minus 119909)

119898+ 119864 (119905)

(81)



int1

0sum119899



119910119894 (119909) asymp 119891119894 (119909)

+

119899

sum

119895=1

int

1

0

119870119894119895 (119909 119905)

119898

sum

119903=0

1

119903(119905 minus 119909)

119903119910(119903)

119895(119909) 119889119905

119894 = 1 2 119899

119910119894 (119909) asymp 119891119894 (119909)

+

119899

sum

119895=1

119898

sum

119903=0

1

119903119910(119903)

119895(119909) int

1

0

119870119894119895 (119909 119905) (119905 minus 119909)119903119889119905

119894 = 1 2 119899

(82)

119910119894 (119909) minus

119899

sum

119895=1

119898

sum

119903=0

1

119903119910(119903)

119895(119909) [int

1

0

119870119894119895 (119909 119905) (119905 minus 119909)119903119889119905]

asymp 119891119894 (119909) 119894 = 1 2 119899

(83)



119910(119904)

119894(119909) = 119891

(119904)

119894(119909) +

119899

sum

119895=1

int

1

0

119870(119904)

119894119895(119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899 119904 = 1 2 119898

(84)

where119870(119904)119894119895(119909 119905) = 120597

(119904)119870119894119895(119909 119905)120597119909


have

119910(119904)

119894(119909) minus [

[

119899

sum

119895=1

int

1

0

119870(119904)

119894119895(119909 119905) 119889119905]

]

119910119895 (119909) asymp 119891(119904)

119894(119909)

119894 = 1 2 119899 119904 = 1 2 119898

(85)




Φ119894 (119905) =

1 (119894 minus 1)119879

119898le 119905 lt 119894

119879

119898

0 otherwise(86)

with 119905 isin [0 119879) 119879119898 = ℎ and 119894 = 1 2 119898




Φ119894 (119905) Φ119895 (119905) = Φ119894 (119905) 119894 = 119895

0 119894 = 119895(87)



int

119879

0

Φ119894 (119905) Φ119895 (119905) 119889119905 = ℎ 119894 = 119895

0 119894 = 119895(88)



int

119879

0

1198912(119905) 119889119905 =

infin

sum

119894=1

1198912

119894

1003817100381710038171003817Φ119894(119905)10038171003817100381710038172 (89)



119891 (119905) =

119898

sum

119894=1

119891119894Φ119894 (119905) (90)



120576 =1

119879int

119879

0

(119891(119905) minus

119898

sum

119895=1

119891119895Φ119895(119905))

2

119889119905 (91)


Now 120597120576

120597119891119894

= minus2

119879int

119879

0

(119891 (119905) minus

119898

sum

119895=1

119891119895Φ119895 (119905))Φ119894 (119905) 119889119905 = 0

997904rArr 119891119894 =1

ℎint

119879

0


(92)


119891 (119905) =

119898

sum

119894=1

119891119894Φ119894 (119905) = 119865119879Φ (119905) = Φ

119879119865

where 119865 = [1198911 1198912 119891119898]119879

Φ (119905) = [Φ1 (119905) Φ2 (119905) Φ119898 (119905)]119879

(93)


119870 (119905 119904) = Φ119879(119905) 119870Ψ (119904) (94)



Φ (119905)Φ119879(119905) = (



d

0 0 sdot sdot sdot Φ119898 (119905)

) (95)


The second case is

Φ119879(119905) Φ (119905) = 1 (96)

because sum119898119894=1(Φ119894(119905))

2= sum119898

119894=1Φ119894(119905) = 1


int

119879

0

Φ (119905) 119889119905 = 119875Φ (119905) (97)

where

Φ (119905) = [Φ1 (119905) Φ2 (119905) Φ119898 (119905)]119879 (98)


119875 =1

2(




d


) (99)


int

119905

0

119891 (119905) 119889119905 = int

119905

0

119865119879Φ (119905) 119889119905 = 119865

119879119875Φ (119905) (100)


119899

sum

119895=1

119910119895 (119909) = 119891119894 (119909) +

119899

sum

119895=1

int

120573

120572

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899

(101)





119891119894 (119909) asymp 119865119879

119894Φ (119909)

119910119895 (119909) asymp 119884119879

119895Φ (119909)

119870119894119895 (119909 119905) asymp Φ119879(119905) 119870119894119895Φ (119909)

(102)


119899

sum

119895=1

119884119879

119895Φ (119909) = 119865

119879

119894Φ (119909)

+

119899

sum

119895=1

int

119898ℎ

0

119884119879

119895Φ (119905)Φ

119879(119905) 119870119894119895Φ (119909) 119889119905

119894 = 1 2 119899

(103)

119899

sum

119895=1

119884119879

119895Φ (119909) = 119865

119879

119894Φ (119909) +

119899

sum

119895=1

119884119879

119895ℎ119868119870119894119895Φ (119909)

119894 = 1 2 119899

(104)

since

int

119898ℎ

0

Φ (119905)Φ119879(119905) 119889119905 = ℎ119868 (105)

From (104) we get

119899

sum

119895=1

(119868 minus ℎ119870119879

119894119895) 119884119895 = 119865119894 119894 = 1 2 119899 (106)

Set 119860 119894119895 = 119868 minus ℎ119870119879

119894119895 then we have from (106)

119899

sum

119895=1

119860 119894119895119884119895 = 119865119894 119894 = 1 2 119899 (107)


(

(

11986011 11986012 1198601119899

11986021 11986022 1198602119899

1198601198991 1198601198992 119860119899119899

)

)

(

(

1198841

1198842

119884119899

)

)

=(

(

1198651

1198652

119865119899

)

)

(108)


119879119884119895 119895 = 1 2 119899



119906 (119909) = 119891 (119909) + int

1

0

119870 (119909 119905) 119865 (119905 119906 (119905)) 119889119905

0 le 119909 le 1

(109)





119910 (119909) = 119865 (119909 119906 (119909))

0 le 119909 le 1

(110)


119906 (119909) = 119860119879Ψ (119909)

119910 (119909) = 119861119879Ψ (119909)

(111)



119865 (119909) = 119863119879Ψ (119909)

119870 (119909 119905) = Ψ119879(119905) ΘΨ (119909)

(112)

where

Θ(119894119895) = int

1

0

[int

1

0

119870 (119909 119905) Ψ119894 (119905) 119889119905]Ψ119895 (119909) 119889119909 (113)


int

1

0

119870 (119909 119905) 119865 (119905 119906 (119905)) 119889119905

= int

1

0

119861119879Ψ (119905) Ψ

119879(119905) ΘΨ (119909) 119889119905

= 119861119879[int

1

0

Ψ (119905) Ψ119879(119905) 119889119905]ΘΨ (119909)

= 119861119879ΘΨ (119909) since int

1

0

Ψ (119905) Ψ119879(119905) 119889119905 = 119868

(114)



119860119879Ψ (119909) = 119863

119879Ψ (119909) + 119861

119879ΘΨ (119909) (115)


119860119879119875 = 119863

119879+ 119861119879Θ

119860119879119875 minus 119863

119879minus 119861119879Θ = 0

(116)


119875 = int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = [

1198751

1198752]

int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = 119868

(117)



119865 (119909 119860119879Ψ (119909)) = 119861

119879Ψ (119909) (118)


+119898minus2) where119894 = 1 2 2

119872+1+ 119898 minus 1







int

119887

119886

119891 (119909) 119889119909 =

119899minus1

sum

119894=1

int

119909119894+1

119909119894minus1

119891 (119909) 119889119909

=ℎ

3119891 (119886) +

4ℎ

3

1198992

sum

119894=1

119891 (1199092119894minus1)

+2ℎ

3

(119899minus1)2

sum

119894=1

119891 (1199092119894)

+ℎ

3119891 (119887)

minus(119887 minus 119886)

180ℎ4119891(4)(120578)

(119)


int

119887

119886

119891 (119909) 119889119909 =

119899

sum

119894=1

int

119909119894

119909119894minus1

119891 (119909) 119889119909

=ℎ

2119891 (119886) + ℎ

119899minus1

sum

119894=1

119891 (119909119894)

+ℎ

2119891 (119887)

+ℎ2

12[1198911015840(119886) minus 119891

1015840(119887)]

(120)


119906 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) 119865 (119906 (119905)) 119889119905

119886 le 119909 le 119887

(121)



+ℎ

3

[

[

119870 (119909 1199050) 119865 (1199060)

+ 4

1198992

sum

119895=1

119870(119909 1199052119895minus1) 119865 (1199062119895minus1)

+ 2

(1198992)minus1

sum

119895=1

119870(119909 1199052119895) 119865 (1199062119895)

+ 119870 (119909 119905119899) 119865 (119906119899)]

]

(122)


+ℎ

3

[

[

119870 (119909119894 1199050) 119865 (1199060)

+ 4

1198992

sum

119895=1

119870(119909119894 1199052119895minus1) 119865 (1199062119895minus1)


+ 2

(1198992)minus1

sum

119895=1

119870(119909119894 1199052119895) 119865 (1199062119895)

+119870 (119909119894 119905119899) 119865 (119906119899)]

]

(123)



119906 (119909) = 119891 (119909)

+ℎ

2119870 (119909 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909 119905119895) 119865 (119906119895)

+ℎ

2119870 (119909 119905119899) 119865 (119906119899)

+ℎ2

12[119869 (119909 1199050) 119865 (1199060)

+ 119870 (119909 1199050) 1199061015840

01198651015840(1199060)

minus 119869 (119909 119905119899) 119865 (119906119899)

minus119870 (119909 119905119899) 1199061015840

1198991198651015840(119906119899)]

(124)


have

119906 (119909119894) = 119891 (119909119894)

+ℎ

2119870 (119909119894 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909119894 119905119895) 119865 (119906119895)

+ℎ

2119870 (119909119894 119905119899) 119865 (119906119899)

+ℎ2

12[119869 (119909119894 1199050) 119865 (1199060)

+ 119870 (119909119894 1199050) 1199061015840

01198651015840(1199060)

minus 119869 (119909119894 119905119899) 119865 (119906119899)

minus119870 (119909119894 119905119899) 1199061015840

1198991198651015840(119906119899)]

(125)

for 119894 = 0 1 119899


120597119870(119909 119905)120597119909 we obtain

1199061015840(119909) = 119891

1015840(119909) + int

119887

119886

119867(119909 119905) 119865 (119906 (119905)) 119889119905

119886 le 119909 le 119887

(126)


1199061015840(119909119894) = 119891

1015840(119909119894)

+ℎ

2119867 (119909119894 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(119909119894 119905119895) 119865 (119906119895)

+ℎ

2119867 (119909119894 119905119899) 119865 (119906119899)

(127)



119906 (119909119894) = (ℎ

2119870 (119909119894 1199050) +

ℎ2

12119869 (119909119894 1199050))119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909119894 119905119895) 119865 (119906119895)

+ (ℎ

2119870 (119909119894 119905119899) minus

ℎ2

12119869 (119909119894 119905119899))119865 (119906119899)

+ℎ2

12(119870 (119909119894 1199050) 119906

1015840

01198651015840(1199060)

minus119870 (119909119894 119905119899) 1199061015840

1198991198651015840(119906119899))

1199061015840(1199090) = 119891

1015840(1199090)

+ℎ

2119867 (1199090 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(1199090 119905119895) 119865 (119906119895)

+ℎ

2119867 (1199090 119905119899) 119865 (119906119899)


1199061015840(119909119899) = 119891

1015840(119909119899)

+ℎ

2119867 (119909119899 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(119909119899 119905119895) 119865 (119906119895)

+ℎ

2119867 (119909119899 119905119899) 119865 (119906119899)

(128)




120595119886119887 (119905) = |119886|minus12120595(

119905 minus 119887

119886)

119886 119887 isin 119877 119886 = 0

(129)



120595119898119899 (119905)

=

(119898 +1

2)

12

21198962119871119898 (2

119896119905 minus 119899)

119899 minus 1

2119896le 119905 lt

119899 + 1

2119896

0 otherwise(130)


1198710 (119905) = 1

1198711 (119905) = 119905

119871119898+1 (119905) =2119898 + 1

119898 + 1119905119871119898 (119905)

minus119898

119898 + 1119871119898minus1 (119905) 119898 = 1 2 3

(131)



119891 (119909) =

infin

sum

119899=1

infin

sum

119898=0

119888119899119898120595119899119898 (119909) (132)

where

119888119899119898 = ⟨119891 (119909) 120595119899119898 (119909)⟩ (133)


119891 (119909) asymp

2119896minus1

sum

119899=1

119872minus1

sum

119898=0

119888119899119898120595119899119898 (119909) = 119862119879Ψ (119909) (134)


119862 = [11988810 11988811 1198881119872minus1 11988820


(135)

Ψ (119909) = [12059510 (119909) 1205951119872minus1 (119909)

12059520 (119909) 1205952119872minus1 (119909)


(136)


approximated as

119896 (119909 119905) asymp Ψ119879(119905) 119870Ψ (119909) (137)


119870119894119895 = ⟨120595119894 (119905) ⟨119896 (119909 119905) 120595119895 (119909)⟩⟩ (138)


[119910 (119909)]119901= [119884119879Ψ (119909)]

119901

= 119884lowast

119901

119879Ψ (119909) (139)





int

119905

0

Ψ(1199051015840) 1198891199051015840= 119875Ψ (119905) (140)



119875 =

[[[[[[

[

119871 119867 sdot sdot sdot 119867 119867

0 119871 sdot sdot sdot 119867 119867

d

0 0 sdot sdot sdot 119871 119867


]]]]]]

]

(141)



119867 =1

2119896

[[[[

[



d


]]]]

]

119871 =1

2119896

[[[[[[[[[[[[[[[[[[[[[[[

[

11


minusradic3

30

radic3


0 minusradic5

5radic30

radic5


0 0 minusradic7


d





]]]]]]]]]]]]]]]]]]]]]]]

]

(142)


int

1

0

Ψ (119905)Ψ119879(119905) 119889119905 = 119868 (143)


defined as

Ψ (119905) Ψ119879(119905) 119862 = 119862

119879Ψ (119905) (144)




119910 (119909) = 119891 (119909) + int

1

0

119896 (119909 119905) [119910 (119905)]119901119889119905 (145)



119891 (119909) asymp 119865119879Ψ (119909)

119910 (119909) asymp 119884119879Ψ (119909)

119896 (119909 119905) asymp Ψ119879(119905) 119870Ψ (119909)

[119910 (119909)]119901asymp 119884lowast119879Ψ (119909)

(146)


119884119879Ψ (119909) = 119865

119879Ψ (119909)

+ int

1

0

119884lowast119879Ψ (119905) Ψ

119879(119905) 119870Ψ (119909) 119889119905

= 119865119879Ψ (119909)

+ 119884lowast119879(int

1

0

Ψ (119905) Ψ119879(119905) 119889119905)119870Ψ (119909)

= 119865119879Ψ (119909) + 119884

lowast119879119870Ψ (119909)

= (119865119879+ 119884lowast119879119870)Ψ (119909)


119879= 0

(147)



119906 (119909) = 119891 (119909) + int

1

0

119870 (119909 119905) 119865 (119906 (119905)) 119889119905

0 le 119909 le 1

(148)


119871 (119906) = 119906 (119909) minus 119891 (119909) minus int

1

0

119870 (119909 119905) 119865 (119906 (119905)) 119889119905 = 0 (149)



119867(119906 0) = 119873 (119906)

119867 (119906 1) = 119871 (119906)

(150)


119867(119906 119901) = (1 minus 119901)119873 (119906) + 119901119871 (119906) = 0 (151)


119906 = 1199060 + 1199011199061 + 11990121199062 + sdot sdot sdot (152a)


119880 = lim119901rarr1

119906 = 1199060 + 1199061 + 1199062 + sdot sdot sdot (152b)


Consider

119873(119906) = 119906 (119909) minus 119891 (119909) (153)


119865 (119906) =

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898)

= 1198670 (1199060) + 1199011198671 (1199060 1199061)


(154)

where119867119898 (1199060 1199061 119906119898)

=1

119898

120597119898

120597119901119898(119865(

119898

sum

119896=0

119901119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816119901=0

119898 ge 0

(155)




minusint

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905) = 0


minus 119901int

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905 = 0

(156)


1199010 1199060 (119909) minus 119891 (119909) = 0 997904rArr 1199060 (119909) = 119891 (119909)

1199011 1199061 (119909) minus int

1

0

119870 (119909 119905)1198670119889119905 = 0 997904rArr 1199061 (119909)

= int

1

0

119870 (119909 119905)1198670119889119905

1199012 1199062 (119909) minus int

1

0

119870 (119909 119905)1198671119889119905 = 0 997904rArr 1199062 (119909)

= int

1

0

119870 (119909 119905)1198671119889119905

(157)


1199060 (119909) = 119891 (119909)

119906119899+1 (119909) = int

1

0

119870 (119909 119905)119867119899119889119905 119899 = 0 1 2

(158)



119906 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) (119871119906 (119905) + 119873119906 (119905)) 119889119905

119886 le 119909 le 119887

(159)



119906 (119909) =

infin

sum

119898=0

119906119898 (119909) (160)


119873119906 =

infin

sum

119899=0

119860119899 (161)

where 119860119899 =1

119899(120597119899

120597120582119899119873(

infin

sum

119896=0

120582119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816120582=0

(162)



infin

sum

119898=0

119906119898 (119909) = 119891 (119909)

+ int

119887

119886

119870 (119909 119905) (119871(

infin

sum

119898=0

119906119898 (119905)) +

infin

sum

119898=0

119860119898)119889119905

(163)


1199060 (119909) = 119891 (119909)

119906119898 (119909) = int

119887

119886

119870 (119909 119905) (119871 (119906119898minus1 (119905)) + 119860119898minus1 (119905)) 119889119905

119898 ge 1

(164)


120593119899 = 1199060 + 1199061 + sdot sdot sdot + 119906119899 (165)

with

119906 (119909) = lim119899rarrinfin

120593119899 (166)




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119898

sum

119895=1

119884119879

119895ℎ(119898) (119909)

= 119891119894 (119909) +

119898

sum

119895=1

int

1

0

119884119879

119895ℎ(119898) (119905) ℎ

119879

(119898)(119905) 119870119894119895ℎ(119898) (119909) 119889119905

119894 = 1 2 119898

(77)


119898

sum

119895=1

119884119879

119895ℎ(119898) (119909) = 119891119894 (119909) +

119898

sum

119895=1

119884119879

119895119863119870119894119895ℎ(119898) (119909)

119894 = 1 2 119898

(78)


119894=1 in the


119898

sum

119895=1

119884119879

119895ℎ(119898) (119909119894) = 119891119894 (119909119894) +

119898

sum

119895=1

119884119879

119895119863119870119894119895ℎ(119898) (119909119894)

119894 = 1 2 119898

(79)


119879

119895ℎ(119898)(119909) 119895 = 1 2 119898



119910119894 (119909) = 119891119894 (119909) +

119899

sum

119895=1

int

1

0

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899 0 le 119909 le 1

(80)


119910119895 (119905) = 119910119895 (119909) + 1199101015840


+1

119898119910(119898)

119895(119909) (119905 minus 119909)

119898+ 119864 (119905)

(81)



int1

0sum119899



119910119894 (119909) asymp 119891119894 (119909)

+

119899

sum

119895=1

int

1

0

119870119894119895 (119909 119905)

119898

sum

119903=0

1

119903(119905 minus 119909)

119903119910(119903)

119895(119909) 119889119905

119894 = 1 2 119899

119910119894 (119909) asymp 119891119894 (119909)

+

119899

sum

119895=1

119898

sum

119903=0

1

119903119910(119903)

119895(119909) int

1

0

119870119894119895 (119909 119905) (119905 minus 119909)119903119889119905

119894 = 1 2 119899

(82)

119910119894 (119909) minus

119899

sum

119895=1

119898

sum

119903=0

1

119903119910(119903)

119895(119909) [int

1

0

119870119894119895 (119909 119905) (119905 minus 119909)119903119889119905]

asymp 119891119894 (119909) 119894 = 1 2 119899

(83)



119910(119904)

119894(119909) = 119891

(119904)

119894(119909) +

119899

sum

119895=1

int

1

0

119870(119904)

119894119895(119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899 119904 = 1 2 119898

(84)

where119870(119904)119894119895(119909 119905) = 120597

(119904)119870119894119895(119909 119905)120597119909


have

119910(119904)

119894(119909) minus [

[

119899

sum

119895=1

int

1

0

119870(119904)

119894119895(119909 119905) 119889119905]

]

119910119895 (119909) asymp 119891(119904)

119894(119909)

119894 = 1 2 119899 119904 = 1 2 119898

(85)




Φ119894 (119905) =

1 (119894 minus 1)119879

119898le 119905 lt 119894

119879

119898

0 otherwise(86)

with 119905 isin [0 119879) 119879119898 = ℎ and 119894 = 1 2 119898




Φ119894 (119905) Φ119895 (119905) = Φ119894 (119905) 119894 = 119895

0 119894 = 119895(87)



int

119879

0

Φ119894 (119905) Φ119895 (119905) 119889119905 = ℎ 119894 = 119895

0 119894 = 119895(88)



int

119879

0

1198912(119905) 119889119905 =

infin

sum

119894=1

1198912

119894

1003817100381710038171003817Φ119894(119905)10038171003817100381710038172 (89)



119891 (119905) =

119898

sum

119894=1

119891119894Φ119894 (119905) (90)



120576 =1

119879int

119879

0

(119891(119905) minus

119898

sum

119895=1

119891119895Φ119895(119905))

2

119889119905 (91)


Now 120597120576

120597119891119894

= minus2

119879int

119879

0

(119891 (119905) minus

119898

sum

119895=1

119891119895Φ119895 (119905))Φ119894 (119905) 119889119905 = 0

997904rArr 119891119894 =1

ℎint

119879

0


(92)


119891 (119905) =

119898

sum

119894=1

119891119894Φ119894 (119905) = 119865119879Φ (119905) = Φ

119879119865

where 119865 = [1198911 1198912 119891119898]119879

Φ (119905) = [Φ1 (119905) Φ2 (119905) Φ119898 (119905)]119879

(93)


119870 (119905 119904) = Φ119879(119905) 119870Ψ (119904) (94)



Φ (119905)Φ119879(119905) = (



d

0 0 sdot sdot sdot Φ119898 (119905)

) (95)


The second case is

Φ119879(119905) Φ (119905) = 1 (96)

because sum119898119894=1(Φ119894(119905))

2= sum119898

119894=1Φ119894(119905) = 1


int

119879

0

Φ (119905) 119889119905 = 119875Φ (119905) (97)

where

Φ (119905) = [Φ1 (119905) Φ2 (119905) Φ119898 (119905)]119879 (98)


119875 =1

2(




d


) (99)


int

119905

0

119891 (119905) 119889119905 = int

119905

0

119865119879Φ (119905) 119889119905 = 119865

119879119875Φ (119905) (100)


119899

sum

119895=1

119910119895 (119909) = 119891119894 (119909) +

119899

sum

119895=1

int

120573

120572

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899

(101)





119891119894 (119909) asymp 119865119879

119894Φ (119909)

119910119895 (119909) asymp 119884119879

119895Φ (119909)

119870119894119895 (119909 119905) asymp Φ119879(119905) 119870119894119895Φ (119909)

(102)


119899

sum

119895=1

119884119879

119895Φ (119909) = 119865

119879

119894Φ (119909)

+

119899

sum

119895=1

int

119898ℎ

0

119884119879

119895Φ (119905)Φ

119879(119905) 119870119894119895Φ (119909) 119889119905

119894 = 1 2 119899

(103)

119899

sum

119895=1

119884119879

119895Φ (119909) = 119865

119879

119894Φ (119909) +

119899

sum

119895=1

119884119879

119895ℎ119868119870119894119895Φ (119909)

119894 = 1 2 119899

(104)

since

int

119898ℎ

0

Φ (119905)Φ119879(119905) 119889119905 = ℎ119868 (105)

From (104) we get

119899

sum

119895=1

(119868 minus ℎ119870119879

119894119895) 119884119895 = 119865119894 119894 = 1 2 119899 (106)

Set 119860 119894119895 = 119868 minus ℎ119870119879

119894119895 then we have from (106)

119899

sum

119895=1

119860 119894119895119884119895 = 119865119894 119894 = 1 2 119899 (107)


(

(

11986011 11986012 1198601119899

11986021 11986022 1198602119899

1198601198991 1198601198992 119860119899119899

)

)

(

(

1198841

1198842

119884119899

)

)

=(

(

1198651

1198652

119865119899

)

)

(108)


119879119884119895 119895 = 1 2 119899



119906 (119909) = 119891 (119909) + int

1

0

119870 (119909 119905) 119865 (119905 119906 (119905)) 119889119905

0 le 119909 le 1

(109)





119910 (119909) = 119865 (119909 119906 (119909))

0 le 119909 le 1

(110)


119906 (119909) = 119860119879Ψ (119909)

119910 (119909) = 119861119879Ψ (119909)

(111)



119865 (119909) = 119863119879Ψ (119909)

119870 (119909 119905) = Ψ119879(119905) ΘΨ (119909)

(112)

where

Θ(119894119895) = int

1

0

[int

1

0

119870 (119909 119905) Ψ119894 (119905) 119889119905]Ψ119895 (119909) 119889119909 (113)


int

1

0

119870 (119909 119905) 119865 (119905 119906 (119905)) 119889119905

= int

1

0

119861119879Ψ (119905) Ψ

119879(119905) ΘΨ (119909) 119889119905

= 119861119879[int

1

0

Ψ (119905) Ψ119879(119905) 119889119905]ΘΨ (119909)

= 119861119879ΘΨ (119909) since int

1

0

Ψ (119905) Ψ119879(119905) 119889119905 = 119868

(114)



119860119879Ψ (119909) = 119863

119879Ψ (119909) + 119861

119879ΘΨ (119909) (115)


119860119879119875 = 119863

119879+ 119861119879Θ

119860119879119875 minus 119863

119879minus 119861119879Θ = 0

(116)


119875 = int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = [

1198751

1198752]

int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = 119868

(117)



119865 (119909 119860119879Ψ (119909)) = 119861

119879Ψ (119909) (118)


+119898minus2) where119894 = 1 2 2

119872+1+ 119898 minus 1







int

119887

119886

119891 (119909) 119889119909 =

119899minus1

sum

119894=1

int

119909119894+1

119909119894minus1

119891 (119909) 119889119909

=ℎ

3119891 (119886) +

4ℎ

3

1198992

sum

119894=1

119891 (1199092119894minus1)

+2ℎ

3

(119899minus1)2

sum

119894=1

119891 (1199092119894)

+ℎ

3119891 (119887)

minus(119887 minus 119886)

180ℎ4119891(4)(120578)

(119)


int

119887

119886

119891 (119909) 119889119909 =

119899

sum

119894=1

int

119909119894

119909119894minus1

119891 (119909) 119889119909

=ℎ

2119891 (119886) + ℎ

119899minus1

sum

119894=1

119891 (119909119894)

+ℎ

2119891 (119887)

+ℎ2

12[1198911015840(119886) minus 119891

1015840(119887)]

(120)


119906 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) 119865 (119906 (119905)) 119889119905

119886 le 119909 le 119887

(121)



+ℎ

3

[

[

119870 (119909 1199050) 119865 (1199060)

+ 4

1198992

sum

119895=1

119870(119909 1199052119895minus1) 119865 (1199062119895minus1)

+ 2

(1198992)minus1

sum

119895=1

119870(119909 1199052119895) 119865 (1199062119895)

+ 119870 (119909 119905119899) 119865 (119906119899)]

]

(122)


+ℎ

3

[

[

119870 (119909119894 1199050) 119865 (1199060)

+ 4

1198992

sum

119895=1

119870(119909119894 1199052119895minus1) 119865 (1199062119895minus1)


+ 2

(1198992)minus1

sum

119895=1

119870(119909119894 1199052119895) 119865 (1199062119895)

+119870 (119909119894 119905119899) 119865 (119906119899)]

]

(123)



119906 (119909) = 119891 (119909)

+ℎ

2119870 (119909 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909 119905119895) 119865 (119906119895)

+ℎ

2119870 (119909 119905119899) 119865 (119906119899)

+ℎ2

12[119869 (119909 1199050) 119865 (1199060)

+ 119870 (119909 1199050) 1199061015840

01198651015840(1199060)

minus 119869 (119909 119905119899) 119865 (119906119899)

minus119870 (119909 119905119899) 1199061015840

1198991198651015840(119906119899)]

(124)


have

119906 (119909119894) = 119891 (119909119894)

+ℎ

2119870 (119909119894 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909119894 119905119895) 119865 (119906119895)

+ℎ

2119870 (119909119894 119905119899) 119865 (119906119899)

+ℎ2

12[119869 (119909119894 1199050) 119865 (1199060)

+ 119870 (119909119894 1199050) 1199061015840

01198651015840(1199060)

minus 119869 (119909119894 119905119899) 119865 (119906119899)

minus119870 (119909119894 119905119899) 1199061015840

1198991198651015840(119906119899)]

(125)

for 119894 = 0 1 119899


120597119870(119909 119905)120597119909 we obtain

1199061015840(119909) = 119891

1015840(119909) + int

119887

119886

119867(119909 119905) 119865 (119906 (119905)) 119889119905

119886 le 119909 le 119887

(126)


1199061015840(119909119894) = 119891

1015840(119909119894)

+ℎ

2119867 (119909119894 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(119909119894 119905119895) 119865 (119906119895)

+ℎ

2119867 (119909119894 119905119899) 119865 (119906119899)

(127)



119906 (119909119894) = (ℎ

2119870 (119909119894 1199050) +

ℎ2

12119869 (119909119894 1199050))119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909119894 119905119895) 119865 (119906119895)

+ (ℎ

2119870 (119909119894 119905119899) minus

ℎ2

12119869 (119909119894 119905119899))119865 (119906119899)

+ℎ2

12(119870 (119909119894 1199050) 119906

1015840

01198651015840(1199060)

minus119870 (119909119894 119905119899) 1199061015840

1198991198651015840(119906119899))

1199061015840(1199090) = 119891

1015840(1199090)

+ℎ

2119867 (1199090 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(1199090 119905119895) 119865 (119906119895)

+ℎ

2119867 (1199090 119905119899) 119865 (119906119899)


1199061015840(119909119899) = 119891

1015840(119909119899)

+ℎ

2119867 (119909119899 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(119909119899 119905119895) 119865 (119906119895)

+ℎ

2119867 (119909119899 119905119899) 119865 (119906119899)

(128)




120595119886119887 (119905) = |119886|minus12120595(

119905 minus 119887

119886)

119886 119887 isin 119877 119886 = 0

(129)



120595119898119899 (119905)

=

(119898 +1

2)

12

21198962119871119898 (2

119896119905 minus 119899)

119899 minus 1

2119896le 119905 lt

119899 + 1

2119896

0 otherwise(130)


1198710 (119905) = 1

1198711 (119905) = 119905

119871119898+1 (119905) =2119898 + 1

119898 + 1119905119871119898 (119905)

minus119898

119898 + 1119871119898minus1 (119905) 119898 = 1 2 3

(131)



119891 (119909) =

infin

sum

119899=1

infin

sum

119898=0

119888119899119898120595119899119898 (119909) (132)

where

119888119899119898 = ⟨119891 (119909) 120595119899119898 (119909)⟩ (133)


119891 (119909) asymp

2119896minus1

sum

119899=1

119872minus1

sum

119898=0

119888119899119898120595119899119898 (119909) = 119862119879Ψ (119909) (134)


119862 = [11988810 11988811 1198881119872minus1 11988820


(135)

Ψ (119909) = [12059510 (119909) 1205951119872minus1 (119909)

12059520 (119909) 1205952119872minus1 (119909)


(136)


approximated as

119896 (119909 119905) asymp Ψ119879(119905) 119870Ψ (119909) (137)


119870119894119895 = ⟨120595119894 (119905) ⟨119896 (119909 119905) 120595119895 (119909)⟩⟩ (138)


[119910 (119909)]119901= [119884119879Ψ (119909)]

119901

= 119884lowast

119901

119879Ψ (119909) (139)





int

119905

0

Ψ(1199051015840) 1198891199051015840= 119875Ψ (119905) (140)



119875 =

[[[[[[

[

119871 119867 sdot sdot sdot 119867 119867

0 119871 sdot sdot sdot 119867 119867

d

0 0 sdot sdot sdot 119871 119867


]]]]]]

]

(141)



119867 =1

2119896

[[[[

[



d


]]]]

]

119871 =1

2119896

[[[[[[[[[[[[[[[[[[[[[[[

[

11


minusradic3

30

radic3


0 minusradic5

5radic30

radic5


0 0 minusradic7


d





]]]]]]]]]]]]]]]]]]]]]]]

]

(142)


int

1

0

Ψ (119905)Ψ119879(119905) 119889119905 = 119868 (143)


defined as

Ψ (119905) Ψ119879(119905) 119862 = 119862

119879Ψ (119905) (144)




119910 (119909) = 119891 (119909) + int

1

0

119896 (119909 119905) [119910 (119905)]119901119889119905 (145)



119891 (119909) asymp 119865119879Ψ (119909)

119910 (119909) asymp 119884119879Ψ (119909)

119896 (119909 119905) asymp Ψ119879(119905) 119870Ψ (119909)

[119910 (119909)]119901asymp 119884lowast119879Ψ (119909)

(146)


119884119879Ψ (119909) = 119865

119879Ψ (119909)

+ int

1

0

119884lowast119879Ψ (119905) Ψ

119879(119905) 119870Ψ (119909) 119889119905

= 119865119879Ψ (119909)

+ 119884lowast119879(int

1

0

Ψ (119905) Ψ119879(119905) 119889119905)119870Ψ (119909)

= 119865119879Ψ (119909) + 119884

lowast119879119870Ψ (119909)

= (119865119879+ 119884lowast119879119870)Ψ (119909)


119879= 0

(147)



119906 (119909) = 119891 (119909) + int

1

0

119870 (119909 119905) 119865 (119906 (119905)) 119889119905

0 le 119909 le 1

(148)


119871 (119906) = 119906 (119909) minus 119891 (119909) minus int

1

0

119870 (119909 119905) 119865 (119906 (119905)) 119889119905 = 0 (149)



119867(119906 0) = 119873 (119906)

119867 (119906 1) = 119871 (119906)

(150)


119867(119906 119901) = (1 minus 119901)119873 (119906) + 119901119871 (119906) = 0 (151)


119906 = 1199060 + 1199011199061 + 11990121199062 + sdot sdot sdot (152a)


119880 = lim119901rarr1

119906 = 1199060 + 1199061 + 1199062 + sdot sdot sdot (152b)


Consider

119873(119906) = 119906 (119909) minus 119891 (119909) (153)


119865 (119906) =

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898)

= 1198670 (1199060) + 1199011198671 (1199060 1199061)


(154)

where119867119898 (1199060 1199061 119906119898)

=1

119898

120597119898

120597119901119898(119865(

119898

sum

119896=0

119901119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816119901=0

119898 ge 0

(155)




minusint

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905) = 0


minus 119901int

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905 = 0

(156)


1199010 1199060 (119909) minus 119891 (119909) = 0 997904rArr 1199060 (119909) = 119891 (119909)

1199011 1199061 (119909) minus int

1

0

119870 (119909 119905)1198670119889119905 = 0 997904rArr 1199061 (119909)

= int

1

0

119870 (119909 119905)1198670119889119905

1199012 1199062 (119909) minus int

1

0

119870 (119909 119905)1198671119889119905 = 0 997904rArr 1199062 (119909)

= int

1

0

119870 (119909 119905)1198671119889119905

(157)


1199060 (119909) = 119891 (119909)

119906119899+1 (119909) = int

1

0

119870 (119909 119905)119867119899119889119905 119899 = 0 1 2

(158)



119906 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) (119871119906 (119905) + 119873119906 (119905)) 119889119905

119886 le 119909 le 119887

(159)



119906 (119909) =

infin

sum

119898=0

119906119898 (119909) (160)


119873119906 =

infin

sum

119899=0

119860119899 (161)

where 119860119899 =1

119899(120597119899

120597120582119899119873(

infin

sum

119896=0

120582119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816120582=0

(162)



infin

sum

119898=0

119906119898 (119909) = 119891 (119909)

+ int

119887

119886

119870 (119909 119905) (119871(

infin

sum

119898=0

119906119898 (119905)) +

infin

sum

119898=0

119860119898)119889119905

(163)


1199060 (119909) = 119891 (119909)

119906119898 (119909) = int

119887

119886

119870 (119909 119905) (119871 (119906119898minus1 (119905)) + 119860119898minus1 (119905)) 119889119905

119898 ge 1

(164)


120593119899 = 1199060 + 1199061 + sdot sdot sdot + 119906119899 (165)

with

119906 (119909) = lim119899rarrinfin

120593119899 (166)




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Φ119894 (119905) Φ119895 (119905) = Φ119894 (119905) 119894 = 119895

0 119894 = 119895(87)



int

119879

0

Φ119894 (119905) Φ119895 (119905) 119889119905 = ℎ 119894 = 119895

0 119894 = 119895(88)



int

119879

0

1198912(119905) 119889119905 =

infin

sum

119894=1

1198912

119894

1003817100381710038171003817Φ119894(119905)10038171003817100381710038172 (89)



119891 (119905) =

119898

sum

119894=1

119891119894Φ119894 (119905) (90)



120576 =1

119879int

119879

0

(119891(119905) minus

119898

sum

119895=1

119891119895Φ119895(119905))

2

119889119905 (91)


Now 120597120576

120597119891119894

= minus2

119879int

119879

0

(119891 (119905) minus

119898

sum

119895=1

119891119895Φ119895 (119905))Φ119894 (119905) 119889119905 = 0

997904rArr 119891119894 =1

ℎint

119879

0


(92)


119891 (119905) =

119898

sum

119894=1

119891119894Φ119894 (119905) = 119865119879Φ (119905) = Φ

119879119865

where 119865 = [1198911 1198912 119891119898]119879

Φ (119905) = [Φ1 (119905) Φ2 (119905) Φ119898 (119905)]119879

(93)


119870 (119905 119904) = Φ119879(119905) 119870Ψ (119904) (94)



Φ (119905)Φ119879(119905) = (



d

0 0 sdot sdot sdot Φ119898 (119905)

) (95)


The second case is

Φ119879(119905) Φ (119905) = 1 (96)

because sum119898119894=1(Φ119894(119905))

2= sum119898

119894=1Φ119894(119905) = 1


int

119879

0

Φ (119905) 119889119905 = 119875Φ (119905) (97)

where

Φ (119905) = [Φ1 (119905) Φ2 (119905) Φ119898 (119905)]119879 (98)


119875 =1

2(




d


) (99)


int

119905

0

119891 (119905) 119889119905 = int

119905

0

119865119879Φ (119905) 119889119905 = 119865

119879119875Φ (119905) (100)


119899

sum

119895=1

119910119895 (119909) = 119891119894 (119909) +

119899

sum

119895=1

int

120573

120572

119870119894119895 (119909 119905) 119910119895 (119905) 119889119905

119894 = 1 2 119899

(101)





119891119894 (119909) asymp 119865119879

119894Φ (119909)

119910119895 (119909) asymp 119884119879

119895Φ (119909)

119870119894119895 (119909 119905) asymp Φ119879(119905) 119870119894119895Φ (119909)

(102)


119899

sum

119895=1

119884119879

119895Φ (119909) = 119865

119879

119894Φ (119909)

+

119899

sum

119895=1

int

119898ℎ

0

119884119879

119895Φ (119905)Φ

119879(119905) 119870119894119895Φ (119909) 119889119905

119894 = 1 2 119899

(103)

119899

sum

119895=1

119884119879

119895Φ (119909) = 119865

119879

119894Φ (119909) +

119899

sum

119895=1

119884119879

119895ℎ119868119870119894119895Φ (119909)

119894 = 1 2 119899

(104)

since

int

119898ℎ

0

Φ (119905)Φ119879(119905) 119889119905 = ℎ119868 (105)

From (104) we get

119899

sum

119895=1

(119868 minus ℎ119870119879

119894119895) 119884119895 = 119865119894 119894 = 1 2 119899 (106)

Set 119860 119894119895 = 119868 minus ℎ119870119879

119894119895 then we have from (106)

119899

sum

119895=1

119860 119894119895119884119895 = 119865119894 119894 = 1 2 119899 (107)


(

(

11986011 11986012 1198601119899

11986021 11986022 1198602119899

1198601198991 1198601198992 119860119899119899

)

)

(

(

1198841

1198842

119884119899

)

)

=(

(

1198651

1198652

119865119899

)

)

(108)


119879119884119895 119895 = 1 2 119899



119906 (119909) = 119891 (119909) + int

1

0

119870 (119909 119905) 119865 (119905 119906 (119905)) 119889119905

0 le 119909 le 1

(109)





119910 (119909) = 119865 (119909 119906 (119909))

0 le 119909 le 1

(110)


119906 (119909) = 119860119879Ψ (119909)

119910 (119909) = 119861119879Ψ (119909)

(111)



119865 (119909) = 119863119879Ψ (119909)

119870 (119909 119905) = Ψ119879(119905) ΘΨ (119909)

(112)

where

Θ(119894119895) = int

1

0

[int

1

0

119870 (119909 119905) Ψ119894 (119905) 119889119905]Ψ119895 (119909) 119889119909 (113)


int

1

0

119870 (119909 119905) 119865 (119905 119906 (119905)) 119889119905

= int

1

0

119861119879Ψ (119905) Ψ

119879(119905) ΘΨ (119909) 119889119905

= 119861119879[int

1

0

Ψ (119905) Ψ119879(119905) 119889119905]ΘΨ (119909)

= 119861119879ΘΨ (119909) since int

1

0

Ψ (119905) Ψ119879(119905) 119889119905 = 119868

(114)



119860119879Ψ (119909) = 119863

119879Ψ (119909) + 119861

119879ΘΨ (119909) (115)


119860119879119875 = 119863

119879+ 119861119879Θ

119860119879119875 minus 119863

119879minus 119861119879Θ = 0

(116)


119875 = int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = [

1198751

1198752]

int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = 119868

(117)



119865 (119909 119860119879Ψ (119909)) = 119861

119879Ψ (119909) (118)


+119898minus2) where119894 = 1 2 2

119872+1+ 119898 minus 1







int

119887

119886

119891 (119909) 119889119909 =

119899minus1

sum

119894=1

int

119909119894+1

119909119894minus1

119891 (119909) 119889119909

=ℎ

3119891 (119886) +

4ℎ

3

1198992

sum

119894=1

119891 (1199092119894minus1)

+2ℎ

3

(119899minus1)2

sum

119894=1

119891 (1199092119894)

+ℎ

3119891 (119887)

minus(119887 minus 119886)

180ℎ4119891(4)(120578)

(119)


int

119887

119886

119891 (119909) 119889119909 =

119899

sum

119894=1

int

119909119894

119909119894minus1

119891 (119909) 119889119909

=ℎ

2119891 (119886) + ℎ

119899minus1

sum

119894=1

119891 (119909119894)

+ℎ

2119891 (119887)

+ℎ2

12[1198911015840(119886) minus 119891

1015840(119887)]

(120)


119906 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) 119865 (119906 (119905)) 119889119905

119886 le 119909 le 119887

(121)



+ℎ

3

[

[

119870 (119909 1199050) 119865 (1199060)

+ 4

1198992

sum

119895=1

119870(119909 1199052119895minus1) 119865 (1199062119895minus1)

+ 2

(1198992)minus1

sum

119895=1

119870(119909 1199052119895) 119865 (1199062119895)

+ 119870 (119909 119905119899) 119865 (119906119899)]

]

(122)


+ℎ

3

[

[

119870 (119909119894 1199050) 119865 (1199060)

+ 4

1198992

sum

119895=1

119870(119909119894 1199052119895minus1) 119865 (1199062119895minus1)


+ 2

(1198992)minus1

sum

119895=1

119870(119909119894 1199052119895) 119865 (1199062119895)

+119870 (119909119894 119905119899) 119865 (119906119899)]

]

(123)



119906 (119909) = 119891 (119909)

+ℎ

2119870 (119909 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909 119905119895) 119865 (119906119895)

+ℎ

2119870 (119909 119905119899) 119865 (119906119899)

+ℎ2

12[119869 (119909 1199050) 119865 (1199060)

+ 119870 (119909 1199050) 1199061015840

01198651015840(1199060)

minus 119869 (119909 119905119899) 119865 (119906119899)

minus119870 (119909 119905119899) 1199061015840

1198991198651015840(119906119899)]

(124)


have

119906 (119909119894) = 119891 (119909119894)

+ℎ

2119870 (119909119894 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909119894 119905119895) 119865 (119906119895)

+ℎ

2119870 (119909119894 119905119899) 119865 (119906119899)

+ℎ2

12[119869 (119909119894 1199050) 119865 (1199060)

+ 119870 (119909119894 1199050) 1199061015840

01198651015840(1199060)

minus 119869 (119909119894 119905119899) 119865 (119906119899)

minus119870 (119909119894 119905119899) 1199061015840

1198991198651015840(119906119899)]

(125)

for 119894 = 0 1 119899


120597119870(119909 119905)120597119909 we obtain

1199061015840(119909) = 119891

1015840(119909) + int

119887

119886

119867(119909 119905) 119865 (119906 (119905)) 119889119905

119886 le 119909 le 119887

(126)


1199061015840(119909119894) = 119891

1015840(119909119894)

+ℎ

2119867 (119909119894 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(119909119894 119905119895) 119865 (119906119895)

+ℎ

2119867 (119909119894 119905119899) 119865 (119906119899)

(127)



119906 (119909119894) = (ℎ

2119870 (119909119894 1199050) +

ℎ2

12119869 (119909119894 1199050))119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909119894 119905119895) 119865 (119906119895)

+ (ℎ

2119870 (119909119894 119905119899) minus

ℎ2

12119869 (119909119894 119905119899))119865 (119906119899)

+ℎ2

12(119870 (119909119894 1199050) 119906

1015840

01198651015840(1199060)

minus119870 (119909119894 119905119899) 1199061015840

1198991198651015840(119906119899))

1199061015840(1199090) = 119891

1015840(1199090)

+ℎ

2119867 (1199090 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(1199090 119905119895) 119865 (119906119895)

+ℎ

2119867 (1199090 119905119899) 119865 (119906119899)


1199061015840(119909119899) = 119891

1015840(119909119899)

+ℎ

2119867 (119909119899 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(119909119899 119905119895) 119865 (119906119895)

+ℎ

2119867 (119909119899 119905119899) 119865 (119906119899)

(128)




120595119886119887 (119905) = |119886|minus12120595(

119905 minus 119887

119886)

119886 119887 isin 119877 119886 = 0

(129)



120595119898119899 (119905)

=

(119898 +1

2)

12

21198962119871119898 (2

119896119905 minus 119899)

119899 minus 1

2119896le 119905 lt

119899 + 1

2119896

0 otherwise(130)


1198710 (119905) = 1

1198711 (119905) = 119905

119871119898+1 (119905) =2119898 + 1

119898 + 1119905119871119898 (119905)

minus119898

119898 + 1119871119898minus1 (119905) 119898 = 1 2 3

(131)



119891 (119909) =

infin

sum

119899=1

infin

sum

119898=0

119888119899119898120595119899119898 (119909) (132)

where

119888119899119898 = ⟨119891 (119909) 120595119899119898 (119909)⟩ (133)


119891 (119909) asymp

2119896minus1

sum

119899=1

119872minus1

sum

119898=0

119888119899119898120595119899119898 (119909) = 119862119879Ψ (119909) (134)


119862 = [11988810 11988811 1198881119872minus1 11988820


(135)

Ψ (119909) = [12059510 (119909) 1205951119872minus1 (119909)

12059520 (119909) 1205952119872minus1 (119909)


(136)


approximated as

119896 (119909 119905) asymp Ψ119879(119905) 119870Ψ (119909) (137)


119870119894119895 = ⟨120595119894 (119905) ⟨119896 (119909 119905) 120595119895 (119909)⟩⟩ (138)


[119910 (119909)]119901= [119884119879Ψ (119909)]

119901

= 119884lowast

119901

119879Ψ (119909) (139)





int

119905

0

Ψ(1199051015840) 1198891199051015840= 119875Ψ (119905) (140)



119875 =

[[[[[[

[

119871 119867 sdot sdot sdot 119867 119867

0 119871 sdot sdot sdot 119867 119867

d

0 0 sdot sdot sdot 119871 119867


]]]]]]

]

(141)



119867 =1

2119896

[[[[

[



d


]]]]

]

119871 =1

2119896

[[[[[[[[[[[[[[[[[[[[[[[

[

11


minusradic3

30

radic3


0 minusradic5

5radic30

radic5


0 0 minusradic7


d





]]]]]]]]]]]]]]]]]]]]]]]

]

(142)


int

1

0

Ψ (119905)Ψ119879(119905) 119889119905 = 119868 (143)


defined as

Ψ (119905) Ψ119879(119905) 119862 = 119862

119879Ψ (119905) (144)




119910 (119909) = 119891 (119909) + int

1

0

119896 (119909 119905) [119910 (119905)]119901119889119905 (145)



119891 (119909) asymp 119865119879Ψ (119909)

119910 (119909) asymp 119884119879Ψ (119909)

119896 (119909 119905) asymp Ψ119879(119905) 119870Ψ (119909)

[119910 (119909)]119901asymp 119884lowast119879Ψ (119909)

(146)


119884119879Ψ (119909) = 119865

119879Ψ (119909)

+ int

1

0

119884lowast119879Ψ (119905) Ψ

119879(119905) 119870Ψ (119909) 119889119905

= 119865119879Ψ (119909)

+ 119884lowast119879(int

1

0

Ψ (119905) Ψ119879(119905) 119889119905)119870Ψ (119909)

= 119865119879Ψ (119909) + 119884

lowast119879119870Ψ (119909)

= (119865119879+ 119884lowast119879119870)Ψ (119909)


119879= 0

(147)



119906 (119909) = 119891 (119909) + int

1

0

119870 (119909 119905) 119865 (119906 (119905)) 119889119905

0 le 119909 le 1

(148)


119871 (119906) = 119906 (119909) minus 119891 (119909) minus int

1

0

119870 (119909 119905) 119865 (119906 (119905)) 119889119905 = 0 (149)



119867(119906 0) = 119873 (119906)

119867 (119906 1) = 119871 (119906)

(150)


119867(119906 119901) = (1 minus 119901)119873 (119906) + 119901119871 (119906) = 0 (151)


119906 = 1199060 + 1199011199061 + 11990121199062 + sdot sdot sdot (152a)


119880 = lim119901rarr1

119906 = 1199060 + 1199061 + 1199062 + sdot sdot sdot (152b)


Consider

119873(119906) = 119906 (119909) minus 119891 (119909) (153)


119865 (119906) =

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898)

= 1198670 (1199060) + 1199011198671 (1199060 1199061)


(154)

where119867119898 (1199060 1199061 119906119898)

=1

119898

120597119898

120597119901119898(119865(

119898

sum

119896=0

119901119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816119901=0

119898 ge 0

(155)




minusint

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905) = 0


minus 119901int

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905 = 0

(156)


1199010 1199060 (119909) minus 119891 (119909) = 0 997904rArr 1199060 (119909) = 119891 (119909)

1199011 1199061 (119909) minus int

1

0

119870 (119909 119905)1198670119889119905 = 0 997904rArr 1199061 (119909)

= int

1

0

119870 (119909 119905)1198670119889119905

1199012 1199062 (119909) minus int

1

0

119870 (119909 119905)1198671119889119905 = 0 997904rArr 1199062 (119909)

= int

1

0

119870 (119909 119905)1198671119889119905

(157)


1199060 (119909) = 119891 (119909)

119906119899+1 (119909) = int

1

0

119870 (119909 119905)119867119899119889119905 119899 = 0 1 2

(158)



119906 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) (119871119906 (119905) + 119873119906 (119905)) 119889119905

119886 le 119909 le 119887

(159)



119906 (119909) =

infin

sum

119898=0

119906119898 (119909) (160)


119873119906 =

infin

sum

119899=0

119860119899 (161)

where 119860119899 =1

119899(120597119899

120597120582119899119873(

infin

sum

119896=0

120582119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816120582=0

(162)



infin

sum

119898=0

119906119898 (119909) = 119891 (119909)

+ int

119887

119886

119870 (119909 119905) (119871(

infin

sum

119898=0

119906119898 (119905)) +

infin

sum

119898=0

119860119898)119889119905

(163)


1199060 (119909) = 119891 (119909)

119906119898 (119909) = int

119887

119886

119870 (119909 119905) (119871 (119906119898minus1 (119905)) + 119860119898minus1 (119905)) 119889119905

119898 ge 1

(164)


120593119899 = 1199060 + 1199061 + sdot sdot sdot + 119906119899 (165)

with

119906 (119909) = lim119899rarrinfin

120593119899 (166)




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119891119894 (119909) asymp 119865119879

119894Φ (119909)

119910119895 (119909) asymp 119884119879

119895Φ (119909)

119870119894119895 (119909 119905) asymp Φ119879(119905) 119870119894119895Φ (119909)

(102)


119899

sum

119895=1

119884119879

119895Φ (119909) = 119865

119879

119894Φ (119909)

+

119899

sum

119895=1

int

119898ℎ

0

119884119879

119895Φ (119905)Φ

119879(119905) 119870119894119895Φ (119909) 119889119905

119894 = 1 2 119899

(103)

119899

sum

119895=1

119884119879

119895Φ (119909) = 119865

119879

119894Φ (119909) +

119899

sum

119895=1

119884119879

119895ℎ119868119870119894119895Φ (119909)

119894 = 1 2 119899

(104)

since

int

119898ℎ

0

Φ (119905)Φ119879(119905) 119889119905 = ℎ119868 (105)

From (104) we get

119899

sum

119895=1

(119868 minus ℎ119870119879

119894119895) 119884119895 = 119865119894 119894 = 1 2 119899 (106)

Set 119860 119894119895 = 119868 minus ℎ119870119879

119894119895 then we have from (106)

119899

sum

119895=1

119860 119894119895119884119895 = 119865119894 119894 = 1 2 119899 (107)


(

(

11986011 11986012 1198601119899

11986021 11986022 1198602119899

1198601198991 1198601198992 119860119899119899

)

)

(

(

1198841

1198842

119884119899

)

)

=(

(

1198651

1198652

119865119899

)

)

(108)


119879119884119895 119895 = 1 2 119899



119906 (119909) = 119891 (119909) + int

1

0

119870 (119909 119905) 119865 (119905 119906 (119905)) 119889119905

0 le 119909 le 1

(109)





119910 (119909) = 119865 (119909 119906 (119909))

0 le 119909 le 1

(110)


119906 (119909) = 119860119879Ψ (119909)

119910 (119909) = 119861119879Ψ (119909)

(111)



119865 (119909) = 119863119879Ψ (119909)

119870 (119909 119905) = Ψ119879(119905) ΘΨ (119909)

(112)

where

Θ(119894119895) = int

1

0

[int

1

0

119870 (119909 119905) Ψ119894 (119905) 119889119905]Ψ119895 (119909) 119889119909 (113)


int

1

0

119870 (119909 119905) 119865 (119905 119906 (119905)) 119889119905

= int

1

0

119861119879Ψ (119905) Ψ

119879(119905) ΘΨ (119909) 119889119905

= 119861119879[int

1

0

Ψ (119905) Ψ119879(119905) 119889119905]ΘΨ (119909)

= 119861119879ΘΨ (119909) since int

1

0

Ψ (119905) Ψ119879(119905) 119889119905 = 119868

(114)



119860119879Ψ (119909) = 119863

119879Ψ (119909) + 119861

119879ΘΨ (119909) (115)


119860119879119875 = 119863

119879+ 119861119879Θ

119860119879119875 minus 119863

119879minus 119861119879Θ = 0

(116)


119875 = int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = [

1198751

1198752]

int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = 119868

(117)



119865 (119909 119860119879Ψ (119909)) = 119861

119879Ψ (119909) (118)


+119898minus2) where119894 = 1 2 2

119872+1+ 119898 minus 1







int

119887

119886

119891 (119909) 119889119909 =

119899minus1

sum

119894=1

int

119909119894+1

119909119894minus1

119891 (119909) 119889119909

=ℎ

3119891 (119886) +

4ℎ

3

1198992

sum

119894=1

119891 (1199092119894minus1)

+2ℎ

3

(119899minus1)2

sum

119894=1

119891 (1199092119894)

+ℎ

3119891 (119887)

minus(119887 minus 119886)

180ℎ4119891(4)(120578)

(119)


int

119887

119886

119891 (119909) 119889119909 =

119899

sum

119894=1

int

119909119894

119909119894minus1

119891 (119909) 119889119909

=ℎ

2119891 (119886) + ℎ

119899minus1

sum

119894=1

119891 (119909119894)

+ℎ

2119891 (119887)

+ℎ2

12[1198911015840(119886) minus 119891

1015840(119887)]

(120)


119906 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) 119865 (119906 (119905)) 119889119905

119886 le 119909 le 119887

(121)



+ℎ

3

[

[

119870 (119909 1199050) 119865 (1199060)

+ 4

1198992

sum

119895=1

119870(119909 1199052119895minus1) 119865 (1199062119895minus1)

+ 2

(1198992)minus1

sum

119895=1

119870(119909 1199052119895) 119865 (1199062119895)

+ 119870 (119909 119905119899) 119865 (119906119899)]

]

(122)


+ℎ

3

[

[

119870 (119909119894 1199050) 119865 (1199060)

+ 4

1198992

sum

119895=1

119870(119909119894 1199052119895minus1) 119865 (1199062119895minus1)


+ 2

(1198992)minus1

sum

119895=1

119870(119909119894 1199052119895) 119865 (1199062119895)

+119870 (119909119894 119905119899) 119865 (119906119899)]

]

(123)



119906 (119909) = 119891 (119909)

+ℎ

2119870 (119909 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909 119905119895) 119865 (119906119895)

+ℎ

2119870 (119909 119905119899) 119865 (119906119899)

+ℎ2

12[119869 (119909 1199050) 119865 (1199060)

+ 119870 (119909 1199050) 1199061015840

01198651015840(1199060)

minus 119869 (119909 119905119899) 119865 (119906119899)

minus119870 (119909 119905119899) 1199061015840

1198991198651015840(119906119899)]

(124)


have

119906 (119909119894) = 119891 (119909119894)

+ℎ

2119870 (119909119894 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909119894 119905119895) 119865 (119906119895)

+ℎ

2119870 (119909119894 119905119899) 119865 (119906119899)

+ℎ2

12[119869 (119909119894 1199050) 119865 (1199060)

+ 119870 (119909119894 1199050) 1199061015840

01198651015840(1199060)

minus 119869 (119909119894 119905119899) 119865 (119906119899)

minus119870 (119909119894 119905119899) 1199061015840

1198991198651015840(119906119899)]

(125)

for 119894 = 0 1 119899


120597119870(119909 119905)120597119909 we obtain

1199061015840(119909) = 119891

1015840(119909) + int

119887

119886

119867(119909 119905) 119865 (119906 (119905)) 119889119905

119886 le 119909 le 119887

(126)


1199061015840(119909119894) = 119891

1015840(119909119894)

+ℎ

2119867 (119909119894 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(119909119894 119905119895) 119865 (119906119895)

+ℎ

2119867 (119909119894 119905119899) 119865 (119906119899)

(127)



119906 (119909119894) = (ℎ

2119870 (119909119894 1199050) +

ℎ2

12119869 (119909119894 1199050))119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909119894 119905119895) 119865 (119906119895)

+ (ℎ

2119870 (119909119894 119905119899) minus

ℎ2

12119869 (119909119894 119905119899))119865 (119906119899)

+ℎ2

12(119870 (119909119894 1199050) 119906

1015840

01198651015840(1199060)

minus119870 (119909119894 119905119899) 1199061015840

1198991198651015840(119906119899))

1199061015840(1199090) = 119891

1015840(1199090)

+ℎ

2119867 (1199090 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(1199090 119905119895) 119865 (119906119895)

+ℎ

2119867 (1199090 119905119899) 119865 (119906119899)


1199061015840(119909119899) = 119891

1015840(119909119899)

+ℎ

2119867 (119909119899 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(119909119899 119905119895) 119865 (119906119895)

+ℎ

2119867 (119909119899 119905119899) 119865 (119906119899)

(128)




120595119886119887 (119905) = |119886|minus12120595(

119905 minus 119887

119886)

119886 119887 isin 119877 119886 = 0

(129)



120595119898119899 (119905)

=

(119898 +1

2)

12

21198962119871119898 (2

119896119905 minus 119899)

119899 minus 1

2119896le 119905 lt

119899 + 1

2119896

0 otherwise(130)


1198710 (119905) = 1

1198711 (119905) = 119905

119871119898+1 (119905) =2119898 + 1

119898 + 1119905119871119898 (119905)

minus119898

119898 + 1119871119898minus1 (119905) 119898 = 1 2 3

(131)



119891 (119909) =

infin

sum

119899=1

infin

sum

119898=0

119888119899119898120595119899119898 (119909) (132)

where

119888119899119898 = ⟨119891 (119909) 120595119899119898 (119909)⟩ (133)


119891 (119909) asymp

2119896minus1

sum

119899=1

119872minus1

sum

119898=0

119888119899119898120595119899119898 (119909) = 119862119879Ψ (119909) (134)


119862 = [11988810 11988811 1198881119872minus1 11988820


(135)

Ψ (119909) = [12059510 (119909) 1205951119872minus1 (119909)

12059520 (119909) 1205952119872minus1 (119909)


(136)


approximated as

119896 (119909 119905) asymp Ψ119879(119905) 119870Ψ (119909) (137)


119870119894119895 = ⟨120595119894 (119905) ⟨119896 (119909 119905) 120595119895 (119909)⟩⟩ (138)


[119910 (119909)]119901= [119884119879Ψ (119909)]

119901

= 119884lowast

119901

119879Ψ (119909) (139)





int

119905

0

Ψ(1199051015840) 1198891199051015840= 119875Ψ (119905) (140)



119875 =

[[[[[[

[

119871 119867 sdot sdot sdot 119867 119867

0 119871 sdot sdot sdot 119867 119867

d

0 0 sdot sdot sdot 119871 119867


]]]]]]

]

(141)



119867 =1

2119896

[[[[

[



d


]]]]

]

119871 =1

2119896

[[[[[[[[[[[[[[[[[[[[[[[

[

11


minusradic3

30

radic3


0 minusradic5

5radic30

radic5


0 0 minusradic7


d





]]]]]]]]]]]]]]]]]]]]]]]

]

(142)


int

1

0

Ψ (119905)Ψ119879(119905) 119889119905 = 119868 (143)


defined as

Ψ (119905) Ψ119879(119905) 119862 = 119862

119879Ψ (119905) (144)




119910 (119909) = 119891 (119909) + int

1

0

119896 (119909 119905) [119910 (119905)]119901119889119905 (145)



119891 (119909) asymp 119865119879Ψ (119909)

119910 (119909) asymp 119884119879Ψ (119909)

119896 (119909 119905) asymp Ψ119879(119905) 119870Ψ (119909)

[119910 (119909)]119901asymp 119884lowast119879Ψ (119909)

(146)


119884119879Ψ (119909) = 119865

119879Ψ (119909)

+ int

1

0

119884lowast119879Ψ (119905) Ψ

119879(119905) 119870Ψ (119909) 119889119905

= 119865119879Ψ (119909)

+ 119884lowast119879(int

1

0

Ψ (119905) Ψ119879(119905) 119889119905)119870Ψ (119909)

= 119865119879Ψ (119909) + 119884

lowast119879119870Ψ (119909)

= (119865119879+ 119884lowast119879119870)Ψ (119909)


119879= 0

(147)



119906 (119909) = 119891 (119909) + int

1

0

119870 (119909 119905) 119865 (119906 (119905)) 119889119905

0 le 119909 le 1

(148)


119871 (119906) = 119906 (119909) minus 119891 (119909) minus int

1

0

119870 (119909 119905) 119865 (119906 (119905)) 119889119905 = 0 (149)



119867(119906 0) = 119873 (119906)

119867 (119906 1) = 119871 (119906)

(150)


119867(119906 119901) = (1 minus 119901)119873 (119906) + 119901119871 (119906) = 0 (151)


119906 = 1199060 + 1199011199061 + 11990121199062 + sdot sdot sdot (152a)


119880 = lim119901rarr1

119906 = 1199060 + 1199061 + 1199062 + sdot sdot sdot (152b)


Consider

119873(119906) = 119906 (119909) minus 119891 (119909) (153)


119865 (119906) =

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898)

= 1198670 (1199060) + 1199011198671 (1199060 1199061)


(154)

where119867119898 (1199060 1199061 119906119898)

=1

119898

120597119898

120597119901119898(119865(

119898

sum

119896=0

119901119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816119901=0

119898 ge 0

(155)




minusint

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905) = 0


minus 119901int

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905 = 0

(156)


1199010 1199060 (119909) minus 119891 (119909) = 0 997904rArr 1199060 (119909) = 119891 (119909)

1199011 1199061 (119909) minus int

1

0

119870 (119909 119905)1198670119889119905 = 0 997904rArr 1199061 (119909)

= int

1

0

119870 (119909 119905)1198670119889119905

1199012 1199062 (119909) minus int

1

0

119870 (119909 119905)1198671119889119905 = 0 997904rArr 1199062 (119909)

= int

1

0

119870 (119909 119905)1198671119889119905

(157)


1199060 (119909) = 119891 (119909)

119906119899+1 (119909) = int

1

0

119870 (119909 119905)119867119899119889119905 119899 = 0 1 2

(158)



119906 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) (119871119906 (119905) + 119873119906 (119905)) 119889119905

119886 le 119909 le 119887

(159)



119906 (119909) =

infin

sum

119898=0

119906119898 (119909) (160)


119873119906 =

infin

sum

119899=0

119860119899 (161)

where 119860119899 =1

119899(120597119899

120597120582119899119873(

infin

sum

119896=0

120582119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816120582=0

(162)



infin

sum

119898=0

119906119898 (119909) = 119891 (119909)

+ int

119887

119886

119870 (119909 119905) (119871(

infin

sum

119898=0

119906119898 (119905)) +

infin

sum

119898=0

119860119898)119889119905

(163)


1199060 (119909) = 119891 (119909)

119906119898 (119909) = int

119887

119886

119870 (119909 119905) (119871 (119906119898minus1 (119905)) + 119860119898minus1 (119905)) 119889119905

119898 ge 1

(164)


120593119899 = 1199060 + 1199061 + sdot sdot sdot + 119906119899 (165)

with

119906 (119909) = lim119899rarrinfin

120593119899 (166)




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119860119879Ψ (119909) = 119863

119879Ψ (119909) + 119861

119879ΘΨ (119909) (115)


119860119879119875 = 119863

119879+ 119861119879Θ

119860119879119875 minus 119863

119879minus 119861119879Θ = 0

(116)


119875 = int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = [

1198751

1198752]

int

1

0

Ψ (119909)Ψ119879(119909) 119889119909 = 119868

(117)



119865 (119909 119860119879Ψ (119909)) = 119861

119879Ψ (119909) (118)


+119898minus2) where119894 = 1 2 2

119872+1+ 119898 minus 1







int

119887

119886

119891 (119909) 119889119909 =

119899minus1

sum

119894=1

int

119909119894+1

119909119894minus1

119891 (119909) 119889119909

=ℎ

3119891 (119886) +

4ℎ

3

1198992

sum

119894=1

119891 (1199092119894minus1)

+2ℎ

3

(119899minus1)2

sum

119894=1

119891 (1199092119894)

+ℎ

3119891 (119887)

minus(119887 minus 119886)

180ℎ4119891(4)(120578)

(119)


int

119887

119886

119891 (119909) 119889119909 =

119899

sum

119894=1

int

119909119894

119909119894minus1

119891 (119909) 119889119909

=ℎ

2119891 (119886) + ℎ

119899minus1

sum

119894=1

119891 (119909119894)

+ℎ

2119891 (119887)

+ℎ2

12[1198911015840(119886) minus 119891

1015840(119887)]

(120)


119906 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) 119865 (119906 (119905)) 119889119905

119886 le 119909 le 119887

(121)



+ℎ

3

[

[

119870 (119909 1199050) 119865 (1199060)

+ 4

1198992

sum

119895=1

119870(119909 1199052119895minus1) 119865 (1199062119895minus1)

+ 2

(1198992)minus1

sum

119895=1

119870(119909 1199052119895) 119865 (1199062119895)

+ 119870 (119909 119905119899) 119865 (119906119899)]

]

(122)


+ℎ

3

[

[

119870 (119909119894 1199050) 119865 (1199060)

+ 4

1198992

sum

119895=1

119870(119909119894 1199052119895minus1) 119865 (1199062119895minus1)


+ 2

(1198992)minus1

sum

119895=1

119870(119909119894 1199052119895) 119865 (1199062119895)

+119870 (119909119894 119905119899) 119865 (119906119899)]

]

(123)



119906 (119909) = 119891 (119909)

+ℎ

2119870 (119909 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909 119905119895) 119865 (119906119895)

+ℎ

2119870 (119909 119905119899) 119865 (119906119899)

+ℎ2

12[119869 (119909 1199050) 119865 (1199060)

+ 119870 (119909 1199050) 1199061015840

01198651015840(1199060)

minus 119869 (119909 119905119899) 119865 (119906119899)

minus119870 (119909 119905119899) 1199061015840

1198991198651015840(119906119899)]

(124)


have

119906 (119909119894) = 119891 (119909119894)

+ℎ

2119870 (119909119894 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909119894 119905119895) 119865 (119906119895)

+ℎ

2119870 (119909119894 119905119899) 119865 (119906119899)

+ℎ2

12[119869 (119909119894 1199050) 119865 (1199060)

+ 119870 (119909119894 1199050) 1199061015840

01198651015840(1199060)

minus 119869 (119909119894 119905119899) 119865 (119906119899)

minus119870 (119909119894 119905119899) 1199061015840

1198991198651015840(119906119899)]

(125)

for 119894 = 0 1 119899


120597119870(119909 119905)120597119909 we obtain

1199061015840(119909) = 119891

1015840(119909) + int

119887

119886

119867(119909 119905) 119865 (119906 (119905)) 119889119905

119886 le 119909 le 119887

(126)


1199061015840(119909119894) = 119891

1015840(119909119894)

+ℎ

2119867 (119909119894 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(119909119894 119905119895) 119865 (119906119895)

+ℎ

2119867 (119909119894 119905119899) 119865 (119906119899)

(127)



119906 (119909119894) = (ℎ

2119870 (119909119894 1199050) +

ℎ2

12119869 (119909119894 1199050))119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909119894 119905119895) 119865 (119906119895)

+ (ℎ

2119870 (119909119894 119905119899) minus

ℎ2

12119869 (119909119894 119905119899))119865 (119906119899)

+ℎ2

12(119870 (119909119894 1199050) 119906

1015840

01198651015840(1199060)

minus119870 (119909119894 119905119899) 1199061015840

1198991198651015840(119906119899))

1199061015840(1199090) = 119891

1015840(1199090)

+ℎ

2119867 (1199090 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(1199090 119905119895) 119865 (119906119895)

+ℎ

2119867 (1199090 119905119899) 119865 (119906119899)


1199061015840(119909119899) = 119891

1015840(119909119899)

+ℎ

2119867 (119909119899 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(119909119899 119905119895) 119865 (119906119895)

+ℎ

2119867 (119909119899 119905119899) 119865 (119906119899)

(128)




120595119886119887 (119905) = |119886|minus12120595(

119905 minus 119887

119886)

119886 119887 isin 119877 119886 = 0

(129)



120595119898119899 (119905)

=

(119898 +1

2)

12

21198962119871119898 (2

119896119905 minus 119899)

119899 minus 1

2119896le 119905 lt

119899 + 1

2119896

0 otherwise(130)


1198710 (119905) = 1

1198711 (119905) = 119905

119871119898+1 (119905) =2119898 + 1

119898 + 1119905119871119898 (119905)

minus119898

119898 + 1119871119898minus1 (119905) 119898 = 1 2 3

(131)



119891 (119909) =

infin

sum

119899=1

infin

sum

119898=0

119888119899119898120595119899119898 (119909) (132)

where

119888119899119898 = ⟨119891 (119909) 120595119899119898 (119909)⟩ (133)


119891 (119909) asymp

2119896minus1

sum

119899=1

119872minus1

sum

119898=0

119888119899119898120595119899119898 (119909) = 119862119879Ψ (119909) (134)


119862 = [11988810 11988811 1198881119872minus1 11988820


(135)

Ψ (119909) = [12059510 (119909) 1205951119872minus1 (119909)

12059520 (119909) 1205952119872minus1 (119909)


(136)


approximated as

119896 (119909 119905) asymp Ψ119879(119905) 119870Ψ (119909) (137)


119870119894119895 = ⟨120595119894 (119905) ⟨119896 (119909 119905) 120595119895 (119909)⟩⟩ (138)


[119910 (119909)]119901= [119884119879Ψ (119909)]

119901

= 119884lowast

119901

119879Ψ (119909) (139)





int

119905

0

Ψ(1199051015840) 1198891199051015840= 119875Ψ (119905) (140)



119875 =

[[[[[[

[

119871 119867 sdot sdot sdot 119867 119867

0 119871 sdot sdot sdot 119867 119867

d

0 0 sdot sdot sdot 119871 119867


]]]]]]

]

(141)



119867 =1

2119896

[[[[

[



d


]]]]

]

119871 =1

2119896

[[[[[[[[[[[[[[[[[[[[[[[

[

11


minusradic3

30

radic3


0 minusradic5

5radic30

radic5


0 0 minusradic7


d





]]]]]]]]]]]]]]]]]]]]]]]

]

(142)


int

1

0

Ψ (119905)Ψ119879(119905) 119889119905 = 119868 (143)


defined as

Ψ (119905) Ψ119879(119905) 119862 = 119862

119879Ψ (119905) (144)




119910 (119909) = 119891 (119909) + int

1

0

119896 (119909 119905) [119910 (119905)]119901119889119905 (145)



119891 (119909) asymp 119865119879Ψ (119909)

119910 (119909) asymp 119884119879Ψ (119909)

119896 (119909 119905) asymp Ψ119879(119905) 119870Ψ (119909)

[119910 (119909)]119901asymp 119884lowast119879Ψ (119909)

(146)


119884119879Ψ (119909) = 119865

119879Ψ (119909)

+ int

1

0

119884lowast119879Ψ (119905) Ψ

119879(119905) 119870Ψ (119909) 119889119905

= 119865119879Ψ (119909)

+ 119884lowast119879(int

1

0

Ψ (119905) Ψ119879(119905) 119889119905)119870Ψ (119909)

= 119865119879Ψ (119909) + 119884

lowast119879119870Ψ (119909)

= (119865119879+ 119884lowast119879119870)Ψ (119909)


119879= 0

(147)



119906 (119909) = 119891 (119909) + int

1

0

119870 (119909 119905) 119865 (119906 (119905)) 119889119905

0 le 119909 le 1

(148)


119871 (119906) = 119906 (119909) minus 119891 (119909) minus int

1

0

119870 (119909 119905) 119865 (119906 (119905)) 119889119905 = 0 (149)



119867(119906 0) = 119873 (119906)

119867 (119906 1) = 119871 (119906)

(150)


119867(119906 119901) = (1 minus 119901)119873 (119906) + 119901119871 (119906) = 0 (151)


119906 = 1199060 + 1199011199061 + 11990121199062 + sdot sdot sdot (152a)


119880 = lim119901rarr1

119906 = 1199060 + 1199061 + 1199062 + sdot sdot sdot (152b)


Consider

119873(119906) = 119906 (119909) minus 119891 (119909) (153)


119865 (119906) =

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898)

= 1198670 (1199060) + 1199011198671 (1199060 1199061)


(154)

where119867119898 (1199060 1199061 119906119898)

=1

119898

120597119898

120597119901119898(119865(

119898

sum

119896=0

119901119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816119901=0

119898 ge 0

(155)




minusint

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905) = 0


minus 119901int

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905 = 0

(156)


1199010 1199060 (119909) minus 119891 (119909) = 0 997904rArr 1199060 (119909) = 119891 (119909)

1199011 1199061 (119909) minus int

1

0

119870 (119909 119905)1198670119889119905 = 0 997904rArr 1199061 (119909)

= int

1

0

119870 (119909 119905)1198670119889119905

1199012 1199062 (119909) minus int

1

0

119870 (119909 119905)1198671119889119905 = 0 997904rArr 1199062 (119909)

= int

1

0

119870 (119909 119905)1198671119889119905

(157)


1199060 (119909) = 119891 (119909)

119906119899+1 (119909) = int

1

0

119870 (119909 119905)119867119899119889119905 119899 = 0 1 2

(158)



119906 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) (119871119906 (119905) + 119873119906 (119905)) 119889119905

119886 le 119909 le 119887

(159)



119906 (119909) =

infin

sum

119898=0

119906119898 (119909) (160)


119873119906 =

infin

sum

119899=0

119860119899 (161)

where 119860119899 =1

119899(120597119899

120597120582119899119873(

infin

sum

119896=0

120582119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816120582=0

(162)



infin

sum

119898=0

119906119898 (119909) = 119891 (119909)

+ int

119887

119886

119870 (119909 119905) (119871(

infin

sum

119898=0

119906119898 (119905)) +

infin

sum

119898=0

119860119898)119889119905

(163)


1199060 (119909) = 119891 (119909)

119906119898 (119909) = int

119887

119886

119870 (119909 119905) (119871 (119906119898minus1 (119905)) + 119860119898minus1 (119905)) 119889119905

119898 ge 1

(164)


120593119899 = 1199060 + 1199061 + sdot sdot sdot + 119906119899 (165)

with

119906 (119909) = lim119899rarrinfin

120593119899 (166)




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+ 2

(1198992)minus1

sum

119895=1

119870(119909119894 1199052119895) 119865 (1199062119895)

+119870 (119909119894 119905119899) 119865 (119906119899)]

]

(123)



119906 (119909) = 119891 (119909)

+ℎ

2119870 (119909 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909 119905119895) 119865 (119906119895)

+ℎ

2119870 (119909 119905119899) 119865 (119906119899)

+ℎ2

12[119869 (119909 1199050) 119865 (1199060)

+ 119870 (119909 1199050) 1199061015840

01198651015840(1199060)

minus 119869 (119909 119905119899) 119865 (119906119899)

minus119870 (119909 119905119899) 1199061015840

1198991198651015840(119906119899)]

(124)


have

119906 (119909119894) = 119891 (119909119894)

+ℎ

2119870 (119909119894 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909119894 119905119895) 119865 (119906119895)

+ℎ

2119870 (119909119894 119905119899) 119865 (119906119899)

+ℎ2

12[119869 (119909119894 1199050) 119865 (1199060)

+ 119870 (119909119894 1199050) 1199061015840

01198651015840(1199060)

minus 119869 (119909119894 119905119899) 119865 (119906119899)

minus119870 (119909119894 119905119899) 1199061015840

1198991198651015840(119906119899)]

(125)

for 119894 = 0 1 119899


120597119870(119909 119905)120597119909 we obtain

1199061015840(119909) = 119891

1015840(119909) + int

119887

119886

119867(119909 119905) 119865 (119906 (119905)) 119889119905

119886 le 119909 le 119887

(126)


1199061015840(119909119894) = 119891

1015840(119909119894)

+ℎ

2119867 (119909119894 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(119909119894 119905119895) 119865 (119906119895)

+ℎ

2119867 (119909119894 119905119899) 119865 (119906119899)

(127)



119906 (119909119894) = (ℎ

2119870 (119909119894 1199050) +

ℎ2

12119869 (119909119894 1199050))119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119870(119909119894 119905119895) 119865 (119906119895)

+ (ℎ

2119870 (119909119894 119905119899) minus

ℎ2

12119869 (119909119894 119905119899))119865 (119906119899)

+ℎ2

12(119870 (119909119894 1199050) 119906

1015840

01198651015840(1199060)

minus119870 (119909119894 119905119899) 1199061015840

1198991198651015840(119906119899))

1199061015840(1199090) = 119891

1015840(1199090)

+ℎ

2119867 (1199090 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(1199090 119905119895) 119865 (119906119895)

+ℎ

2119867 (1199090 119905119899) 119865 (119906119899)


1199061015840(119909119899) = 119891

1015840(119909119899)

+ℎ

2119867 (119909119899 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(119909119899 119905119895) 119865 (119906119895)

+ℎ

2119867 (119909119899 119905119899) 119865 (119906119899)

(128)




120595119886119887 (119905) = |119886|minus12120595(

119905 minus 119887

119886)

119886 119887 isin 119877 119886 = 0

(129)



120595119898119899 (119905)

=

(119898 +1

2)

12

21198962119871119898 (2

119896119905 minus 119899)

119899 minus 1

2119896le 119905 lt

119899 + 1

2119896

0 otherwise(130)


1198710 (119905) = 1

1198711 (119905) = 119905

119871119898+1 (119905) =2119898 + 1

119898 + 1119905119871119898 (119905)

minus119898

119898 + 1119871119898minus1 (119905) 119898 = 1 2 3

(131)



119891 (119909) =

infin

sum

119899=1

infin

sum

119898=0

119888119899119898120595119899119898 (119909) (132)

where

119888119899119898 = ⟨119891 (119909) 120595119899119898 (119909)⟩ (133)


119891 (119909) asymp

2119896minus1

sum

119899=1

119872minus1

sum

119898=0

119888119899119898120595119899119898 (119909) = 119862119879Ψ (119909) (134)


119862 = [11988810 11988811 1198881119872minus1 11988820


(135)

Ψ (119909) = [12059510 (119909) 1205951119872minus1 (119909)

12059520 (119909) 1205952119872minus1 (119909)


(136)


approximated as

119896 (119909 119905) asymp Ψ119879(119905) 119870Ψ (119909) (137)


119870119894119895 = ⟨120595119894 (119905) ⟨119896 (119909 119905) 120595119895 (119909)⟩⟩ (138)


[119910 (119909)]119901= [119884119879Ψ (119909)]

119901

= 119884lowast

119901

119879Ψ (119909) (139)





int

119905

0

Ψ(1199051015840) 1198891199051015840= 119875Ψ (119905) (140)



119875 =

[[[[[[

[

119871 119867 sdot sdot sdot 119867 119867

0 119871 sdot sdot sdot 119867 119867

d

0 0 sdot sdot sdot 119871 119867


]]]]]]

]

(141)



119867 =1

2119896

[[[[

[



d


]]]]

]

119871 =1

2119896

[[[[[[[[[[[[[[[[[[[[[[[

[

11


minusradic3

30

radic3


0 minusradic5

5radic30

radic5


0 0 minusradic7


d





]]]]]]]]]]]]]]]]]]]]]]]

]

(142)


int

1

0

Ψ (119905)Ψ119879(119905) 119889119905 = 119868 (143)


defined as

Ψ (119905) Ψ119879(119905) 119862 = 119862

119879Ψ (119905) (144)




119910 (119909) = 119891 (119909) + int

1

0

119896 (119909 119905) [119910 (119905)]119901119889119905 (145)



119891 (119909) asymp 119865119879Ψ (119909)

119910 (119909) asymp 119884119879Ψ (119909)

119896 (119909 119905) asymp Ψ119879(119905) 119870Ψ (119909)

[119910 (119909)]119901asymp 119884lowast119879Ψ (119909)

(146)


119884119879Ψ (119909) = 119865

119879Ψ (119909)

+ int

1

0

119884lowast119879Ψ (119905) Ψ

119879(119905) 119870Ψ (119909) 119889119905

= 119865119879Ψ (119909)

+ 119884lowast119879(int

1

0

Ψ (119905) Ψ119879(119905) 119889119905)119870Ψ (119909)

= 119865119879Ψ (119909) + 119884

lowast119879119870Ψ (119909)

= (119865119879+ 119884lowast119879119870)Ψ (119909)


119879= 0

(147)



119906 (119909) = 119891 (119909) + int

1

0

119870 (119909 119905) 119865 (119906 (119905)) 119889119905

0 le 119909 le 1

(148)


119871 (119906) = 119906 (119909) minus 119891 (119909) minus int

1

0

119870 (119909 119905) 119865 (119906 (119905)) 119889119905 = 0 (149)



119867(119906 0) = 119873 (119906)

119867 (119906 1) = 119871 (119906)

(150)


119867(119906 119901) = (1 minus 119901)119873 (119906) + 119901119871 (119906) = 0 (151)


119906 = 1199060 + 1199011199061 + 11990121199062 + sdot sdot sdot (152a)


119880 = lim119901rarr1

119906 = 1199060 + 1199061 + 1199062 + sdot sdot sdot (152b)


Consider

119873(119906) = 119906 (119909) minus 119891 (119909) (153)


119865 (119906) =

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898)

= 1198670 (1199060) + 1199011198671 (1199060 1199061)


(154)

where119867119898 (1199060 1199061 119906119898)

=1

119898

120597119898

120597119901119898(119865(

119898

sum

119896=0

119901119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816119901=0

119898 ge 0

(155)




minusint

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905) = 0


minus 119901int

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905 = 0

(156)


1199010 1199060 (119909) minus 119891 (119909) = 0 997904rArr 1199060 (119909) = 119891 (119909)

1199011 1199061 (119909) minus int

1

0

119870 (119909 119905)1198670119889119905 = 0 997904rArr 1199061 (119909)

= int

1

0

119870 (119909 119905)1198670119889119905

1199012 1199062 (119909) minus int

1

0

119870 (119909 119905)1198671119889119905 = 0 997904rArr 1199062 (119909)

= int

1

0

119870 (119909 119905)1198671119889119905

(157)


1199060 (119909) = 119891 (119909)

119906119899+1 (119909) = int

1

0

119870 (119909 119905)119867119899119889119905 119899 = 0 1 2

(158)



119906 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) (119871119906 (119905) + 119873119906 (119905)) 119889119905

119886 le 119909 le 119887

(159)



119906 (119909) =

infin

sum

119898=0

119906119898 (119909) (160)


119873119906 =

infin

sum

119899=0

119860119899 (161)

where 119860119899 =1

119899(120597119899

120597120582119899119873(

infin

sum

119896=0

120582119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816120582=0

(162)



infin

sum

119898=0

119906119898 (119909) = 119891 (119909)

+ int

119887

119886

119870 (119909 119905) (119871(

infin

sum

119898=0

119906119898 (119905)) +

infin

sum

119898=0

119860119898)119889119905

(163)


1199060 (119909) = 119891 (119909)

119906119898 (119909) = int

119887

119886

119870 (119909 119905) (119871 (119906119898minus1 (119905)) + 119860119898minus1 (119905)) 119889119905

119898 ge 1

(164)


120593119899 = 1199060 + 1199061 + sdot sdot sdot + 119906119899 (165)

with

119906 (119909) = lim119899rarrinfin

120593119899 (166)




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1199061015840(119909119899) = 119891

1015840(119909119899)

+ℎ

2119867 (119909119899 1199050) 119865 (1199060)

+ ℎ

119899minus1

sum

119895=1

119867(119909119899 119905119895) 119865 (119906119895)

+ℎ

2119867 (119909119899 119905119899) 119865 (119906119899)

(128)




120595119886119887 (119905) = |119886|minus12120595(

119905 minus 119887

119886)

119886 119887 isin 119877 119886 = 0

(129)



120595119898119899 (119905)

=

(119898 +1

2)

12

21198962119871119898 (2

119896119905 minus 119899)

119899 minus 1

2119896le 119905 lt

119899 + 1

2119896

0 otherwise(130)


1198710 (119905) = 1

1198711 (119905) = 119905

119871119898+1 (119905) =2119898 + 1

119898 + 1119905119871119898 (119905)

minus119898

119898 + 1119871119898minus1 (119905) 119898 = 1 2 3

(131)



119891 (119909) =

infin

sum

119899=1

infin

sum

119898=0

119888119899119898120595119899119898 (119909) (132)

where

119888119899119898 = ⟨119891 (119909) 120595119899119898 (119909)⟩ (133)


119891 (119909) asymp

2119896minus1

sum

119899=1

119872minus1

sum

119898=0

119888119899119898120595119899119898 (119909) = 119862119879Ψ (119909) (134)


119862 = [11988810 11988811 1198881119872minus1 11988820


(135)

Ψ (119909) = [12059510 (119909) 1205951119872minus1 (119909)

12059520 (119909) 1205952119872minus1 (119909)


(136)


approximated as

119896 (119909 119905) asymp Ψ119879(119905) 119870Ψ (119909) (137)


119870119894119895 = ⟨120595119894 (119905) ⟨119896 (119909 119905) 120595119895 (119909)⟩⟩ (138)


[119910 (119909)]119901= [119884119879Ψ (119909)]

119901

= 119884lowast

119901

119879Ψ (119909) (139)





int

119905

0

Ψ(1199051015840) 1198891199051015840= 119875Ψ (119905) (140)



119875 =

[[[[[[

[

119871 119867 sdot sdot sdot 119867 119867

0 119871 sdot sdot sdot 119867 119867

d

0 0 sdot sdot sdot 119871 119867


]]]]]]

]

(141)



119867 =1

2119896

[[[[

[



d


]]]]

]

119871 =1

2119896

[[[[[[[[[[[[[[[[[[[[[[[

[

11


minusradic3

30

radic3


0 minusradic5

5radic30

radic5


0 0 minusradic7


d





]]]]]]]]]]]]]]]]]]]]]]]

]

(142)


int

1

0

Ψ (119905)Ψ119879(119905) 119889119905 = 119868 (143)


defined as

Ψ (119905) Ψ119879(119905) 119862 = 119862

119879Ψ (119905) (144)




119910 (119909) = 119891 (119909) + int

1

0

119896 (119909 119905) [119910 (119905)]119901119889119905 (145)



119891 (119909) asymp 119865119879Ψ (119909)

119910 (119909) asymp 119884119879Ψ (119909)

119896 (119909 119905) asymp Ψ119879(119905) 119870Ψ (119909)

[119910 (119909)]119901asymp 119884lowast119879Ψ (119909)

(146)


119884119879Ψ (119909) = 119865

119879Ψ (119909)

+ int

1

0

119884lowast119879Ψ (119905) Ψ

119879(119905) 119870Ψ (119909) 119889119905

= 119865119879Ψ (119909)

+ 119884lowast119879(int

1

0

Ψ (119905) Ψ119879(119905) 119889119905)119870Ψ (119909)

= 119865119879Ψ (119909) + 119884

lowast119879119870Ψ (119909)

= (119865119879+ 119884lowast119879119870)Ψ (119909)


119879= 0

(147)



119906 (119909) = 119891 (119909) + int

1

0

119870 (119909 119905) 119865 (119906 (119905)) 119889119905

0 le 119909 le 1

(148)


119871 (119906) = 119906 (119909) minus 119891 (119909) minus int

1

0

119870 (119909 119905) 119865 (119906 (119905)) 119889119905 = 0 (149)



119867(119906 0) = 119873 (119906)

119867 (119906 1) = 119871 (119906)

(150)


119867(119906 119901) = (1 minus 119901)119873 (119906) + 119901119871 (119906) = 0 (151)


119906 = 1199060 + 1199011199061 + 11990121199062 + sdot sdot sdot (152a)


119880 = lim119901rarr1

119906 = 1199060 + 1199061 + 1199062 + sdot sdot sdot (152b)


Consider

119873(119906) = 119906 (119909) minus 119891 (119909) (153)


119865 (119906) =

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898)

= 1198670 (1199060) + 1199011198671 (1199060 1199061)


(154)

where119867119898 (1199060 1199061 119906119898)

=1

119898

120597119898

120597119901119898(119865(

119898

sum

119896=0

119901119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816119901=0

119898 ge 0

(155)




minusint

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905) = 0


minus 119901int

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905 = 0

(156)


1199010 1199060 (119909) minus 119891 (119909) = 0 997904rArr 1199060 (119909) = 119891 (119909)

1199011 1199061 (119909) minus int

1

0

119870 (119909 119905)1198670119889119905 = 0 997904rArr 1199061 (119909)

= int

1

0

119870 (119909 119905)1198670119889119905

1199012 1199062 (119909) minus int

1

0

119870 (119909 119905)1198671119889119905 = 0 997904rArr 1199062 (119909)

= int

1

0

119870 (119909 119905)1198671119889119905

(157)


1199060 (119909) = 119891 (119909)

119906119899+1 (119909) = int

1

0

119870 (119909 119905)119867119899119889119905 119899 = 0 1 2

(158)



119906 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) (119871119906 (119905) + 119873119906 (119905)) 119889119905

119886 le 119909 le 119887

(159)



119906 (119909) =

infin

sum

119898=0

119906119898 (119909) (160)


119873119906 =

infin

sum

119899=0

119860119899 (161)

where 119860119899 =1

119899(120597119899

120597120582119899119873(

infin

sum

119896=0

120582119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816120582=0

(162)



infin

sum

119898=0

119906119898 (119909) = 119891 (119909)

+ int

119887

119886

119870 (119909 119905) (119871(

infin

sum

119898=0

119906119898 (119905)) +

infin

sum

119898=0

119860119898)119889119905

(163)


1199060 (119909) = 119891 (119909)

119906119898 (119909) = int

119887

119886

119870 (119909 119905) (119871 (119906119898minus1 (119905)) + 119860119898minus1 (119905)) 119889119905

119898 ge 1

(164)


120593119899 = 1199060 + 1199061 + sdot sdot sdot + 119906119899 (165)

with

119906 (119909) = lim119899rarrinfin

120593119899 (166)




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119867 =1

2119896

[[[[

[



d


]]]]

]

119871 =1

2119896

[[[[[[[[[[[[[[[[[[[[[[[

[

11


minusradic3

30

radic3


0 minusradic5

5radic30

radic5


0 0 minusradic7


d





]]]]]]]]]]]]]]]]]]]]]]]

]

(142)


int

1

0

Ψ (119905)Ψ119879(119905) 119889119905 = 119868 (143)


defined as

Ψ (119905) Ψ119879(119905) 119862 = 119862

119879Ψ (119905) (144)




119910 (119909) = 119891 (119909) + int

1

0

119896 (119909 119905) [119910 (119905)]119901119889119905 (145)



119891 (119909) asymp 119865119879Ψ (119909)

119910 (119909) asymp 119884119879Ψ (119909)

119896 (119909 119905) asymp Ψ119879(119905) 119870Ψ (119909)

[119910 (119909)]119901asymp 119884lowast119879Ψ (119909)

(146)


119884119879Ψ (119909) = 119865

119879Ψ (119909)

+ int

1

0

119884lowast119879Ψ (119905) Ψ

119879(119905) 119870Ψ (119909) 119889119905

= 119865119879Ψ (119909)

+ 119884lowast119879(int

1

0

Ψ (119905) Ψ119879(119905) 119889119905)119870Ψ (119909)

= 119865119879Ψ (119909) + 119884

lowast119879119870Ψ (119909)

= (119865119879+ 119884lowast119879119870)Ψ (119909)


119879= 0

(147)



119906 (119909) = 119891 (119909) + int

1

0

119870 (119909 119905) 119865 (119906 (119905)) 119889119905

0 le 119909 le 1

(148)


119871 (119906) = 119906 (119909) minus 119891 (119909) minus int

1

0

119870 (119909 119905) 119865 (119906 (119905)) 119889119905 = 0 (149)



119867(119906 0) = 119873 (119906)

119867 (119906 1) = 119871 (119906)

(150)


119867(119906 119901) = (1 minus 119901)119873 (119906) + 119901119871 (119906) = 0 (151)


119906 = 1199060 + 1199011199061 + 11990121199062 + sdot sdot sdot (152a)


119880 = lim119901rarr1

119906 = 1199060 + 1199061 + 1199062 + sdot sdot sdot (152b)


Consider

119873(119906) = 119906 (119909) minus 119891 (119909) (153)


119865 (119906) =

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898)

= 1198670 (1199060) + 1199011198671 (1199060 1199061)


(154)

where119867119898 (1199060 1199061 119906119898)

=1

119898

120597119898

120597119901119898(119865(

119898

sum

119896=0

119901119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816119901=0

119898 ge 0

(155)




minusint

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905) = 0


minus 119901int

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905 = 0

(156)


1199010 1199060 (119909) minus 119891 (119909) = 0 997904rArr 1199060 (119909) = 119891 (119909)

1199011 1199061 (119909) minus int

1

0

119870 (119909 119905)1198670119889119905 = 0 997904rArr 1199061 (119909)

= int

1

0

119870 (119909 119905)1198670119889119905

1199012 1199062 (119909) minus int

1

0

119870 (119909 119905)1198671119889119905 = 0 997904rArr 1199062 (119909)

= int

1

0

119870 (119909 119905)1198671119889119905

(157)


1199060 (119909) = 119891 (119909)

119906119899+1 (119909) = int

1

0

119870 (119909 119905)119867119899119889119905 119899 = 0 1 2

(158)



119906 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) (119871119906 (119905) + 119873119906 (119905)) 119889119905

119886 le 119909 le 119887

(159)



119906 (119909) =

infin

sum

119898=0

119906119898 (119909) (160)


119873119906 =

infin

sum

119899=0

119860119899 (161)

where 119860119899 =1

119899(120597119899

120597120582119899119873(

infin

sum

119896=0

120582119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816120582=0

(162)



infin

sum

119898=0

119906119898 (119909) = 119891 (119909)

+ int

119887

119886

119870 (119909 119905) (119871(

infin

sum

119898=0

119906119898 (119905)) +

infin

sum

119898=0

119860119898)119889119905

(163)


1199060 (119909) = 119891 (119909)

119906119898 (119909) = int

119887

119886

119870 (119909 119905) (119871 (119906119898minus1 (119905)) + 119860119898minus1 (119905)) 119889119905

119898 ge 1

(164)


120593119899 = 1199060 + 1199061 + sdot sdot sdot + 119906119899 (165)

with

119906 (119909) = lim119899rarrinfin

120593119899 (166)




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119867(119906 0) = 119873 (119906)

119867 (119906 1) = 119871 (119906)

(150)


119867(119906 119901) = (1 minus 119901)119873 (119906) + 119901119871 (119906) = 0 (151)


119906 = 1199060 + 1199011199061 + 11990121199062 + sdot sdot sdot (152a)


119880 = lim119901rarr1

119906 = 1199060 + 1199061 + 1199062 + sdot sdot sdot (152b)


Consider

119873(119906) = 119906 (119909) minus 119891 (119909) (153)


119865 (119906) =

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898)

= 1198670 (1199060) + 1199011198671 (1199060 1199061)


(154)

where119867119898 (1199060 1199061 119906119898)

=1

119898

120597119898

120597119901119898(119865(

119898

sum

119896=0

119901119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816119901=0

119898 ge 0

(155)




minusint

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905) = 0


minus 119901int

1

0

119870 (119909 119905)

infin

sum

119898=0

119901119898119867119898 (1199060 1199061 119906119898) 119889119905 = 0

(156)


1199010 1199060 (119909) minus 119891 (119909) = 0 997904rArr 1199060 (119909) = 119891 (119909)

1199011 1199061 (119909) minus int

1

0

119870 (119909 119905)1198670119889119905 = 0 997904rArr 1199061 (119909)

= int

1

0

119870 (119909 119905)1198670119889119905

1199012 1199062 (119909) minus int

1

0

119870 (119909 119905)1198671119889119905 = 0 997904rArr 1199062 (119909)

= int

1

0

119870 (119909 119905)1198671119889119905

(157)


1199060 (119909) = 119891 (119909)

119906119899+1 (119909) = int

1

0

119870 (119909 119905)119867119899119889119905 119899 = 0 1 2

(158)



119906 (119909) = 119891 (119909) + int

119887

119886

119870 (119909 119905) (119871119906 (119905) + 119873119906 (119905)) 119889119905

119886 le 119909 le 119887

(159)



119906 (119909) =

infin

sum

119898=0

119906119898 (119909) (160)


119873119906 =

infin

sum

119899=0

119860119899 (161)

where 119860119899 =1

119899(120597119899

120597120582119899119873(

infin

sum

119896=0

120582119896119906119896))

1003816100381610038161003816100381610038161003816100381610038161003816120582=0

(162)



infin

sum

119898=0

119906119898 (119909) = 119891 (119909)

+ int

119887

119886

119870 (119909 119905) (119871(

infin

sum

119898=0

119906119898 (119905)) +

infin

sum

119898=0

119860119898)119889119905

(163)


1199060 (119909) = 119891 (119909)

119906119898 (119909) = int

119887

119886

119870 (119909 119905) (119871 (119906119898minus1 (119905)) + 119860119898minus1 (119905)) 119889119905

119898 ge 1

(164)


120593119899 = 1199060 + 1199061 + sdot sdot sdot + 119906119899 (165)

with

119906 (119909) = lim119899rarrinfin

120593119899 (166)




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











infin

sum

119898=0

119906119898 (119909) = 119891 (119909)

+ int

119887

119886

119870 (119909 119905) (119871(

infin

sum

119898=0

119906119898 (119905)) +

infin

sum

119898=0

119860119898)119889119905

(163)


1199060 (119909) = 119891 (119909)

119906119898 (119909) = int

119887

119886

119870 (119909 119905) (119871 (119906119898minus1 (119905)) + 119860119898minus1 (119905)) 119889119905

119898 ge 1

(164)


120593119899 = 1199060 + 1199061 + sdot sdot sdot + 119906119899 (165)

with

119906 (119909) = lim119899rarrinfin

120593119899 (166)




References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Numerical Methods for Solving Fredholm Integral … · 2019. 7. 31. · Numerical Methods for Solving Fredholm Integral Equations of Second Kind S.SahaRayandP.K.Sahu

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