Research Article Spectral-Collocation Methods for...

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2013 Article ID 821327 14 pageshttpdxdoiorg1011552013821327

Research ArticleSpectral-Collocation Methods for Fractional PantographDelay-Integrodifferential Equations

Yin Yang1 and Yunqing Huang2

1 College of Civil Engineering and Mechanics Hunan Key Laboratory for Computation and Simulation in Science and EngineeringXiangtan University Xiangtan Hunan 411105 China

2Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtan Hunan 411105 China

Correspondence should be addressed to Yin Yang yangyinxtuxtueducn

Received 15 May 2013 Accepted 15 September 2013

Academic Editor Varsha Daftardar-Gejji

Copyright copy 2013 Y Yang and Y Huang 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

We propose and analyze a spectral Jacobi-collocation approximation for fractional order integrodifferential equations of Volterratype with pantograph delay The fractional derivative is described in the Caputo sense We provide a rigorous error analysis for thecollocation method which shows that the error of approximate solution decays exponentially in 119871

infin norm and weighted 1198712-norm

The numerical examples are given to illustrate the theoretical results

1 Introduction

Many phenomena in engineering physics chemistry andother sciences can be described very successfully by modelsusing mathematical tools from fractional calculus that is thetheory of derivatives and integrals of fractional nonintegerorder This allows one to describe physical phenomenamore accurately Moreover fractional calculus is appliedto the model frequency dependent damping behavior ofmany viscoelastic materials economics and dynamics ofinterfaces between nanoparticles and substrates Recentlyseveral numerical methods to solve fractional differentialequations (FDEs) and fractional integrodifferential equations(FIDEs) have been proposed

In this paper we consider the general linear fractionalpantograph delay-integrodifferential equations (FDIDEs)with proportional delays

119863120574

119910 (119905) = 119910 (119905) + 119910 (119902119905) + 119892 (119905) + int

119905

0

1(119905 119904) 119910 (119904) 119889119904

+ int

119902119905

0

2(119905 120591) 119910 (120591) 119889120591

0 lt 120574 le 1 119905 isin 119868 = [0 119879]

119910 (0) = 1199100

(1)

with 0 lt 119902 lt 1 where 119892 119868 rarr 119877 1 119863 rarr 119877 (119863 =

(119905 120591) 0 le 119904 le 119905 le 119879) and 2 119863 rarr 119877 (119863 = (119905 120591)

0 le 120591 le 119902119905 119905 isin 119868) are given functions and are assumed tobe sufficiently smooth in the respective domains In (1) 119863120574

denotes the fractional derivative of fractional order 120574Differential and integral equations involving derivatives

of noninteger order have shown to be adequate modelsfor various phenomena arising in damping laws diffusionprocesses models of earthquake [1] fluid-dynamics trafficmodel [2] mathematical physics and engineering [3] fluidand continuum mechanics [4] chemistry acoustics andpsychology [5]

Let Γ(sdot) denote the Gamma function For any positiveinteger 119899 and 119899 minus 1 lt 120574 lt 119899 the Caputo derivative is definedas follows

119863120574

119910 (119905) =1

Γ (119899 minus 120574)int

119905

0

119910(119899)

(119904)

(119905 minus 119904)(120574minus119899+1)

119889119904 119905 isin [0 119879] (2)

The Riemann-Liouville fractional integral 119868120574 of order 120574 isdefined as

119868120574

119910 (119905) =1

Γ (120574)int

119905

0

(119905 minus 119904)120574minus1

119910 (119904) 119889119904 (3)

2 Advances in Mathematical Physics

we note that

119868120574

(119863120574

119910 (119905)) = 119910 (119905) minus

119899minus1

sum

119896=0

119910(119896)

(0)119905119896

119896 (4)

From (4) fractional integrodifferential equation (1) can bedescribed as

119863120574

119910 (119905) = 119910 (119905) + 119910 (119902119905) + 119892 (119905) + int

119905

0

1(119905 119904) 119910 (119904) 119889119904

+ int

119902119905

0

2(119905 120591) 119910 (120591) 119889120591

0 lt 120574 le 1 119905 isin 119868 = [0 119879]

119910 (119905) =1

Γ (120574)int

119905

0

(119905 minus 120591)120574minus1

119863120574

119910 (120591) 119889120591 + 119910 (0)

(5)

Several analytical methods have been introduced to solveFDEs including various transformation techniques [6] oper-ational calculus methods [7] the Adomian decompositionmethod [8] and the iterative and series-based methods [9]A small number of algorithms for the numerical solutionof FDEs have been suggested [10] and most of them arefinite difference methods which are generally limited to lowdimensions and are of limited accuracy

As we know fractional derivatives are global (they aredefined by an integral over the whole interval [0 119879]) andtherefore global methods such as spectral methods areperhaps better suited for FDEs Standard spectral methodspossess an infinite order of accuracy for the equations withregular solutions while failing for many complicated prob-lems with singular solutions So it is relevant to be interestedin how to enlarge the adaptability of spectral methods andconstruct certain simple approximation schemes without aloss of accuracy for more complicated problems

Spectral methods have been proposed to solve frac-tional differential equations such as the Legendre collocationmethod [11 12] Legendre wavelets method [13 14] andJacobi-Gauss-Lobatto collocation method [15] The authorsin [16ndash18] constructed an efficient spectral method for thenumerical approximation of fractional integrodifferentialequations based on tau and pseudospectral methods More-over Bhrawy et al [19] introduced a quadrature shiftedLegendre tau method based on the Gauss-Lobatto interpo-lation for solving multiorder FDEs with variable coefficientsand in [20] shifted Legendre spectral methods have beendeveloped for solving fractional-order multipoint boundaryvalue problems In [21] truncated Legendre series togetherwith the Legendre operationalmatrix of fractional derivativesare used for the numerical integration of fractional differ-ential equations In [22] the authors derived a new explicitformula for the integral of shifted Chebyshev polynomials ofany degree for any fractional-order The shifted Chebyshevoperational matrix [23] and shifted Jacobi operational matrix[24] of fractional derivatives have been developed which areapplied together with the spectral tau method for numericalsolution of general linear and nonlinear multiterm fractionaldifferential equations However very few theoretical results

were provided to justify the high accuracy numericallyobtained Recently Chen and Tang [25 26] developed anovel spectral Jacobi-collocation method to solve secondkindVolterra integral equationswith aweakly singular kerneland provided a rigorous error analysis which theoreticallyjustifies the spectral rate of convergence Inspired by thework of [26] we extend the approach to fractional orderdelay-integrodifferential equations (1) However it is difficultto apply the spectral approximations to the initial valueproblem and fractional order derivatives To facilitate the useof the spectral methods we restate the initial condition as anequivalent integral equation with singular kernel Then weget the discrete scheme by using Gauss quadrature formulaIn this paper we will provide a rigorous error analysis notonly for approximate solutions but also for approximatefractional derivatives which theoretically justifies the spectralrate of convergence

For ease of analysis we will describe the spectral methodson the standard interval 119868 = [minus1 1] Hence we employ thetransformation

119905 =119879

2(1 + 119909) 119904 =

119879 (1 + 120579)

2 120591 =

119879 (1 + 120578)

2

(6)

then the previous problem (5) becomes

119863120574

119906 (119909) = 119906 (119909) + 119906 (119902119909 + 119902 minus 1) + 119892 (119909)

+ int

119909

minus1

1198701(119909 120579) 119906 (120579) 119889120579

+ int

119902119909+119902minus1

minus1

1198702(119909 120578) 119906 (120578) 119889120578 119909 isin 119868

(7)

119906 (119909) =1

Γ (120574)(119879

2)

120574

int

119909

minus1

(119909 minus 119904)120574minus1

119863120574

119906 (119904) 119889119904 + 119906 (minus1)

(8)

where

119906 (119909) = 119910 (119879

2(1 + 119909)) 119863

120574

119906 (119909) = 119863120574

119910(119879

2(1 + 119909))

119892 (119909) =119879

2119892 (

119879

2(1 + 119909))

(9)

1198701(119909 120579) =

119879

41(119879

2(1 + 119909)

119879

2(1 + 120579))

1198702(119909 120578) =

119879

42(119879

2(1 + 119909)

119879

2(1 + 120578))

(10)

This paper is organized as follows In Section 2 we intro-duce the spectral approaches for pantograph FDIDEs Someuseful lemmas are provided in Section 3 These lemmas willplay a key role in the derivation of the convergence analysisWe provide a rigorous error analysis for the spectralmethodswhich shows that both the errors of approximate solutionsand the errors of approximate fractional derivatives of thesolutions decay exponentially in 119871

infin norm and weighted 1198712-

norm in Section 4 and Section 5 contains numerical results

Advances in Mathematical Physics 3

which will be used to verify the theoretical results obtained inSection 4

Throughout the paper 119862 will denote a generic positiveconstant that is independent of 119873 but which will depend on119902 119879 and on the bounds for the given functions 119886 119887 and 119870

119895

119895 = 1 2

2 Jacobi-Collocation Method

Let 120596120572120573(119909) = (1 minus 119909)120572

(1 + 119909)120573 be a weight function in the

usual sense for 120572 120573 gt minus1 The set of Jacobi polynomials119869120572120573

119899(119909)

infin

119899=0forms a complete1198712

120596120572120573(minus1 1)-orthogonal system

where 1198712120596120572120573(minus1 1) is a weighted space defined by

1198712

120596120572120573 (minus1 1) = V V is measurable and V

120596120572120573 lt infin

(11)

equipped with the norm

V120596120572120573 = (int

1

minus1

|V (119909)|2120596120572120573 (119909) 119889119909)12

(12)

and the inner product

(119906 V)120596120572120573 = int

1

minus1

119906 (119909) V (119909) 120596120572120573 (119909) 119889119909 forall119906 V isin 1198712

120596120572120573 (minus1 1)

(13)

For a given 119873 ge 0 we denote by 120579119896119873

119896=0the Legendre

points and by 120596119896119873

119896=0the corresponding Legendre weights

(ie Jacobi weights 12059600

119896119873

119896=0) Then the Legendre-Gauss

integration formula is

int

1

minus1

119891 (119909) 119889119909 asymp

119873

sum

119896=0

119891 (120579119896) 120596

119896 (14)

Similarly we denote by 120579119896119873

119896=0the Jacobi-Gauss points and by

120596120572120573

119896119873

119896=0the corresponding Jacobi weights Then the Jacobi-

Gauss integration formula is

int

1

minus1

119891 (119909) 120596120572120573

(119909) 119889119909 asymp

119873

sum

119896=0

119891 (120579119896) 120596

120572120573

119896 (15)

For a given positive integer119873 we denote the collocationpoints by 119909120572120573

119894119873

119894=0 which is the set of (119873 + 1) Jacobi-Gauss

points corresponding to the weight 120596120572120573(119909) Let P119873denote

the space of all polynomials of degree not exceeding 119873 Forany V isin 119862[minus1 1] we can define the Lagrange interpolatingpolynomial 119868120572120573

119873V isin P

119873 satisfying

119868120572120573

119873V (119909

119894) = V (119909

119894) 0 le 119894 le 119873 (16)

The Lagrange interpolating polynomial can be written in theform

119868120572120573

119873V (119909) =

119873

sum

119894minus0

V (119909119894) 119865

119894(119909) 0 le 119894 le 119873 (17)

where 119865119894(119909) is the Lagrange interpolation basis function

associated with 119909119894119873

119894=0

Let minus120583 = 120574 minus 1 In order to obtain high order accuracyof the approximate solution the main difficulty is to computethe integral terms in (7) and (8) In particular for small valuesof 119909 there is little information available for 119906 To overcomethis difficulty we transfer the integration interval to a fixedinterval [minus1 1] by using the following variable changes

120579 =1 + 119909

2] +

119909 minus 1

2≜ 120579 (119909 ])

120578 =119902 (1 + 119909)

2120585 +

119902 (1 + 119909)

2minus 1 ≜ 120578 (119909 120585)

(18)

119863120574

119906 (119909) = 119906 (119909) + 119906 (119902119909 + 119902 minus 1) + 119892 (119909)

+1 + 119909

2int

1

minus1

1198701(119909 120579 (119909 ])) 119906 (120579 (119909 ])) 119889]

+119902 (1 + 119909)

2int

1

minus1

1198702(119909 120578 (119909 120585)) 119906 (120578 (119909 120585)) 119889120585

(19)

119906 (119909) =1

Γ (120574)(1 + 119909

2)

120574

int

1

minus1

(1 minus ])minus120583119863120574

119906

times (120579 (119909 ])) 119889] + 119906 (minus1)

(20)

Set the collocation points as the set of (119873 + 1) Jacobi-Gauss points 119909minus120583minus120583

119894119873

119894=0associated with 120596

minus120583minus120583 Assume that(19) and (20) holds at 119909minus120583minus120583

119894

119863120574

119906 (119909minus120583minus120583

119894) = 119906 (119909

minus120583minus120583

119894) + 119906 (119902119909


119894+ 119902 minus 1)

+ 119892 (119909minus120583minus120583

119894) +

1 + 119909minus120583minus120583

119894

2

times int

1

minus1

1198701(119909


119894 120579 (119909


119894 ]))

times 119906 (120579 (119909minus120583minus120583

119894 ])) 119889]

+

119902 (1 + 119909minus120583minus120583

119894)

2

times int

1

minus1

1198702(119909


119894 120578 (119909


119894 120585))


119894 120585)) 119889120585

(21)

119906 (119909minus120583minus120583

119894) =

1

Γ (120574)(

(1 + 119909minus120583minus120583

119894) 119879

4)

120574

times int

1

minus1

(1 minus ])minus120583119863120574

119906 (120579 (119909minus120583minus120583

119894 ])) 119889]

+ 119906 (minus1)

(22)


Next using a (119873 + 1)-point Gauss quadrature formularelative to the Jacobi weight 12059600

119896119873

119894=0 the integration term in

(21) can be approximated by

int

1

minus1

1198701(119909


120579 (119909minus120583minus120583

119894 ])) 119906 (120579 (119909minus120583minus120583

119894 ])) 119889]

=

119873

sum

119896=0

1198701(119909

119894 120579 (119909

119894 ]119896)) 119906 (120579 (119909

119894 ]119896)) 120596

00

119896

int

1

minus1

1198702(119909


119894 120578 (119909


119894 120585)) 119906 (120578 (119909


119894 120585)) 119889120585

=

119873

sum

119896=0

1198702(119909

119894 120578 (119909

119894 120585119896)) 119906 (120578 (119909

119894 120585119896)) 120596

00

119896

(23)

The sets ]119896119873

119896=0and 120585

119896119873

119896=0coincide with the Jacobi-Gauss

points corresponding Jacobi weights 12059600119896119873

119896=0 that is ]

119896119873

119896=0

and 120585119896119873

119896=0are Legendre-Gauss points

Using a (119873 + 1)-point Gauss quadrature formula relativeto the Jacobi weight 120596minus1205830 the integration term in (22) can beapproximated by

int

1

minus1

(1 minus ])120583119863120574

119906 (120579 (119909minus120583minus120583

119894 ])) 119889120579

=

119873

sum

119896=0

119863120574

119906 (120579 (119909minus120583minus120583

119894 ]119896)) 120596

minus1205830

119896

(24)

where the set ]119896119873

119896=0is the Jacobi-Gauss points correspond-

ing to the weight 120596minus1205830We use 119906

119894 119906120574

119894to approximate the function value 119906(119909minus120583minus120583

119894)

119863120574

119906(119909minus120583minus120583

119894) 0 le 119894 le 119873 and expand 119906 and 119906120574 using Lagrange

interpolation polynomials that is

119880 (119909) =

119873

sum

119895=0

119906119895119865119895(119909) 119880

120574

(119909) =

119873

sum

119895=0

119906120574

119895119865119895(119909) (25)


associated with 119909minus120583minus120583

119894119873

119894=0which is the set of (119873 + 1)

Jacobi-Gauss points The Jacobi collocation methods are toseek 119906

120574

119894119873

119894=0and 119906

119894119873

119894=0such that the following collocation

equations hold

119906120574

119894= 119906

119894+ (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1))

+ 119892 (119909minus120583minus120583

119894)

+1 + 119909


119894

2

119873

sum

119895=0

119906119895(

119873

sum

119896=0

1198701(119909


119894 120579 (119909

119894 ]119896))

times119865119895(120579 (119909


119894 ]119896)) 120596

00

119896)

+

119902 (1 + 119909minus120583minus120583

119894)

2

119873

sum

119895=0

119906119895(

119873

sum

119896=0

1198702(119909


119894 120578 (119909

119894 ]119896))

times 119865119895(120578 (119909


119894 ]119896)) 120596

00

119896)

(26)

119906119894=

1

Γ (120574)(

(1 + 119909minus120583minus120583

119894) 119879

4)

120574

119873

sum

119895=0

119906120574

119895

times (

119873

sum

119896=0

119865119895(119904 (119909


119894 ]119896)) 120596

minus1205830

119896) + 119906 (minus1)

(27)

Writing 119880119873

= (1199060 119906

1 119906

119873)119879 and 119880

120574

119873= (119906

120574

0 119906

120574

1

119906120574

119873)119879 we obtain the following of the matrix form from (26)-

(27)

119880120574

119873= (119864 + 119871 + 119862 + 119863)119880

119873+ 119866

119880119873= 119880

minus1+ 119861119880

120574

119873

(28)

where

119864 is the identity matrix

119866 = (119892 (119909minus120583minus120583

0) 119892 (119909


1) 119892 (119909


119873))

119879

119871119894119895= 119865

119895(119902119909


119894+ 119902 minus 1)

119862119894119895=1 + 119909


119894

2

119873

sum

119896=0

1(119909


119894 120579 (119909


119894 ]119896)) 119865

119895

times (120579 (119909119894 ]119896)) 120596

00

119896

119863119894119895=

119902 (1 + 119909minus120583minus120583

119894)

2

119873

sum

119896=0

2(119909


119894 120578 (119909


119894 ]119896)) 119865

119895

times (120578 (119909minus120583minus120583

119894 ]119896)) 120596

00

119896

119861119894119895=

1

Γ (120574)(

(1 + 119909minus120583minus120583

119894) 119879

4)

120574

times

119873

sum

119896=0

119865119895(119904 (119909


119894 ]119896)) 120596

minus1205830

119896

119880minus1

= (119906 (minus1) 119906 (minus1) 119906 (minus1))119879

(29)

We can get the values of 119906119894119873

119894=0and 119906

120574

119894119873

119894=0by solving the

system of linear system (28) Therefore the expressions of119880(119909) and 119880

120574

(119909) can be obtained

3 Some Useful Lemmas

In this section we will provide some elementary lemmaswhich are important for the derivation of the main results inthe subsequent section Let 119868 = (minus1 1)


Lemma 1 (see [27]) Assume that an (119873 + 1)-point Gaussquadrature formula relative to the Jacobi weight is used tointegrate the product 119906120593 where 119906 isin 119867

119898

(119868)with 119868 for some119898 ge

1 and 120593 isin P119873 Then there exists a constant 119862 independent of

N such that100381610038161003816100381610038161003816100381610038161003816

int

1

minus1

119906 (119909) 120593 (119909) 119889119909 minus (119906 120593)119873

100381610038161003816100381610038161003816100381610038161003816

le 119862119873minus119898

|119906|119867119898119873

120596120572120573(119868)

100381710038171003817100381712059310038171003817100381710038171198712120596120572120573(119868)

(30)

where

|119906|119867119898119873

120596120572120573(119868)

= (

119898

sum

119895=min(119898119873+1)

10038171003817100381710038171003817119906(119895)10038171003817100381710038171003817

2

1198712

120596120572120573(119868)

)

12

(119906 120593)119873=

119873

sum

119895=0

119906 (119909119895) 120593 (119909

119895) 120596

119895

(31)

Lemma 2 (see [26 27]) Assume that 119906 isin 119867119898119873

120596minus120583minus120583

(119868) anddenote by 119868minus120583minus120583

119873119906 its interpolation polynomial associated with

the (119873 + 1) Jacobi-Gauss points 119909119895119873

119895=0 namely

119868minus120583minus120583

119873119906 =

119873

sum

119894=0

119906 (119909119894) 119865 (119909

119894) (32)

Then the following estimates hold

10038171003817100381710038171003817119906 minus 119868


119873119906100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

le 119862119873minus119898

|119906|119867119898119873

120596120572120573(119868) (33a)

10038171003817100381710038171003817119906 minus 119868


11987311990610038171003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus119898

|119906|119867119898119873

120596119888(119868) 0 le 120583 lt

1

2

11986211987312minus119898 log119873|119906|

119867119898119873

120596119888(119868)

1

2le 120583 lt 1

(33b)

where 120596119888 = 120596minus12minus12 denotes the Chebyshev weight function

Lemma 3 (see [28]) Assume that 119865119895(119909)

119873

119895=0are the 119873-th

degree Lagrange basis polynomials associated with the Gausspoints of the Jacobi polynomials Then

100381710038171003817100381710038171003817119868120572120573

119873

100381710038171003817100381710038171003817119871infin(119868)le max

119909isin[minus11]

119873

sum

119895=0

10038161003816100381610038161003816119865119895(119909)

10038161003816100381610038161003816

=

O (log119873) minus1 lt 120572 120573 le minus1

2

O (119873120574+12

) 120574 = max (120572 120573) 119900119905ℎ119890119903119908119894119904119890(34)

Lemma 4 (Gronwall inequality see [29] Lemma 711)Suppose that 119871 ge 0 0 lt 120583 lt 1 and 119906 and V are a nonnegativelocally integrable functions defined on [minus1 1] satisfying

119906 (119909) le V (119909) + 119871int

119909

minus1

(119909 minus 120591)minus120583

119906 (120591) 119889120591 (35)

Then there exists a constant 119862 = 119862(120583) such that

119906 (119909) le V (119909) + 119862119871int

119909

minus1

(119909 minus 120591)minus120583V (120591) 119889120591 119891119900119903 minus 1 le 119909 lt 1

(36)

Lemma 5 (see [30 31]) For a nonnegative integer 119903 and 120581 isin

(0 1) there exists a constant119862119903120581

gt 0 such that for any functionV isin 119862

119903120581

([minus1 1]) there exists a polynomial function T119873V isin

P119873such that

1003817100381710038171003817V minusT119873V1003817100381710038171003817119871infin(119868) le 119862

119903120581119873minus(119903+120581)

V119903120581 (37)

where sdot 119903120581

is the standard norm in 119862119903120581

([minus1 1]) which isdenoted by the space of functions whose 119903th derivatives areHolder continuous with exponent 120581 endowed with the usualnorm

V119903120581

= max0le120581le119903

max119909isin[minus11]

1003816100381610038161003816120597120581

119909V (119909)1003816100381610038161003816

+ max0le120581le119903

sup119909119910isin[minus11]119909 = 119910

1003816100381610038161003816120597120581

119909V (119909) minus 120597

120581

119909V (119909)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

120581

(38)

T119873is a linear operator from 119862

119903120581

([minus1 1]) intoP119873

Lemma 6 (see [32]) Let 120581 isin (0 1) and letM be defined by

(MV) (119909) = int

119909

minus1

(119909 minus 120591)minus120583

119870 (119909 120591) V (120591) 119889120591 (39)

Then for any function V isin 119862([minus1 1]) there exists a positiveconstant 119862 such that

10038161003816100381610038161003816MV (1199091015840) minusMV (11990910158401015840)

1003816100381610038161003816100381610038161003816100381610038161199091015840 minus 11990910158401015840

1003816100381610038161003816

le 119862 max119909isin[minus11]

|V (119909)| (40)

under the assumption that 0 lt 120581 lt 1 minus 120583 for any 1199091015840

11990910158401015840

isin

[minus1 1] and 1199091015840 = 11990910158401015840 This implies that

MV0120581


|V (119909)| 0 lt 120581 lt 1 minus 120583 (41)

Lemma7 (see [33]) For every bounded function V there existsa constant 119862 independent of V such that

sup119873

10038171003817100381710038171003817100381710038171003817100381710038171003817

119873

sum

119895=0

V (119909119895) 119865

119895(119909)

100381710038171003817100381710038171003817100381710038171003817100381710038171198712120596120572120573(119868)


|V (119909)| (42)

where 119865119895(119909) 119895 = 0 1 119873 are the Lagrange interpolation

basis functions associated with the Jacobi collocation points119909

119895119873

119895=0

Lemma 8 (see [34]) For all measurable function 119891 ge 0 thefollowing generalized Hardyrsquos inequality

(int

119887

119886

1003816100381610038161003816(119879119891) (119909)1003816100381610038161003816

119902

119906 (119909) 119889119909)

1119902

le (int

119887

119886

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901

V (119909) 119889119909)1119901

(43)


holds if and only if

sup119886lt119909lt119887

(int

119887

119909

119906 (119905) 119889119905)

1119902

(int

119909

119886

V1minus1199011015840

(119905) 119889119905)

11199011015840

lt infin 1199011015840

=119901

119901 minus 1

(44)

for the case 1 lt 119901 le 119902 lt infin Here 119879 is an operator of the form

(119879119865) (119909) = int

119909

119886

119896 (119909 119905) 119891 (119905) 119889119905 (45)

with 119896(119909 119905) a given kernel 119906 V are nonnegative weightfunctions and minusinfin le 119886 lt 119887 le infin

4 Convergence Analysis

This section is devoted to provide a convergence analysis forthe numerical scheme The goal is to show that the rate ofconvergence is exponential that is the spectral accuracy canbe obtained for the proposed approximations Firstly we willcarry our convergence analysis in 119871

infin space

Theorem 9 Let 119906(119909) be the exact solution of the fractionaldelay-integrodifferential equation (7)-(8) which is assumedto be sufficiently smooth Assume that 119880(119909) and 119880

120574

(119909) areobtained by using the spectral collocation scheme (26)-(27)together with a polynomial interpolation (25) If 120574 associatedwith the weakly singular kernel satisfies 0 lt 120574 lt 1 and119906 isin 119867

119898+1

120596minus120583minus120583(119868) then

1003817100381710038171003817119880120574

minus 119863120574

1199061003817100381710038171003817119871infin(119868)

le

119862119873120574minus12minus119898

(119870lowast

1199061198712(119868)

+ 11987312U)

1

2lt 120574 lt 1

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 11987312U) 0 lt 120574 le

1

2

(46)

119880 minus 119906119871infin(119868)

le

119862119873120574minus12minus119898

(119870lowast

1199061198712(119868)

+ 11987312U)

1

2lt 120574 lt 1

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 11987312U) 0 lt 120574 le

1

2

(47)

provided that 119873 is sufficiently large where 119862 is a constantindependent of 119873 but which will depend on the bounds of thefunctions 119870(119909 119904) and the index 120583

119870lowast

= max119909isin[minus11]

10038161003816100381610038161198701(119909 120579 (119909 sdot))

100381610038161003816100381611989811987312059600(119868)

+ max119909isin[minus11]

10038161003816100381610038161198702(119909 120578 (119909 sdot))

100381610038161003816100381611989811987312059600(119868)

(48)

U =1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868) (49)

Proof We let

(119870 (119904 (119909119894 ])) 120601 (119904 (119909

119894 ])))

119873

=

119873

sum

119896=0

119870(119904 (119909119894 ]119896)) 120601 (119904 (119909

119894 ]119896)) 120596

00

119896

(50)

The numerical scheme (26)-(27) can be written as

119906120574

119894= 119906

119894+ (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1)) + 119892 (119909


119894)

+1 + 119909


119894

2(119870

1(119909


119894 120579 (119909


119894 ]))

119880 (120579 (119909minus120583minus120583

119894 ])))

119873

+

119902 (1 + 119909minus120583minus120583

119894)

2(119870

2(119909


119894 120578 (119909


119894 ]))

119880 (120578 (119909minus120583minus120583

119894 ])))

119873

(51)

119906119894=

1

Γ (120574)(119879

2)

120574

int

119909119894

minus1

(119909119894minus 120579)

minus120583

119880120574

(120579) 119889120579 + 119906minus1 (52)

which gives

119906120574

119894= 119906

119894+ (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1)) + 119892 (119909


119894)

+1 + 119909


119894

2int

1

minus1

1198701(119909


119894 120579 (119909


119894 ]))


119894 ])) 119889] + 119868

1(119909


119894)

+

119902 (1 + 119909minus120583minus120583

119894)

2int

1

minus1

1198702(119909


119894 120578 (119909


119894 120585))


119894 120585)) 119889120585 + 119868

2(119909


119894)

(53)

where

1198681(119909) =

1 + 119909

2int

1

minus1

1198701(119909 120579 (119909)) 119880 (120579 (119909 ])) 119889]

minus1 + 119909

2(119870

1(119909 120579 (119909 ])) 119880 (120579 (119909 ])))

119873

1198682(119909) =

119902 (1 + 119909)

2int

1

minus1

1198702(119909 120578 (119909 120585)) 119880 (120578 (119909 120585)) 119889120585

minus119902 (1 + 119909)

2(119870

2(119909 120578 (119909 ])) 119880 (120578 (119909 ])))

119873

(54)


Using the integration error estimates from Jacobi-Gausspolynomials quadrature in Lemma 1 we have

10038161003816100381610038161198681 (119909)1003816100381610038161003816 le 119862119873

minus119898100381610038161003816100381610038161(119909 120579 (119909 sdot))

1003816100381610038161003816100381611986711989811987312059600(119868)

119880 (120579 (119909 sdot))1198712(119868)

le 119862119873minus119898

119870lowast

1119880

1198712(119868)

(55)

10038161003816100381610038161198682 (119909)1003816100381610038161003816 le 119862119873

minus119898100381610038161003816100381610038162(119909 120578 (119909 sdot))

1003816100381610038161003816100381611986711989811987312059600(119868)

1003817100381710038171003817119880 (120578 (119909 sdot))10038171003817100381710038171198712(119868)

le 119862119873minus119898

119870lowast

2119880

1198712(119868)

(56)

119870lowast

1= max

119909isin[minus11]

10038161003816100381610038161198701(119909 120579 (119909 sdot))

100381610038161003816100381611989811987312059600(119868)

119870lowast

2= max

119909isin[minus11]

10038161003816100381610038161198701(119909 120579 (119909 sdot))

100381610038161003816100381611989811987312059600(119868)

(57)

From (18) (53) can be rewritten as

119906120574

119894= 119886 (119909


119894) 119906

119894+ 119887 (119909


119894)

times (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1)) + 119892 (119909


119894)

+ int

119909minus120583minus120583

119894

minus1

1198701(119909


119894 120579)119880 (120579) 119889120579 + 119868

1(119909


119894)

+ int

119902119909minus120583minus120583

119894+119902minus1

minus1

1198702(119909


119894 120578)119880 (120578) 119889120578

+ 1198682(119909


119894)

(58)

Let 119890 and 119890120574 denote the error function

119890 (119909) = 119880 (119909) minus 119906 (119909) 119890120574

(119909) = 119880120574

(119909) minus 119906120574

(119909)

(59)

we have

119906120574

119894= 119886 (119909


119894) 119906

119894+ 119887 (119909


119894)

times (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1)) + 119892 (119909


119894)

+ int

119909minus120583minus120583

119894

minus1

1198701(119909


119894 120579) 119906 (120579) 119889120579

+ int

119909minus120583minus120583

119894

minus1

1198701(119909


119894 120579) 119890 (120579) 119889120579

+ int

119902119909minus120583minus120583

119894+119902minus1

minus1

1198702(119909


119894 120578) 119906 (120578) 119889120578

+ int

119902119909minus120583minus120583

119894+119902minus1

minus1

1198702(119909


119894 120578) 119890 (120578) 119889120578

+ 1198681(119909


119894) + 119868

2(119909


119894)

(60)

Multiplying119865119894(119909)onboth sides of (62) and (52) summing

up from 0 to119873 and using (7)-(8) yield

119880120574

(119909) = 119880 (119909) + 119868minus120583minus120583

119873(

119873

sum

119895=0

119906119895119865119895(119902119909 + 119902 minus 1))

+ 119868minus120583minus120583

119873119892 (119909)

+ 119868minus120583minus120583

119873[119863

120574

119906 (119909) minus 119906 (119909)

minus119906 (119902 (119909) + 119902 minus 1) minus 119892 (119909)]

+ 119868minus120583minus120583

119873int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

+ 119868minus120583minus120583

119873int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

+ 1198691(119909) + 119869

2(119909)

(61)

119880 (119909) = 119868minus120583minus120583

119873(119906 (119909) minus 119906

minus1)

+ 119868minus120583minus120583

119873(

1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579)

+ 119868minus120583minus120583

119873(119906

minus1)

(62)

where

1198691(119909) = 119868


1198731198681(119909) 119869

2(119909) = 119868


1198731198682(119909) (63)

From (61)-(62) we have

119890120574

(119909) = 119890 (119909) + int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

+ int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

+ 1198691(119909) + 119869

2(119909) + 119869

3(119909) + 119869

4(119909)

+ 1198695(119909) + 119869

6(119909) + 119869

7(119909)

(64)

119890 (119909) =1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

+ 1198694(119909) + 119869

8(119909)

(65)

where

1198693(119909) = 119868


119873119863120574

119906 (119909) minus 119863120574

119906 (119909)

1198694(119909) = 119868


119873119906 (119909) minus 119906 (119909)

1198695(119909) = 119868


119873119906 (119902119909 + 119902 minus 1) minus 119906 (119902119909 + 119902 minus 1)

1198696(119909) = 119868


119873int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

minus int

119909

minus1

119870 (119909 119904) 119890 (119904) 119889119904


1198697(119909) = 119868


119873int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

minus int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

1198698(119909) = 119868


119873(

1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579)

minus1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

(66)

Due to (64) we obtain

119890120574

(119909) =1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

+1

Γ (120574)int

119909

minus1

1198701(119909 120579) int

120579

minus1

(120579 minus 120591)minus120583

119890120574

(120591) 119889120591 119889120579

+1

Γ (120574)int

119902119909+119902minus1

minus1

1198702(119909 120578) int

120578

minus1

(120578 minus 119911)minus120583

times 119890120574

(119911) 119889119911 119889120578 + 119867 (119909)

+ 1198691(119909) + 119869

2(119909) + 119869

3(119909) + 119869

4(119909)

+ 1198695(119909) + 119869

6(119909) + 119869

7(119909)

(67)

where

119867(119909) = int

119909

minus1

1198701(119909 120579) int

120579

minus1

(1198694(120579) + 119869

8(120579)) 119889120579

+1

Γ (120574)int

119902119909+119902minus1

minus1

1198702(119909 120578) int

120578

minus1

(1198694(120578) + 119869

8(120578)) 119889120578

+ 1198694(119909) + 119869

8(119909)

(68)

Using the Dirichletrsquos formula which sates that

int

119909

minus1

int

120579

minus1

Φ (120579 120591) 119889120591 119889120579 = int

119909

minus1

int

119909

120591

Φ (120579 120591) 119889120579 119889120591 (69)

provided that the integral exists we obtain

1

Γ (120574)int

119909

minus1

1198701(119909 120579) int

120579

minus1

(120579 minus 120591)minus120583

119890120574

(120591) 119889120591 119889120579

=1

Γ (120574)int

119909

minus1

(int

119909

120579

1198701(119909 120579) (120579 minus 120591)

minus120583

119889120579) 119890120574

(120591) 119889120591

le1

Γ (120574)21minus120583 max

119909120591isin[minus11]

100381610038161003816100381610038161(119909 120591)

10038161003816100381610038161003816int

119909

minus1

1003816100381610038161003816119890120574

(120591)1003816100381610038161003816 119889120591

≜ 1198721int

119909

minus1

1003816100381610038161003816119890120574

(120591)1003816100381610038161003816 119889120591

(70)

letting 120578 = 119902120591 + 119902 minus 1 119911 = 119902120579 + 119902 minus 1 we have

1

Γ (120574)int

119902119909+119902minus1

minus1

1198702(119909 120578) int

120578

minus1

(120578 minus 119911)minus120583

119890120574

(119911) 119889119911 119889120578

=1

Γ (120574)119902int

119909

minus1

1198702(119909 119902120591 + 119902 minus 1)

times (int

119902120591+119902minus1

minus1

(119909 minus 119911)minus120583

119890120574

(119911) 119889119911) 119889120591

=1

Γ (120574)1199021+120574

int

119909

minus1

1198702(119909 119902120591 + 119902 minus 1)

times (int

120591

minus1

(1 minus 120579)minus120583

119890120574

(119902120579 + 119902 minus 1) 119889120579)119889120591

=1

Γ (120574)1199021+120574

int

119909

minus1

(int

119909

120579

1198702(119909 119902120591 + 119902 minus 1)

times (1 minus 120579)

minus120583

119889120591) 119890120574

times (119902120579 + 119902 minus 1) 119889120579

le1

Γ (120574)21minus120583

1199021+120574 max

119909120591isin[minus11]

10038161003816100381610038161198702(119909 120591)

1003816100381610038161003816

times int

119909

minus1

1003816100381610038161003816119890120574

(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579

≜ 1198722int

119909

minus1

1003816100381610038161003816119890120574

(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579

(71)

Then (67) gives

119890120574

(119909) le1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

+1198721int

119909

minus1

1003816100381610038161003816119890120574

(120591)1003816100381610038161003816 119889120591

+1198722int

119909

minus1

1003816100381610038161003816119890120574

(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579

+ 119867 (119909) + 1198691(119909) + 119869

2(119909) + 119869

3(119909)

+ 1198694(119909) + 119869

5(119909) + 119869

6(119909) + 119869

7(119909)

(72)

It follows from the Gronwall inequality that

1003817100381710038171003817119890120574

(119909)1003817100381710038171003817119871infin(119868)

le

8

sum

119894=1

1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)

(73)

From (65) we have

119890 (119909)119871infin(119868)

le

8

sum

119894=1

1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)

(74)


We now apply Lemma 7 to obtain that

100381710038171003817100381711986911003817100381710038171003817119871infin(119868)

le

119862119873(12)minus120583max

0le119894le119873

100381610038161003816100381611986811989411003816100381610038161003816 0 lt 120583 lt

1

2

119862 log119873max0le119894le119873

100381610038161003816100381611986811989411003816100381610038161003816

1

2le 120583 lt 1

le

119862119873(12)minus120583minus119898

119870lowast

1(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 0 lt 120583 lt1

2

119862119873minus119898 log119873119870

lowast

1(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 1

2le 120583 lt 1

100381710038171003817100381711986921003817100381710038171003817119871infin(119868)

le

119862119873(12)minus120583max

0le119894le119873

100381610038161003816100381611986811989421003816100381610038161003816 0 lt 120583 lt

1

2

119862 log119873max0le119894le119873

100381610038161003816100381611986811989411003816100381610038161003816

1

2le 120583 lt 1

le

119862119873(12)minus120583minus119898

119870lowast

2(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 0 lt 120583 lt1

2

119862119873minus119898 log119873119870

lowast

2(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 1

2le 120583 lt 1

(75)

Due to Lemma 2

100381710038171003817100381711986931003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus1198981003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868) 0 lt 120583 lt

1

2

119862119873(12)minus119898 log1198731003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

1

2le 120583 lt 1

100381710038171003817100381711986941003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus119898

|119906|119867119898119873

120596119888(119868) 0 lt 120583 lt

1

2

119862119873(12)minus119898 log119873|119906|

119867119898119873

120596119888(119868)

1

2le 120583 lt 1

100381710038171003817100381711986951003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus119898

|119906|119867119898119873

120596119888(119868) 0 lt 120583 lt

1

2

119862119873(12)minus119898 log119873|119906|

119867119898119873

120596119888(119868)

1

2le 120583 lt 1

(76)

By virtue of Lemma 2 (33b) with119898 = 1

100381710038171003817100381711986961003817100381710038171003817119871infin(119868)

le

119862119873minus120583

119890119871infin(119868) 0 lt 120583 lt

1

2

119862119873minus(12)

119890119871infin(119868)

1

2le 120583 lt 1

100381710038171003817100381711986971003817100381710038171003817119871infin(119868)

le

119862119873minus120583

119890119871infin(119868) 0 lt 120583 lt

1

2

119862119873minus(12)

119890119871infin(119868)

1

2le 120583 lt 1

(77)

We now estimate the term 1198698(119909) It follows from Lemmas

5 and 6 with 119870(119909 120591) = 1Γ(120574) that

100381710038171003817100381711986951003817100381710038171003817119871infin(119868)

=10038171003817100381710038171003817(119868


119873minus 119868)M119890

12057410038171003817100381710038171003817119871infin(119868)

=10038171003817100381710038171003817(119868


119873minus 119868) (M119890

120574

minusT119873M119890

120574

)10038171003817100381710038171003817119871infin(119868)

le (1 +10038171003817100381710038171003817119868minus120583minus120583

119873

10038171003817100381710038171003817119871infin(119868))119862119873

minus1198961003817100381710038171003817M11989012057410038171003817100381710038170120581

le

119862119873(12)minus120583minus1205811003817100381710038171003817119890

1205741003817100381710038171003817119871infin(119868) 0 lt 120583 lt

1

2

119862119873minus120581 log1198731003817100381710038171003817119890

1205741003817100381710038171003817119871infin(119868)

1

2le 120583 lt 1

(78)

where in the last step we have used Lemma 6 under thefollowing assumption

1

2minus 120583 lt 120581 lt 1 minus 120583 when 0 lt 120583 lt

1

2

0 lt 120581 lt 1 minus 120583 when 1

2le 120583 lt 1

(79)

Provided that119873 is sufficiently large combining (75) (76)(77) and (78) gives1003817100381710038171003817119880

120574

(119909) minus 119906120574

(119909)1003817100381710038171003817119871infin(119868)

le

119862119873(12)minus120583minus119898

(119870lowast

1199061198712(119868)

+ 119873(12)U) 0 lt 120583 lt

1

2

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 119873(12)U)

1

2le 120583 lt 1

119880 (119909) minus 119906 (119909)119871infin(119868)

le

119862119873(12)minus120583minus119898

(119870lowast

1199061198712(119868)

+ 119873(12)U) 0 lt 120583 lt

1

2

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 119873(12)U)

1

2le 120583 lt 1

(80)

Using 120574 = 1 minus 120583 we have the desired estimates (46) and (47)

Next we will give the error estimates in 1198712

120596minus120583minus120583 space

Theorem 10 If the hypotheses given in Theorem 9 hold then1003817100381710038171003817119880

120574

(119909) minus 119906120574

(119909)10038171003817100381710038171198712120596minus120583minus120583

(119868)

le

119862119873minus119898

(1198811+ 119873

120574minus(12)minus120581

1198812+ 119873

120574minus120581U) 1

2lt 120574 lt 1

119862119873minus119898

(1198811+ 119873

minus120581 log1198731198812+ 119873

(12)minus120581 log119873U) 0 lt 120574 le1

2

119880(119909) minus 119906(119909)1198712

120596minus120583minus120583

(119868)

le

119862119873minus119898

(1198811+ 119873

120574minus(12)minus120581

1198812+ 119873

120574minus120581U) 1

2lt 120574 lt 1

119862119873minus119898

(1198811+ 119873

minus120581 log1198731198812+ 119873

(12)minus120581 log119873U) 0 lt 120574 le1

2

(81)


for any 120581 isin (0 120574) provided that119873 is sufficiently large and 119862 isa constant independent of119873 where

1198811= 119870

lowast

(1199061198712(119868)

+1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868))

1198812= 119870

lowast

1199061198712(119868)

(82)

Proof Byusing the generalization ofGronwalls Lemma 4 andthe Hardy inequality Lemma 8 it follows from (72) that

100381710038171003817100381711989012057410038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862

8

sum

119894=1

100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583

(119868)

1198901198712

120596minus120583minus120583

(119868)le 119862

8

sum

119894=1

100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583

(119868)

(83)

Now using Lemma 7 we have10038171003817100381710038171198691

10038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862 max

119909isin[minus11]

|119868 (119909)|

le 119862119873minus119898

119870lowast

1(119906

1198712(119868)

+ 119890119871infin(119868))

1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862 max

119909isin[minus11]

|119868 (119909)|

le 119862119873minus119898

119870lowast

2(119906

1198712(119868)

+ 119890119871infin(119868))

(84)

By the convergence result in Theorem 9 (119898 = 1) we have

119890119871infin(119868)

le 119862(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868)) (85)

So that10038171003817100381710038171198691

10038171003817100381710038171198712120596minus120583minus120583

(119868)

le 119862119873minus119898

119870lowast

1(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868))

1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583

(119868)

le 119862119873minus119898

119870lowast

2(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868))

(86)

Due to Lemma 2 (33a)10038171003817100381710038171198693

10038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1198981003816100381610038161003816119863120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

1003817100381710038171003817119869410038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus119898

|119906|119867119898119873

120596119888(119868)

1003817100381710038171003817119869510038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus119898

|119906|119867119898119873

120596119888(119868)

(87)

By virtue of Lemma 2 (33a) with119898 = 1

1003817100381710038171003817119869610038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1

10038161003816100381610038161003816100381610038161003816

int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583

(119868)

le 119862119873minus1

1198901198712

120596minus120583minus120583

(119868)

1003817100381710038171003817119869710038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1

100381610038161003816100381610038161003816100381610038161003816

int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

1003816100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583

(119868)

le 119862119873minus1

1198901198712

120596minus120583minus120583

(119868)

(88)

Finally it follows from Lemmas 5 and 7 that

1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583

(119868)=10038171003817100381710038171003817(119868


119873minus 119868)M119890

120574100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

=10038171003817100381710038171003817(119868


119873minus 119868) (M119890

120574

minusT119873119890120574

)100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

le10038171003817100381710038171003817119868minus120583minus120583

119873(M119890

120574

minusT119873119890120574

)100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

+1003817100381710038171003817M119890

120574

minusT11987311989012057410038171003817100381710038171198712120596minus120583minus120583

(119868)

le 1198621003817100381710038171003817M119890

120574

minusT1198731198901205741003817100381710038171003817119871infin(119868)

le 119862119873minus1205811003817100381710038171003817M119890

12057410038171003817100381710038170120581

le 119862119873minus1205811003817100381710038171003817119890

1205741003817100381710038171003817119871infin(119868)

(89)

where in the last step we used Lemma 6 for any 120581 isin (0 1minus120583)By the convergence result in Theorem 9 we obtain that

1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583

(119868)

le

11986211987312minus120583minus119898minus120581

(119870lowast

1199061198712(119868)

+ 11987312U) 0 lt 120583 lt

1

2

119862119873minus119898minus120581 log119873(119870

lowast

1199061198712(119868)

+ 11987312U)

1

2le 120583 lt 1

(90)

for119873 sufficiently large and for any 120581 isin (0 1 minus 120583) The desiredestimates (81) follows from the previous estimates and (83)with 120574 = 1 minus 120583

5 Algorithm Implementation andNumerical Experiments

In this subsection we present the numerical results obtainedby the proposed collocation spectral method The estimatesin Section 4 indicates that the convergence of numericalsolutions is exponential if the exact solution is smooth Toconfirm the theoretical prediction a numerical experimentis carried out by considering the following example

Example 1 Consider the following fractional integrodifferen-tial equation

119863075

119910 (119905) =6119905225

Γ (325)minus1199052

119890119905

5119910 (119905) minus

(119902119905)2

5119910 (119902119905)

+ int

119905

0

119890119905

119904119910 (119904) 119889119904 + int

119902119905

0

120591119910 (120591) 119889120591 119905 isin [0 1]

119910 (0) = 0

(91)

The corresponding exact solution is given by 119910(119905) = 1199053

Figure 1 presents the approximate and exact solutionson the left-hand side and presents the approximate and


0 02 04 06 08 10

02

04

06

08

1

12

14

Exact solutionApproximate solution

(a)

0 02 04 06 08 10

05

1

15

2

25


(b)

Figure 1 Example 1 comparison between approximate solution and exact solution 119910(119905) (a) and approximate fraction derivative and exactderivative119863075

119910(119905) (b)

2 4 6 8 10 12 14 16

100

10minus5

10minus10

2 le N le 16

Linfin

L2

120596minus120583minus120583

-error-error

(a)

2 4 6 8 10 12 14 16

100

10minus5

10minus10

2 le N le 16

Linfin

L2

120596minus120583minus120583

-error-error

(b)

Figure 2 Example 1 the errors of numerical and exact solution 119910(119905) (a) and the errors of numerical and exact solution 119863075

119910(119905) (b) versusthe number of collocation points in 119871

infin and 1198712

120596norms

exact derivatives on the right-hand side which are foundin excellent agreement In Figure 2 the numerical errors areplotted for 2 le 119873 le 20 in both 119871

infin and 1198712

120596minus120583minus120583 norms As

expected an exponential rate of convergence is observed forthe problem which confirmed our theoretical predictions

Example 2 Consider the following fractional integrodiffer-ential equation

119863120572

119910 (119905) = 2119905 minus 119910 (119905) + 119910 (119902119905) + 119905 (1 + 2119905) int

119905

0

119890119904(119905minus119904)

119910 (119904) 119889119904

+ int

119902119905

0

119902119905119890120591(119902119905minus120591)

119910 (120591) 119889120591 119905 isin [0 1]

119910 (0) = 1

(92)

when 120572 = 1 the exact solution of (92) is 119910(119905) = 1198901199052


0 02 04 06 08 105

1

15

2

25

3

35

4

Approximate solution 120572 = 025Approximate solution 120572 = 05Approximate solution 120572 = 075Exact solution 120572 = 1Approximate solution 120572 = 1

(a)

0 02 04 06 08 10

1

2

3

4

5

6

Approximate derivativeExact derivative

(b)

Figure 3 Example 2 approximation solutions with different 120572 and exact solution of 119910(119905) with 120572 = 1 (a) Comparison between approximatesolution and exact solution of 1199101015840(119905)

2 4 6 8 10 12 14 16 18 20

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

2 le N le 20

Linfin

L2

120596

(a)

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

2 4 6 8 10 12 14 16 18 202 le N le 20

Linfin

L2

120596

(b)


(119905) (b) versus thenumber of collocation points in 119871

infin and 1198712

120596norms

In the only case of 120572 = 1 we know the exact solutionWe have reported the obtained numerical results for119873 = 20

and 120572 = 025 05 075 and 1 in Figure 3 We can see thatas 120572 approaches 1 the numerical solutions converge to theanalytical solution 119910(119905) = 119890

1199052

that is in the limit the solutionof fractional integrodifferential equations approach to that ofthe integer order integrodifferential equations In Figure 4we plot the resulting errors versus the number 119873 of the

steps This figure shows the exponential rate of convergencepredicted by the proposed method

6 Conclusions and Future Work

This paper proposes a spectral Jacobi-collocation approxi-mation for fractional order integrodifferential equations ofVolterra type with pantograph delay The most important


contribution of this work is that we are able to demonstraterigorously that the errors of spectral approximations decayexponentially in both infinity and weighted norms which isa desired feature for a spectral method

We only investigated the case of pantograph delay inthe present work with the availability of this methodologyit will be possible to generalize this algorithm to solve thesame problem in semi-infinite interval based on generalizedLaguerre [35] and modified generalized Laguerre polynomi-als

Acknowledgments

The authors are grateful to the referees for many usefulsuggestions The work was supported by NSFC Project(11301446 11031006) China Postdoctoral Science Founda-tion Grant (2013M531789) Project of Scientific ResearchFund of Hunan Provincial Science and Technology Depart-ment (2013RS4057) and the Research Foundation of HunanProvincial Education Department (13B116)

References

[1] J HHe ldquoNonlinear oscillationwith fractional derivative and itsCahpinpalicationsrdquo in Proceedings of the International Confer-ence on Vibrating Engineering pp 288ndash291 Dalian China 1998

[2] J H He ldquoSome applications of nonlinear fractional differentialequations and therir approximationsrdquo Bulletin of Science Tech-nology and Society vol 15 pp 86ndash90 1999

[3] I Podlubny Fractional Differential Equations Academic PressNewYork NY USA 1999

[4] F Mainardi Fractional Calculus Continuum MechanicsSpringer Berlin Germany 1997

[5] W M Ahmad and R El-Khazali ldquoFractional-order dynamicalmodels of loverdquo Chaos Solitons and Fractals vol 33 no 4 pp1367ndash1375 2007

[6] F Huang and F Liu ldquoThe time fractional diffusion equation andthe advection-dispersion equationrdquoThe ANZIAM Journal vol46 no 3 pp 317ndash330 2005

[7] Y Luchko and R Gorenflo The Initial Value Problem for SomeFractional Differential Equations with the Caputo DerivativesPreprint series A08-98 Fachbreich Mathematik und Infor-matik Freic Universitat Berlin Germany 1998

[8] N T Shawagfeh ldquoAnalytical approximate solutions for nonlin-ear fractional differential equationsrdquo Applied Mathematics andComputation vol 131 no 2-3 pp 517ndash529 2002

[9] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Yverdon Switzerland 1993

[10] O P Agrawal and P Kumar ldquoComparison of five numericalschemes for fractional differential equationsrdquo in Advances inFractional Calculus Theoretical Developments and Applicationsin Physics and Engineering J Sabatier et al Ed pp 43ndash60Springer Berlin Germany 2007

[11] M M Khader and A S Hendy ldquoThe approximate and exactsolutions of the fractional-order delay differential equationsusing Legendre seudospectral methodrdquo International Journal ofPure and Applied Mathematics vol 74 no 3 pp 287ndash297 2012

[12] A Saadatmandi and M Dehghan ldquoA Legendre collocationmethod for fractional integro-differential equationsrdquo Journal ofVibration and Control vol 17 no 13 pp 2050ndash2058 2011

[13] E A Rawashdeh ldquoLegendre wavelets method for fractionalintegro-differential equationsrdquo Applied Mathematical Sciencesvol 5 no 2 pp 2467ndash2474 2011

[14] M Rehman and R A Khan ldquoThe Legendre wavelet methodfor solving fractional differential equationsrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 16 no 11pp 4163ndash4173 2011

[15] A H Bhrawy and M A Alghamdi ldquoA shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractionalLangevin equation involving two fractional orders in differentintervalsrdquoBoundaryValue Problems vol 1 no 62 pp 1ndash13 2012

[16] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoEfficientChebyshev spectral methods for solving multi-term fractionalorders differential equationsrdquo Applied Mathematical Modellingvol 35 no 12 pp 5662ndash5672 2011

[17] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA Chebyshevspectral method based on operational matrix for initial andboundary value problems of fractional orderrdquo Computers ampMathematics with Applications vol 62 no 5 pp 2364ndash23732011

[18] N H Sweilam and M M Khader ldquoA Chebyshev pseudo-spectral method for solving fractional-order integro-differen-tial equationsrdquoTheANZIAM Journal vol 51 no 4 pp 464ndash4752010

[19] A H Bhrawy A S Alofi and S S Ezz-Eldien ldquoA quadraturetau method for fractional differential equations with variablecoefficientsrdquo Applied Mathematics Letters vol 24 no 12 pp2146ndash2152 2011

[20] A H Bhrawy andM Alshomrani ldquoA shifted Legendre spectralmethod for fractional-order multi-point boundary value prob-lemsrdquoAdvances inDifference Equations vol 2012 article 8 2012

[21] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010

[22] A H Bhrawy M MTharwat and A Yildirim ldquoA new formulafor fractional integrals of Chebyshev polynomials applicationfor solvingmulti-term fractional differential equationsrdquoAppliedMathematical Modelling vol 37 no 6 pp 4245ndash4252 2013

[23] A H Bhrawy and A S Alofi ldquoThe operational matrix of frac-tional integration for shifted Chebyshev polynomialsrdquo AppliedMathematics Letters vol 26 no 1 pp 25ndash31 2013

[24] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA newJacobi operational matrix an application for solving fractionaldifferential equationsrdquoAppliedMathematical Modelling vol 36no 10 pp 4931ndash4943 2012

[25] Y Chen and T Tang ldquoConvergence analysis of the Jacobispectral-collocation methods for Volterra integral equationswith a weakly singular kernelrdquo Mathematics of Computationvol 79 no 269 pp 147ndash167 2010

[26] YWei andY Chen ldquoConvergence analysis of the spectralmeth-ods for weakly singular Volterra integro-differential equationswith smooth solutionsrdquo Advances in Applied Mathematics andMechanics vol 4 no 1 pp 1ndash20 2012

[27] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Fundamentals in Single Domains SpringerBerlin Germany 2006


[28] G Mastroianni and D Occorsio ldquoOptimal systems of nodes forLagrange interpolation on bounded intervals A surveyrdquo Journalof Computational and Applied Mathematics vol 134 no 1-2 pp325ndash341 2001

[29] D Henry Geometric Theory of Semilinear Parabolic EquationsSpringer Berlin Germany 1989

[30] D L Ragozin ldquoPolynomial approximation on compact mani-folds and homogeneous spacesrdquo Transactions of the AmericanMathematical Society vol 150 pp 41ndash53 1970

[31] D L Ragozin ldquoConstructive polynomial approximation onspheres and projective spacesrdquo Transactions of the AmericanMathematical Society vol 162 pp 157ndash170 1971

[32] D Colton and R Kress Inverse Coustic and ElectromagneticScattering Theory Applied Mathematical Sciences SpringerBerlin Germany 2nd edition 1998

[33] P Nevai ldquoMean convergence of Lagrange interpolation IIIrdquoTransactions of the AmericanMathematical Society vol 282 no2 pp 669ndash698 1984

[34] A Kufner and L E Persson Weighted Inequalities of HardyType World Scientific New York NY USA 2003

[35] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficientgeneralized Laguerre spectral algorithms for fractional initialvalue problemsrdquoAbstract andAppliedAnalysis vol 2013 ArticleID 546502 10 pages 2013

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we note that

119868120574

(119863120574

119910 (119905)) = 119910 (119905) minus

119899minus1

sum

119896=0

119910(119896)

(0)119905119896

119896 (4)

From (4) fractional integrodifferential equation (1) can bedescribed as

119863120574

119910 (119905) = 119910 (119905) + 119910 (119902119905) + 119892 (119905) + int

119905

0

1(119905 119904) 119910 (119904) 119889119904

+ int

119902119905

0

2(119905 120591) 119910 (120591) 119889120591

0 lt 120574 le 1 119905 isin 119868 = [0 119879]

119910 (119905) =1

Γ (120574)int

119905

0

(119905 minus 120591)120574minus1

119863120574

119910 (120591) 119889120591 + 119910 (0)

(5)

Several analytical methods have been introduced to solveFDEs including various transformation techniques [6] oper-ational calculus methods [7] the Adomian decompositionmethod [8] and the iterative and series-based methods [9]A small number of algorithms for the numerical solutionof FDEs have been suggested [10] and most of them arefinite difference methods which are generally limited to lowdimensions and are of limited accuracy

As we know fractional derivatives are global (they aredefined by an integral over the whole interval [0 119879]) andtherefore global methods such as spectral methods areperhaps better suited for FDEs Standard spectral methodspossess an infinite order of accuracy for the equations withregular solutions while failing for many complicated prob-lems with singular solutions So it is relevant to be interestedin how to enlarge the adaptability of spectral methods andconstruct certain simple approximation schemes without aloss of accuracy for more complicated problems

Spectral methods have been proposed to solve frac-tional differential equations such as the Legendre collocationmethod [11 12] Legendre wavelets method [13 14] andJacobi-Gauss-Lobatto collocation method [15] The authorsin [16ndash18] constructed an efficient spectral method for thenumerical approximation of fractional integrodifferentialequations based on tau and pseudospectral methods More-over Bhrawy et al [19] introduced a quadrature shiftedLegendre tau method based on the Gauss-Lobatto interpo-lation for solving multiorder FDEs with variable coefficientsand in [20] shifted Legendre spectral methods have beendeveloped for solving fractional-order multipoint boundaryvalue problems In [21] truncated Legendre series togetherwith the Legendre operationalmatrix of fractional derivativesare used for the numerical integration of fractional differ-ential equations In [22] the authors derived a new explicitformula for the integral of shifted Chebyshev polynomials ofany degree for any fractional-order The shifted Chebyshevoperational matrix [23] and shifted Jacobi operational matrix[24] of fractional derivatives have been developed which areapplied together with the spectral tau method for numericalsolution of general linear and nonlinear multiterm fractionaldifferential equations However very few theoretical results

were provided to justify the high accuracy numericallyobtained Recently Chen and Tang [25 26] developed anovel spectral Jacobi-collocation method to solve secondkindVolterra integral equationswith aweakly singular kerneland provided a rigorous error analysis which theoreticallyjustifies the spectral rate of convergence Inspired by thework of [26] we extend the approach to fractional orderdelay-integrodifferential equations (1) However it is difficultto apply the spectral approximations to the initial valueproblem and fractional order derivatives To facilitate the useof the spectral methods we restate the initial condition as anequivalent integral equation with singular kernel Then weget the discrete scheme by using Gauss quadrature formulaIn this paper we will provide a rigorous error analysis notonly for approximate solutions but also for approximatefractional derivatives which theoretically justifies the spectralrate of convergence

For ease of analysis we will describe the spectral methodson the standard interval 119868 = [minus1 1] Hence we employ thetransformation

119905 =119879

2(1 + 119909) 119904 =

119879 (1 + 120579)

2 120591 =

119879 (1 + 120578)

2

(6)

then the previous problem (5) becomes

119863120574

119906 (119909) = 119906 (119909) + 119906 (119902119909 + 119902 minus 1) + 119892 (119909)

+ int

119909

minus1

1198701(119909 120579) 119906 (120579) 119889120579

+ int

119902119909+119902minus1

minus1

1198702(119909 120578) 119906 (120578) 119889120578 119909 isin 119868

(7)

119906 (119909) =1

Γ (120574)(119879

2)

120574

int

119909

minus1

(119909 minus 119904)120574minus1

119863120574

119906 (119904) 119889119904 + 119906 (minus1)

(8)

where

119906 (119909) = 119910 (119879

2(1 + 119909)) 119863

120574

119906 (119909) = 119863120574

119910(119879

2(1 + 119909))

119892 (119909) =119879

2119892 (

119879

2(1 + 119909))

(9)

1198701(119909 120579) =

119879

41(119879

2(1 + 119909)

119879

2(1 + 120579))

1198702(119909 120578) =

119879

42(119879

2(1 + 119909)

119879

2(1 + 120578))

(10)

This paper is organized as follows In Section 2 we intro-duce the spectral approaches for pantograph FDIDEs Someuseful lemmas are provided in Section 3 These lemmas willplay a key role in the derivation of the convergence analysisWe provide a rigorous error analysis for the spectralmethodswhich shows that both the errors of approximate solutionsand the errors of approximate fractional derivatives of thesolutions decay exponentially in 119871

infin norm and weighted 1198712-

norm in Section 4 and Section 5 contains numerical results




119895

119895 = 1 2


Let 120596120572120573(119909) = (1 minus 119909)120572



119899(119909)

infin




1198712


120596120572120573 lt infin

(11)


V120596120572120573 = (int

1

minus1

|V (119909)|2120596120572120573 (119909) 119889119909)12

(12)


(119906 V)120596120572120573 = int

1

minus1

119906 (119909) V (119909) 120596120572120573 (119909) 119889119909 forall119906 V isin 1198712

120596120572120573 (minus1 1)

(13)



points and by 120596119896119873



119896119873



int

1

minus1

119891 (119909) 119889119909 asymp

119873

sum

119896=0

119891 (120579119896) 120596

119896 (14)



120596120572120573

119896119873



int

1

minus1

119891 (119909) 120596120572120573

(119909) 119889119909 asymp

119873

sum

119896=0

119891 (120579119896) 120596

120572120573

119896 (15)


119894119873




119873V isin P

119873 satisfying

119868120572120573

119873V (119909

119894) = V (119909

119894) 0 le 119894 le 119873 (16)


119868120572120573

119873V (119909) =

119873

sum

119894minus0

V (119909119894) 119865

119894(119909) 0 le 119894 le 119873 (17)



119894=0


120579 =1 + 119909

2] +

119909 minus 1

2≜ 120579 (119909 ])

120578 =119902 (1 + 119909)

2120585 +

119902 (1 + 119909)

2minus 1 ≜ 120578 (119909 120585)

(18)

119863120574

119906 (119909) = 119906 (119909) + 119906 (119902119909 + 119902 minus 1) + 119892 (119909)

+1 + 119909

2int

1

minus1

1198701(119909 120579 (119909 ])) 119906 (120579 (119909 ])) 119889]

+119902 (1 + 119909)

2int

1

minus1

1198702(119909 120578 (119909 120585)) 119906 (120578 (119909 120585)) 119889120585

(19)

119906 (119909) =1

Γ (120574)(1 + 119909

2)

120574

int

1

minus1

(1 minus ])minus120583119863120574

119906

times (120579 (119909 ])) 119889] + 119906 (minus1)

(20)


119894119873



119894

119863120574

119906 (119909minus120583minus120583

119894) = 119906 (119909


119894) + 119906 (119902119909


119894+ 119902 minus 1)

+ 119892 (119909minus120583minus120583

119894) +

1 + 119909minus120583minus120583

119894

2

times int

1

minus1

1198701(119909


119894 120579 (119909


119894 ]))


119894 ])) 119889]

+

119902 (1 + 119909minus120583minus120583

119894)

2

times int

1

minus1

1198702(119909


119894 120578 (119909


119894 120585))


119894 120585)) 119889120585

(21)

119906 (119909minus120583minus120583

119894) =

1

Γ (120574)(

(1 + 119909minus120583minus120583

119894) 119879

4)

120574

times int

1

minus1

(1 minus ])minus120583119863120574

119906 (120579 (119909minus120583minus120583

119894 ])) 119889]

+ 119906 (minus1)

(22)



119896119873



int

1

minus1

1198701(119909


120579 (119909minus120583minus120583

119894 ])) 119906 (120579 (119909minus120583minus120583

119894 ])) 119889]

=

119873

sum

119896=0

1198701(119909

119894 120579 (119909

119894 ]119896)) 119906 (120579 (119909

119894 ]119896)) 120596

00

119896

int

1

minus1

1198702(119909


119894 120578 (119909


119894 120585)) 119906 (120578 (119909


119894 120585)) 119889120585

=

119873

sum

119896=0

1198702(119909

119894 120578 (119909

119894 120585119896)) 119906 (120578 (119909

119894 120585119896)) 120596

00

119896

(23)

The sets ]119896119873

119896=0and 120585

119896119873



119896=0 that is ]

119896119873

119896=0

and 120585119896119873



int

1

minus1

(1 minus ])120583119863120574

119906 (120579 (119909minus120583minus120583

119894 ])) 119889120579

=

119873

sum

119896=0

119863120574

119906 (120579 (119909minus120583minus120583

119894 ]119896)) 120596

minus1205830

119896

(24)




119894 119906120574


119894)

119863120574

119906(119909minus120583minus120583



119880 (119909) =

119873

sum

119895=0

119906119895119865119895(119909) 119880

120574

(119909) =

119873

sum

119895=0

119906120574

119895119865119895(119909) (25)



119894119873



120574

119894119873

119894=0and 119906

119894119873


equations hold

119906120574

119894= 119906

119894+ (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1))

+ 119892 (119909minus120583minus120583

119894)

+1 + 119909


119894

2

119873

sum

119895=0

119906119895(

119873

sum

119896=0

1198701(119909


119894 120579 (119909

119894 ]119896))

times119865119895(120579 (119909


119894 ]119896)) 120596

00

119896)

+

119902 (1 + 119909minus120583minus120583

119894)

2

119873

sum

119895=0

119906119895(

119873

sum

119896=0

1198702(119909


119894 120578 (119909

119894 ]119896))

times 119865119895(120578 (119909


119894 ]119896)) 120596

00

119896)

(26)

119906119894=

1

Γ (120574)(

(1 + 119909minus120583minus120583

119894) 119879

4)

120574

119873

sum

119895=0

119906120574

119895

times (

119873

sum

119896=0

119865119895(119904 (119909


119894 ]119896)) 120596

minus1205830

119896) + 119906 (minus1)

(27)

Writing 119880119873

= (1199060 119906

1 119906

119873)119879 and 119880

120574

119873= (119906

120574

0 119906

120574

1

119906120574


(27)

119880120574

119873= (119864 + 119871 + 119862 + 119863)119880

119873+ 119866

119880119873= 119880

minus1+ 119861119880

120574

119873

(28)

where


119866 = (119892 (119909minus120583minus120583

0) 119892 (119909


1) 119892 (119909


119873))

119879

119871119894119895= 119865

119895(119902119909


119894+ 119902 minus 1)

119862119894119895=1 + 119909


119894

2

119873

sum

119896=0

1(119909


119894 120579 (119909


119894 ]119896)) 119865

119895

times (120579 (119909119894 ]119896)) 120596

00

119896

119863119894119895=

119902 (1 + 119909minus120583minus120583

119894)

2

119873

sum

119896=0

2(119909


119894 120578 (119909


119894 ]119896)) 119865

119895


119894 ]119896)) 120596

00

119896

119861119894119895=

1

Γ (120574)(

(1 + 119909minus120583minus120583

119894) 119879

4)

120574

times

119873

sum

119896=0

119865119895(119904 (119909


119894 ]119896)) 120596

minus1205830

119896

119880minus1


(29)


119894=0and 119906

120574

119894119873



120574






119898

(119868)with 119868 for some119898 ge


N such that100381610038161003816100381610038161003816100381610038161003816

int

1

minus1

119906 (119909) 120593 (119909) 119889119909 minus (119906 120593)119873

100381610038161003816100381610038161003816100381610038161003816

le 119862119873minus119898

|119906|119867119898119873

120596120572120573(119868)

100381710038171003817100381712059310038171003817100381710038171198712120596120572120573(119868)

(30)

where

|119906|119867119898119873

120596120572120573(119868)

= (

119898

sum

119895=min(119898119873+1)

10038171003817100381710038171003817119906(119895)10038171003817100381710038171003817

2

1198712

120596120572120573(119868)

)

12

(119906 120593)119873=

119873

sum

119895=0

119906 (119909119895) 120593 (119909

119895) 120596

119895

(31)


120596minus120583minus120583




119895=0 namely

119868minus120583minus120583

119873119906 =

119873

sum

119894=0

119906 (119909119894) 119865 (119909

119894) (32)


10038171003817100381710038171003817119906 minus 119868


119873119906100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

le 119862119873minus119898

|119906|119867119898119873

120596120572120573(119868) (33a)

10038171003817100381710038171003817119906 minus 119868


11987311990610038171003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus119898

|119906|119867119898119873

120596119888(119868) 0 le 120583 lt

1

2

11986211987312minus119898 log119873|119906|

119867119898119873

120596119888(119868)

1

2le 120583 lt 1

(33b)



119873

119895=0are the 119873-th


100381710038171003817100381710038171003817119868120572120573

119873

100381710038171003817100381710038171003817119871infin(119868)le max

119909isin[minus11]

119873

sum

119895=0

10038161003816100381610038161003816119865119895(119909)

10038161003816100381610038161003816

=


2

O (119873120574+12

) 120574 = max (120572 120573) 119900119905ℎ119890119903119908119894119904119890(34)


119906 (119909) le V (119909) + 119871int

119909

minus1

(119909 minus 120591)minus120583

119906 (120591) 119889120591 (35)


119906 (119909) le V (119909) + 119862119871int

119909

minus1


(36)




119903120581


P119873such that


119903120581119873minus(119903+120581)

V119903120581 (37)

where sdot 119903120581



V119903120581

= max0le120581le119903


1003816100381610038161003816120597120581

119909V (119909)1003816100381610038161003816

+ max0le120581le119903

sup119909119910isin[minus11]119909 = 119910

1003816100381610038161003816120597120581

119909V (119909) minus 120597

120581

119909V (119909)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

120581

(38)


119903120581



(MV) (119909) = int

119909

minus1

(119909 minus 120591)minus120583

119870 (119909 120591) V (120591) 119889120591 (39)


10038161003816100381610038161003816MV (1199091015840) minusMV (11990910158401015840)

1003816100381610038161003816100381610038161003816100381610038161199091015840 minus 11990910158401015840

1003816100381610038161003816


|V (119909)| (40)


11990910158401015840

isin


MV0120581


|V (119909)| 0 lt 120581 lt 1 minus 120583 (41)


sup119873

10038171003817100381710038171003817100381710038171003817100381710038171003817

119873

sum

119895=0

V (119909119895) 119865

119895(119909)

100381710038171003817100381710038171003817100381710038171003817100381710038171198712120596120572120573(119868)


|V (119909)| (42)



119895119873

119895=0


(int

119887

119886

1003816100381610038161003816(119879119891) (119909)1003816100381610038161003816

119902

119906 (119909) 119889119909)

1119902

le (int

119887

119886

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901

V (119909) 119889119909)1119901

(43)



sup119886lt119909lt119887

(int

119887

119909

119906 (119905) 119889119905)

1119902

(int

119909

119886

V1minus1199011015840

(119905) 119889119905)

11199011015840

lt infin 1199011015840

=119901

119901 minus 1

(44)


(119879119865) (119909) = int

119909

119886

119896 (119909 119905) 119891 (119905) 119889119905 (45)




infin space


120574


119898+1

120596minus120583minus120583(119868) then

1003817100381710038171003817119880120574

minus 119863120574

1199061003817100381710038171003817119871infin(119868)

le

119862119873120574minus12minus119898

(119870lowast

1199061198712(119868)

+ 11987312U)

1

2lt 120574 lt 1

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 11987312U) 0 lt 120574 le

1

2

(46)

119880 minus 119906119871infin(119868)

le

119862119873120574minus12minus119898

(119870lowast

1199061198712(119868)

+ 11987312U)

1

2lt 120574 lt 1

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 11987312U) 0 lt 120574 le

1

2

(47)


119870lowast


10038161003816100381610038161198701(119909 120579 (119909 sdot))

100381610038161003816100381611989811987312059600(119868)


10038161003816100381610038161198702(119909 120578 (119909 sdot))

100381610038161003816100381611989811987312059600(119868)

(48)

U =1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868) (49)

Proof We let

(119870 (119904 (119909119894 ])) 120601 (119904 (119909

119894 ])))

119873

=

119873

sum

119896=0

119870(119904 (119909119894 ]119896)) 120601 (119904 (119909

119894 ]119896)) 120596

00

119896

(50)


119906120574

119894= 119906

119894+ (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1)) + 119892 (119909


119894)

+1 + 119909


119894

2(119870

1(119909


119894 120579 (119909


119894 ]))

119880 (120579 (119909minus120583minus120583

119894 ])))

119873

+

119902 (1 + 119909minus120583minus120583

119894)

2(119870

2(119909


119894 120578 (119909


119894 ]))

119880 (120578 (119909minus120583minus120583

119894 ])))

119873

(51)

119906119894=

1

Γ (120574)(119879

2)

120574

int

119909119894

minus1

(119909119894minus 120579)

minus120583

119880120574

(120579) 119889120579 + 119906minus1 (52)

which gives

119906120574

119894= 119906

119894+ (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1)) + 119892 (119909


119894)

+1 + 119909


119894

2int

1

minus1

1198701(119909


119894 120579 (119909


119894 ]))


119894 ])) 119889] + 119868

1(119909


119894)

+

119902 (1 + 119909minus120583minus120583

119894)

2int

1

minus1

1198702(119909


119894 120578 (119909


119894 120585))


119894 120585)) 119889120585 + 119868

2(119909


119894)

(53)

where

1198681(119909) =

1 + 119909

2int

1

minus1

1198701(119909 120579 (119909)) 119880 (120579 (119909 ])) 119889]

minus1 + 119909

2(119870

1(119909 120579 (119909 ])) 119880 (120579 (119909 ])))

119873

1198682(119909) =

119902 (1 + 119909)

2int

1

minus1

1198702(119909 120578 (119909 120585)) 119880 (120578 (119909 120585)) 119889120585

minus119902 (1 + 119909)

2(119870

2(119909 120578 (119909 ])) 119880 (120578 (119909 ])))

119873

(54)



10038161003816100381610038161198681 (119909)1003816100381610038161003816 le 119862119873

minus119898100381610038161003816100381610038161(119909 120579 (119909 sdot))

1003816100381610038161003816100381611986711989811987312059600(119868)

119880 (120579 (119909 sdot))1198712(119868)

le 119862119873minus119898

119870lowast

1119880

1198712(119868)

(55)

10038161003816100381610038161198682 (119909)1003816100381610038161003816 le 119862119873

minus119898100381610038161003816100381610038162(119909 120578 (119909 sdot))

1003816100381610038161003816100381611986711989811987312059600(119868)

1003817100381710038171003817119880 (120578 (119909 sdot))10038171003817100381710038171198712(119868)

le 119862119873minus119898

119870lowast

2119880

1198712(119868)

(56)

119870lowast

1= max

119909isin[minus11]

10038161003816100381610038161198701(119909 120579 (119909 sdot))

100381610038161003816100381611989811987312059600(119868)

119870lowast

2= max

119909isin[minus11]

10038161003816100381610038161198701(119909 120579 (119909 sdot))

100381610038161003816100381611989811987312059600(119868)

(57)


119906120574

119894= 119886 (119909


119894) 119906

119894+ 119887 (119909


119894)

times (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1)) + 119892 (119909


119894)

+ int

119909minus120583minus120583

119894

minus1

1198701(119909


119894 120579)119880 (120579) 119889120579 + 119868

1(119909


119894)

+ int

119902119909minus120583minus120583

119894+119902minus1

minus1

1198702(119909


119894 120578)119880 (120578) 119889120578

+ 1198682(119909


119894)

(58)


119890 (119909) = 119880 (119909) minus 119906 (119909) 119890120574

(119909) = 119880120574

(119909) minus 119906120574

(119909)

(59)

we have

119906120574

119894= 119886 (119909


119894) 119906

119894+ 119887 (119909


119894)

times (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1)) + 119892 (119909


119894)

+ int

119909minus120583minus120583

119894

minus1

1198701(119909


119894 120579) 119906 (120579) 119889120579

+ int

119909minus120583minus120583

119894

minus1

1198701(119909


119894 120579) 119890 (120579) 119889120579

+ int

119902119909minus120583minus120583

119894+119902minus1

minus1

1198702(119909


119894 120578) 119906 (120578) 119889120578

+ int

119902119909minus120583minus120583

119894+119902minus1

minus1

1198702(119909


119894 120578) 119890 (120578) 119889120578

+ 1198681(119909


119894) + 119868

2(119909


119894)

(60)



119880120574

(119909) = 119880 (119909) + 119868minus120583minus120583

119873(

119873

sum

119895=0

119906119895119865119895(119902119909 + 119902 minus 1))

+ 119868minus120583minus120583

119873119892 (119909)

+ 119868minus120583minus120583

119873[119863

120574

119906 (119909) minus 119906 (119909)


+ 119868minus120583minus120583

119873int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

+ 119868minus120583minus120583

119873int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

+ 1198691(119909) + 119869

2(119909)

(61)

119880 (119909) = 119868minus120583minus120583

119873(119906 (119909) minus 119906

minus1)

+ 119868minus120583minus120583

119873(

1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579)

+ 119868minus120583minus120583

119873(119906

minus1)

(62)

where

1198691(119909) = 119868


1198731198681(119909) 119869

2(119909) = 119868


1198731198682(119909) (63)


119890120574

(119909) = 119890 (119909) + int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

+ int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

+ 1198691(119909) + 119869

2(119909) + 119869

3(119909) + 119869

4(119909)

+ 1198695(119909) + 119869

6(119909) + 119869

7(119909)

(64)

119890 (119909) =1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

+ 1198694(119909) + 119869

8(119909)

(65)

where

1198693(119909) = 119868


119873119863120574

119906 (119909) minus 119863120574

119906 (119909)

1198694(119909) = 119868


119873119906 (119909) minus 119906 (119909)

1198695(119909) = 119868



1198696(119909) = 119868


119873int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

minus int

119909

minus1

119870 (119909 119904) 119890 (119904) 119889119904


1198697(119909) = 119868


119873int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

minus int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

1198698(119909) = 119868


119873(

1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579)

minus1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

(66)


119890120574

(119909) =1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

+1

Γ (120574)int

119909

minus1

1198701(119909 120579) int

120579

minus1

(120579 minus 120591)minus120583

119890120574

(120591) 119889120591 119889120579

+1

Γ (120574)int

119902119909+119902minus1

minus1

1198702(119909 120578) int

120578

minus1

(120578 minus 119911)minus120583

times 119890120574

(119911) 119889119911 119889120578 + 119867 (119909)

+ 1198691(119909) + 119869

2(119909) + 119869

3(119909) + 119869

4(119909)

+ 1198695(119909) + 119869

6(119909) + 119869

7(119909)

(67)

where

119867(119909) = int

119909

minus1

1198701(119909 120579) int

120579

minus1

(1198694(120579) + 119869

8(120579)) 119889120579

+1

Γ (120574)int

119902119909+119902minus1

minus1

1198702(119909 120578) int

120578

minus1

(1198694(120578) + 119869

8(120578)) 119889120578

+ 1198694(119909) + 119869

8(119909)

(68)


int

119909

minus1

int

120579

minus1

Φ (120579 120591) 119889120591 119889120579 = int

119909

minus1

int

119909

120591

Φ (120579 120591) 119889120579 119889120591 (69)


1

Γ (120574)int

119909

minus1

1198701(119909 120579) int

120579

minus1

(120579 minus 120591)minus120583

119890120574

(120591) 119889120591 119889120579

=1

Γ (120574)int

119909

minus1

(int

119909

120579

1198701(119909 120579) (120579 minus 120591)

minus120583

119889120579) 119890120574

(120591) 119889120591

le1

Γ (120574)21minus120583 max

119909120591isin[minus11]

100381610038161003816100381610038161(119909 120591)

10038161003816100381610038161003816int

119909

minus1

1003816100381610038161003816119890120574

(120591)1003816100381610038161003816 119889120591

≜ 1198721int

119909

minus1

1003816100381610038161003816119890120574

(120591)1003816100381610038161003816 119889120591

(70)


1

Γ (120574)int

119902119909+119902minus1

minus1

1198702(119909 120578) int

120578

minus1

(120578 minus 119911)minus120583

119890120574

(119911) 119889119911 119889120578

=1

Γ (120574)119902int

119909

minus1

1198702(119909 119902120591 + 119902 minus 1)

times (int

119902120591+119902minus1

minus1

(119909 minus 119911)minus120583

119890120574

(119911) 119889119911) 119889120591

=1

Γ (120574)1199021+120574

int

119909

minus1

1198702(119909 119902120591 + 119902 minus 1)

times (int

120591

minus1

(1 minus 120579)minus120583

119890120574

(119902120579 + 119902 minus 1) 119889120579)119889120591

=1

Γ (120574)1199021+120574

int

119909

minus1

(int

119909

120579

1198702(119909 119902120591 + 119902 minus 1)


minus120583

119889120591) 119890120574

times (119902120579 + 119902 minus 1) 119889120579

le1

Γ (120574)21minus120583

1199021+120574 max

119909120591isin[minus11]

10038161003816100381610038161198702(119909 120591)

1003816100381610038161003816

times int

119909

minus1

1003816100381610038161003816119890120574

(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579

≜ 1198722int

119909

minus1

1003816100381610038161003816119890120574

(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579

(71)

Then (67) gives

119890120574

(119909) le1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

+1198721int

119909

minus1

1003816100381610038161003816119890120574

(120591)1003816100381610038161003816 119889120591

+1198722int

119909

minus1

1003816100381610038161003816119890120574

(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579

+ 119867 (119909) + 1198691(119909) + 119869

2(119909) + 119869

3(119909)

+ 1198694(119909) + 119869

5(119909) + 119869

6(119909) + 119869

7(119909)

(72)


1003817100381710038171003817119890120574

(119909)1003817100381710038171003817119871infin(119868)

le

8

sum

119894=1

1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)

(73)

From (65) we have

119890 (119909)119871infin(119868)

le

8

sum

119894=1

1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)

(74)



100381710038171003817100381711986911003817100381710038171003817119871infin(119868)

le

119862119873(12)minus120583max

0le119894le119873

100381610038161003816100381611986811989411003816100381610038161003816 0 lt 120583 lt

1

2

119862 log119873max0le119894le119873

100381610038161003816100381611986811989411003816100381610038161003816

1

2le 120583 lt 1

le

119862119873(12)minus120583minus119898

119870lowast

1(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 0 lt 120583 lt1

2

119862119873minus119898 log119873119870

lowast

1(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 1

2le 120583 lt 1

100381710038171003817100381711986921003817100381710038171003817119871infin(119868)

le

119862119873(12)minus120583max

0le119894le119873

100381610038161003816100381611986811989421003816100381610038161003816 0 lt 120583 lt

1

2

119862 log119873max0le119894le119873

100381610038161003816100381611986811989411003816100381610038161003816

1

2le 120583 lt 1

le

119862119873(12)minus120583minus119898

119870lowast

2(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 0 lt 120583 lt1

2

119862119873minus119898 log119873119870

lowast

2(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 1

2le 120583 lt 1

(75)

Due to Lemma 2

100381710038171003817100381711986931003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus1198981003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868) 0 lt 120583 lt

1

2

119862119873(12)minus119898 log1198731003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

1

2le 120583 lt 1

100381710038171003817100381711986941003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus119898

|119906|119867119898119873

120596119888(119868) 0 lt 120583 lt

1

2

119862119873(12)minus119898 log119873|119906|

119867119898119873

120596119888(119868)

1

2le 120583 lt 1

100381710038171003817100381711986951003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus119898

|119906|119867119898119873

120596119888(119868) 0 lt 120583 lt

1

2

119862119873(12)minus119898 log119873|119906|

119867119898119873

120596119888(119868)

1

2le 120583 lt 1

(76)


100381710038171003817100381711986961003817100381710038171003817119871infin(119868)

le

119862119873minus120583

119890119871infin(119868) 0 lt 120583 lt

1

2

119862119873minus(12)

119890119871infin(119868)

1

2le 120583 lt 1

100381710038171003817100381711986971003817100381710038171003817119871infin(119868)

le

119862119873minus120583

119890119871infin(119868) 0 lt 120583 lt

1

2

119862119873minus(12)

119890119871infin(119868)

1

2le 120583 lt 1

(77)


5 and 6 with 119870(119909 120591) = 1Γ(120574) that

100381710038171003817100381711986951003817100381710038171003817119871infin(119868)

=10038171003817100381710038171003817(119868


119873minus 119868)M119890

12057410038171003817100381710038171003817119871infin(119868)

=10038171003817100381710038171003817(119868


119873minus 119868) (M119890

120574

minusT119873M119890

120574

)10038171003817100381710038171003817119871infin(119868)

le (1 +10038171003817100381710038171003817119868minus120583minus120583

119873

10038171003817100381710038171003817119871infin(119868))119862119873

minus1198961003817100381710038171003817M11989012057410038171003817100381710038170120581

le

119862119873(12)minus120583minus1205811003817100381710038171003817119890

1205741003817100381710038171003817119871infin(119868) 0 lt 120583 lt

1

2

119862119873minus120581 log1198731003817100381710038171003817119890

1205741003817100381710038171003817119871infin(119868)

1

2le 120583 lt 1

(78)


1


1

2

0 lt 120581 lt 1 minus 120583 when 1

2le 120583 lt 1

(79)


120574

(119909) minus 119906120574

(119909)1003817100381710038171003817119871infin(119868)

le

119862119873(12)minus120583minus119898

(119870lowast

1199061198712(119868)

+ 119873(12)U) 0 lt 120583 lt

1

2

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 119873(12)U)

1

2le 120583 lt 1

119880 (119909) minus 119906 (119909)119871infin(119868)

le

119862119873(12)minus120583minus119898

(119870lowast

1199061198712(119868)

+ 119873(12)U) 0 lt 120583 lt

1

2

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 119873(12)U)

1

2le 120583 lt 1

(80)





120574

(119909) minus 119906120574

(119909)10038171003817100381710038171198712120596minus120583minus120583

(119868)

le

119862119873minus119898

(1198811+ 119873

120574minus(12)minus120581

1198812+ 119873

120574minus120581U) 1

2lt 120574 lt 1

119862119873minus119898

(1198811+ 119873

minus120581 log1198731198812+ 119873

(12)minus120581 log119873U) 0 lt 120574 le1

2

119880(119909) minus 119906(119909)1198712

120596minus120583minus120583

(119868)

le

119862119873minus119898

(1198811+ 119873

120574minus(12)minus120581

1198812+ 119873

120574minus120581U) 1

2lt 120574 lt 1

119862119873minus119898

(1198811+ 119873

minus120581 log1198731198812+ 119873

(12)minus120581 log119873U) 0 lt 120574 le1

2

(81)



1198811= 119870

lowast

(1199061198712(119868)

+1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868))

1198812= 119870

lowast

1199061198712(119868)

(82)


100381710038171003817100381711989012057410038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862

8

sum

119894=1

100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583

(119868)

1198901198712

120596minus120583minus120583

(119868)le 119862

8

sum

119894=1

100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583

(119868)

(83)


10038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862 max

119909isin[minus11]

|119868 (119909)|

le 119862119873minus119898

119870lowast

1(119906

1198712(119868)

+ 119890119871infin(119868))

1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862 max

119909isin[minus11]

|119868 (119909)|

le 119862119873minus119898

119870lowast

2(119906

1198712(119868)

+ 119890119871infin(119868))

(84)


119890119871infin(119868)

le 119862(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868)) (85)

So that10038171003817100381710038171198691

10038171003817100381710038171198712120596minus120583minus120583

(119868)

le 119862119873minus119898

119870lowast

1(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868))

1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583

(119868)

le 119862119873minus119898

119870lowast

2(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868))

(86)

Due to Lemma 2 (33a)10038171003817100381710038171198693

10038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1198981003816100381610038161003816119863120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

1003817100381710038171003817119869410038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus119898

|119906|119867119898119873

120596119888(119868)

1003817100381710038171003817119869510038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus119898

|119906|119867119898119873

120596119888(119868)

(87)


1003817100381710038171003817119869610038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1

10038161003816100381610038161003816100381610038161003816

int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583

(119868)

le 119862119873minus1

1198901198712

120596minus120583minus120583

(119868)

1003817100381710038171003817119869710038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1

100381610038161003816100381610038161003816100381610038161003816

int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

1003816100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583

(119868)

le 119862119873minus1

1198901198712

120596minus120583minus120583

(119868)

(88)


1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583

(119868)=10038171003817100381710038171003817(119868


119873minus 119868)M119890

120574100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

=10038171003817100381710038171003817(119868


119873minus 119868) (M119890

120574

minusT119873119890120574

)100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

le10038171003817100381710038171003817119868minus120583minus120583

119873(M119890

120574

minusT119873119890120574

)100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

+1003817100381710038171003817M119890

120574


(119868)

le 1198621003817100381710038171003817M119890

120574

minusT1198731198901205741003817100381710038171003817119871infin(119868)

le 119862119873minus1205811003817100381710038171003817M119890

12057410038171003817100381710038170120581

le 119862119873minus1205811003817100381710038171003817119890

1205741003817100381710038171003817119871infin(119868)

(89)


1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583

(119868)

le


(119870lowast

1199061198712(119868)

+ 11987312U) 0 lt 120583 lt

1

2

119862119873minus119898minus120581 log119873(119870

lowast

1199061198712(119868)

+ 11987312U)

1

2le 120583 lt 1

(90)





119863075

119910 (119905) =6119905225

Γ (325)minus1199052

119890119905

5119910 (119905) minus

(119902119905)2

5119910 (119902119905)

+ int

119905

0

119890119905

119904119910 (119904) 119889119904 + int

119902119905

0

120591119910 (120591) 119889120591 119905 isin [0 1]

119910 (0) = 0

(91)




0 02 04 06 08 10

02

04

06

08

1

12

14


(a)

0 02 04 06 08 10

05

1

15

2

25


(b)


119910(119905) (b)

2 4 6 8 10 12 14 16

100

10minus5

10minus10

2 le N le 16

Linfin

L2

120596minus120583minus120583

-error-error

(a)

2 4 6 8 10 12 14 16

100

10minus5

10minus10

2 le N le 16

Linfin

L2

120596minus120583minus120583

-error-error

(b)



infin and 1198712

120596norms


infin and 1198712




119863120572

119910 (119905) = 2119905 minus 119910 (119905) + 119910 (119902119905) + 119905 (1 + 2119905) int

119905

0

119890119904(119905minus119904)

119910 (119904) 119889119904

+ int

119902119905

0

119902119905119890120591(119902119905minus120591)

119910 (120591) 119889120591 119905 isin [0 1]

119910 (0) = 1

(92)



0 02 04 06 08 105

1

15

2

25

3

35

4


(a)

0 02 04 06 08 10

1

2

3

4

5

6


(b)


2 4 6 8 10 12 14 16 18 20

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

2 le N le 20

Linfin

L2

120596

(a)

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

2 4 6 8 10 12 14 16 18 202 le N le 20

Linfin

L2

120596

(b)



infin and 1198712

120596norms



1199052








Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119895

119895 = 1 2


Let 120596120572120573(119909) = (1 minus 119909)120572



119899(119909)

infin




1198712


120596120572120573 lt infin

(11)


V120596120572120573 = (int

1

minus1

|V (119909)|2120596120572120573 (119909) 119889119909)12

(12)


(119906 V)120596120572120573 = int

1

minus1

119906 (119909) V (119909) 120596120572120573 (119909) 119889119909 forall119906 V isin 1198712

120596120572120573 (minus1 1)

(13)



points and by 120596119896119873



119896119873



int

1

minus1

119891 (119909) 119889119909 asymp

119873

sum

119896=0

119891 (120579119896) 120596

119896 (14)



120596120572120573

119896119873



int

1

minus1

119891 (119909) 120596120572120573

(119909) 119889119909 asymp

119873

sum

119896=0

119891 (120579119896) 120596

120572120573

119896 (15)


119894119873




119873V isin P

119873 satisfying

119868120572120573

119873V (119909

119894) = V (119909

119894) 0 le 119894 le 119873 (16)


119868120572120573

119873V (119909) =

119873

sum

119894minus0

V (119909119894) 119865

119894(119909) 0 le 119894 le 119873 (17)



119894=0


120579 =1 + 119909

2] +

119909 minus 1

2≜ 120579 (119909 ])

120578 =119902 (1 + 119909)

2120585 +

119902 (1 + 119909)

2minus 1 ≜ 120578 (119909 120585)

(18)

119863120574

119906 (119909) = 119906 (119909) + 119906 (119902119909 + 119902 minus 1) + 119892 (119909)

+1 + 119909

2int

1

minus1

1198701(119909 120579 (119909 ])) 119906 (120579 (119909 ])) 119889]

+119902 (1 + 119909)

2int

1

minus1

1198702(119909 120578 (119909 120585)) 119906 (120578 (119909 120585)) 119889120585

(19)

119906 (119909) =1

Γ (120574)(1 + 119909

2)

120574

int

1

minus1

(1 minus ])minus120583119863120574

119906

times (120579 (119909 ])) 119889] + 119906 (minus1)

(20)


119894119873



119894

119863120574

119906 (119909minus120583minus120583

119894) = 119906 (119909


119894) + 119906 (119902119909


119894+ 119902 minus 1)

+ 119892 (119909minus120583minus120583

119894) +

1 + 119909minus120583minus120583

119894

2

times int

1

minus1

1198701(119909


119894 120579 (119909


119894 ]))


119894 ])) 119889]

+

119902 (1 + 119909minus120583minus120583

119894)

2

times int

1

minus1

1198702(119909


119894 120578 (119909


119894 120585))


119894 120585)) 119889120585

(21)

119906 (119909minus120583minus120583

119894) =

1

Γ (120574)(

(1 + 119909minus120583minus120583

119894) 119879

4)

120574

times int

1

minus1

(1 minus ])minus120583119863120574

119906 (120579 (119909minus120583minus120583

119894 ])) 119889]

+ 119906 (minus1)

(22)



119896119873



int

1

minus1

1198701(119909


120579 (119909minus120583minus120583

119894 ])) 119906 (120579 (119909minus120583minus120583

119894 ])) 119889]

=

119873

sum

119896=0

1198701(119909

119894 120579 (119909

119894 ]119896)) 119906 (120579 (119909

119894 ]119896)) 120596

00

119896

int

1

minus1

1198702(119909


119894 120578 (119909


119894 120585)) 119906 (120578 (119909


119894 120585)) 119889120585

=

119873

sum

119896=0

1198702(119909

119894 120578 (119909

119894 120585119896)) 119906 (120578 (119909

119894 120585119896)) 120596

00

119896

(23)

The sets ]119896119873

119896=0and 120585

119896119873



119896=0 that is ]

119896119873

119896=0

and 120585119896119873



int

1

minus1

(1 minus ])120583119863120574

119906 (120579 (119909minus120583minus120583

119894 ])) 119889120579

=

119873

sum

119896=0

119863120574

119906 (120579 (119909minus120583minus120583

119894 ]119896)) 120596

minus1205830

119896

(24)




119894 119906120574


119894)

119863120574

119906(119909minus120583minus120583



119880 (119909) =

119873

sum

119895=0

119906119895119865119895(119909) 119880

120574

(119909) =

119873

sum

119895=0

119906120574

119895119865119895(119909) (25)



119894119873



120574

119894119873

119894=0and 119906

119894119873


equations hold

119906120574

119894= 119906

119894+ (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1))

+ 119892 (119909minus120583minus120583

119894)

+1 + 119909


119894

2

119873

sum

119895=0

119906119895(

119873

sum

119896=0

1198701(119909


119894 120579 (119909

119894 ]119896))

times119865119895(120579 (119909


119894 ]119896)) 120596

00

119896)

+

119902 (1 + 119909minus120583minus120583

119894)

2

119873

sum

119895=0

119906119895(

119873

sum

119896=0

1198702(119909


119894 120578 (119909

119894 ]119896))

times 119865119895(120578 (119909


119894 ]119896)) 120596

00

119896)

(26)

119906119894=

1

Γ (120574)(

(1 + 119909minus120583minus120583

119894) 119879

4)

120574

119873

sum

119895=0

119906120574

119895

times (

119873

sum

119896=0

119865119895(119904 (119909


119894 ]119896)) 120596

minus1205830

119896) + 119906 (minus1)

(27)

Writing 119880119873

= (1199060 119906

1 119906

119873)119879 and 119880

120574

119873= (119906

120574

0 119906

120574

1

119906120574


(27)

119880120574

119873= (119864 + 119871 + 119862 + 119863)119880

119873+ 119866

119880119873= 119880

minus1+ 119861119880

120574

119873

(28)

where


119866 = (119892 (119909minus120583minus120583

0) 119892 (119909


1) 119892 (119909


119873))

119879

119871119894119895= 119865

119895(119902119909


119894+ 119902 minus 1)

119862119894119895=1 + 119909


119894

2

119873

sum

119896=0

1(119909


119894 120579 (119909


119894 ]119896)) 119865

119895

times (120579 (119909119894 ]119896)) 120596

00

119896

119863119894119895=

119902 (1 + 119909minus120583minus120583

119894)

2

119873

sum

119896=0

2(119909


119894 120578 (119909


119894 ]119896)) 119865

119895


119894 ]119896)) 120596

00

119896

119861119894119895=

1

Γ (120574)(

(1 + 119909minus120583minus120583

119894) 119879

4)

120574

times

119873

sum

119896=0

119865119895(119904 (119909


119894 ]119896)) 120596

minus1205830

119896

119880minus1


(29)


119894=0and 119906

120574

119894119873



120574






119898

(119868)with 119868 for some119898 ge


N such that100381610038161003816100381610038161003816100381610038161003816

int

1

minus1

119906 (119909) 120593 (119909) 119889119909 minus (119906 120593)119873

100381610038161003816100381610038161003816100381610038161003816

le 119862119873minus119898

|119906|119867119898119873

120596120572120573(119868)

100381710038171003817100381712059310038171003817100381710038171198712120596120572120573(119868)

(30)

where

|119906|119867119898119873

120596120572120573(119868)

= (

119898

sum

119895=min(119898119873+1)

10038171003817100381710038171003817119906(119895)10038171003817100381710038171003817

2

1198712

120596120572120573(119868)

)

12

(119906 120593)119873=

119873

sum

119895=0

119906 (119909119895) 120593 (119909

119895) 120596

119895

(31)


120596minus120583minus120583




119895=0 namely

119868minus120583minus120583

119873119906 =

119873

sum

119894=0

119906 (119909119894) 119865 (119909

119894) (32)


10038171003817100381710038171003817119906 minus 119868


119873119906100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

le 119862119873minus119898

|119906|119867119898119873

120596120572120573(119868) (33a)

10038171003817100381710038171003817119906 minus 119868


11987311990610038171003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus119898

|119906|119867119898119873

120596119888(119868) 0 le 120583 lt

1

2

11986211987312minus119898 log119873|119906|

119867119898119873

120596119888(119868)

1

2le 120583 lt 1

(33b)



119873

119895=0are the 119873-th


100381710038171003817100381710038171003817119868120572120573

119873

100381710038171003817100381710038171003817119871infin(119868)le max

119909isin[minus11]

119873

sum

119895=0

10038161003816100381610038161003816119865119895(119909)

10038161003816100381610038161003816

=


2

O (119873120574+12

) 120574 = max (120572 120573) 119900119905ℎ119890119903119908119894119904119890(34)


119906 (119909) le V (119909) + 119871int

119909

minus1

(119909 minus 120591)minus120583

119906 (120591) 119889120591 (35)


119906 (119909) le V (119909) + 119862119871int

119909

minus1


(36)




119903120581


P119873such that


119903120581119873minus(119903+120581)

V119903120581 (37)

where sdot 119903120581



V119903120581

= max0le120581le119903


1003816100381610038161003816120597120581

119909V (119909)1003816100381610038161003816

+ max0le120581le119903

sup119909119910isin[minus11]119909 = 119910

1003816100381610038161003816120597120581

119909V (119909) minus 120597

120581

119909V (119909)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

120581

(38)


119903120581



(MV) (119909) = int

119909

minus1

(119909 minus 120591)minus120583

119870 (119909 120591) V (120591) 119889120591 (39)


10038161003816100381610038161003816MV (1199091015840) minusMV (11990910158401015840)

1003816100381610038161003816100381610038161003816100381610038161199091015840 minus 11990910158401015840

1003816100381610038161003816


|V (119909)| (40)


11990910158401015840

isin


MV0120581


|V (119909)| 0 lt 120581 lt 1 minus 120583 (41)


sup119873

10038171003817100381710038171003817100381710038171003817100381710038171003817

119873

sum

119895=0

V (119909119895) 119865

119895(119909)

100381710038171003817100381710038171003817100381710038171003817100381710038171198712120596120572120573(119868)


|V (119909)| (42)



119895119873

119895=0


(int

119887

119886

1003816100381610038161003816(119879119891) (119909)1003816100381610038161003816

119902

119906 (119909) 119889119909)

1119902

le (int

119887

119886

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901

V (119909) 119889119909)1119901

(43)



sup119886lt119909lt119887

(int

119887

119909

119906 (119905) 119889119905)

1119902

(int

119909

119886

V1minus1199011015840

(119905) 119889119905)

11199011015840

lt infin 1199011015840

=119901

119901 minus 1

(44)


(119879119865) (119909) = int

119909

119886

119896 (119909 119905) 119891 (119905) 119889119905 (45)




infin space


120574


119898+1

120596minus120583minus120583(119868) then

1003817100381710038171003817119880120574

minus 119863120574

1199061003817100381710038171003817119871infin(119868)

le

119862119873120574minus12minus119898

(119870lowast

1199061198712(119868)

+ 11987312U)

1

2lt 120574 lt 1

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 11987312U) 0 lt 120574 le

1

2

(46)

119880 minus 119906119871infin(119868)

le

119862119873120574minus12minus119898

(119870lowast

1199061198712(119868)

+ 11987312U)

1

2lt 120574 lt 1

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 11987312U) 0 lt 120574 le

1

2

(47)


119870lowast


10038161003816100381610038161198701(119909 120579 (119909 sdot))

100381610038161003816100381611989811987312059600(119868)


10038161003816100381610038161198702(119909 120578 (119909 sdot))

100381610038161003816100381611989811987312059600(119868)

(48)

U =1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868) (49)

Proof We let

(119870 (119904 (119909119894 ])) 120601 (119904 (119909

119894 ])))

119873

=

119873

sum

119896=0

119870(119904 (119909119894 ]119896)) 120601 (119904 (119909

119894 ]119896)) 120596

00

119896

(50)


119906120574

119894= 119906

119894+ (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1)) + 119892 (119909


119894)

+1 + 119909


119894

2(119870

1(119909


119894 120579 (119909


119894 ]))

119880 (120579 (119909minus120583minus120583

119894 ])))

119873

+

119902 (1 + 119909minus120583minus120583

119894)

2(119870

2(119909


119894 120578 (119909


119894 ]))

119880 (120578 (119909minus120583minus120583

119894 ])))

119873

(51)

119906119894=

1

Γ (120574)(119879

2)

120574

int

119909119894

minus1

(119909119894minus 120579)

minus120583

119880120574

(120579) 119889120579 + 119906minus1 (52)

which gives

119906120574

119894= 119906

119894+ (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1)) + 119892 (119909


119894)

+1 + 119909


119894

2int

1

minus1

1198701(119909


119894 120579 (119909


119894 ]))


119894 ])) 119889] + 119868

1(119909


119894)

+

119902 (1 + 119909minus120583minus120583

119894)

2int

1

minus1

1198702(119909


119894 120578 (119909


119894 120585))


119894 120585)) 119889120585 + 119868

2(119909


119894)

(53)

where

1198681(119909) =

1 + 119909

2int

1

minus1

1198701(119909 120579 (119909)) 119880 (120579 (119909 ])) 119889]

minus1 + 119909

2(119870

1(119909 120579 (119909 ])) 119880 (120579 (119909 ])))

119873

1198682(119909) =

119902 (1 + 119909)

2int

1

minus1

1198702(119909 120578 (119909 120585)) 119880 (120578 (119909 120585)) 119889120585

minus119902 (1 + 119909)

2(119870

2(119909 120578 (119909 ])) 119880 (120578 (119909 ])))

119873

(54)



10038161003816100381610038161198681 (119909)1003816100381610038161003816 le 119862119873

minus119898100381610038161003816100381610038161(119909 120579 (119909 sdot))

1003816100381610038161003816100381611986711989811987312059600(119868)

119880 (120579 (119909 sdot))1198712(119868)

le 119862119873minus119898

119870lowast

1119880

1198712(119868)

(55)

10038161003816100381610038161198682 (119909)1003816100381610038161003816 le 119862119873

minus119898100381610038161003816100381610038162(119909 120578 (119909 sdot))

1003816100381610038161003816100381611986711989811987312059600(119868)

1003817100381710038171003817119880 (120578 (119909 sdot))10038171003817100381710038171198712(119868)

le 119862119873minus119898

119870lowast

2119880

1198712(119868)

(56)

119870lowast

1= max

119909isin[minus11]

10038161003816100381610038161198701(119909 120579 (119909 sdot))

100381610038161003816100381611989811987312059600(119868)

119870lowast

2= max

119909isin[minus11]

10038161003816100381610038161198701(119909 120579 (119909 sdot))

100381610038161003816100381611989811987312059600(119868)

(57)


119906120574

119894= 119886 (119909


119894) 119906

119894+ 119887 (119909


119894)

times (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1)) + 119892 (119909


119894)

+ int

119909minus120583minus120583

119894

minus1

1198701(119909


119894 120579)119880 (120579) 119889120579 + 119868

1(119909


119894)

+ int

119902119909minus120583minus120583

119894+119902minus1

minus1

1198702(119909


119894 120578)119880 (120578) 119889120578

+ 1198682(119909


119894)

(58)


119890 (119909) = 119880 (119909) minus 119906 (119909) 119890120574

(119909) = 119880120574

(119909) minus 119906120574

(119909)

(59)

we have

119906120574

119894= 119886 (119909


119894) 119906

119894+ 119887 (119909


119894)

times (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1)) + 119892 (119909


119894)

+ int

119909minus120583minus120583

119894

minus1

1198701(119909


119894 120579) 119906 (120579) 119889120579

+ int

119909minus120583minus120583

119894

minus1

1198701(119909


119894 120579) 119890 (120579) 119889120579

+ int

119902119909minus120583minus120583

119894+119902minus1

minus1

1198702(119909


119894 120578) 119906 (120578) 119889120578

+ int

119902119909minus120583minus120583

119894+119902minus1

minus1

1198702(119909


119894 120578) 119890 (120578) 119889120578

+ 1198681(119909


119894) + 119868

2(119909


119894)

(60)



119880120574

(119909) = 119880 (119909) + 119868minus120583minus120583

119873(

119873

sum

119895=0

119906119895119865119895(119902119909 + 119902 minus 1))

+ 119868minus120583minus120583

119873119892 (119909)

+ 119868minus120583minus120583

119873[119863

120574

119906 (119909) minus 119906 (119909)


+ 119868minus120583minus120583

119873int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

+ 119868minus120583minus120583

119873int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

+ 1198691(119909) + 119869

2(119909)

(61)

119880 (119909) = 119868minus120583minus120583

119873(119906 (119909) minus 119906

minus1)

+ 119868minus120583minus120583

119873(

1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579)

+ 119868minus120583minus120583

119873(119906

minus1)

(62)

where

1198691(119909) = 119868


1198731198681(119909) 119869

2(119909) = 119868


1198731198682(119909) (63)


119890120574

(119909) = 119890 (119909) + int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

+ int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

+ 1198691(119909) + 119869

2(119909) + 119869

3(119909) + 119869

4(119909)

+ 1198695(119909) + 119869

6(119909) + 119869

7(119909)

(64)

119890 (119909) =1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

+ 1198694(119909) + 119869

8(119909)

(65)

where

1198693(119909) = 119868


119873119863120574

119906 (119909) minus 119863120574

119906 (119909)

1198694(119909) = 119868


119873119906 (119909) minus 119906 (119909)

1198695(119909) = 119868



1198696(119909) = 119868


119873int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

minus int

119909

minus1

119870 (119909 119904) 119890 (119904) 119889119904


1198697(119909) = 119868


119873int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

minus int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

1198698(119909) = 119868


119873(

1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579)

minus1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

(66)


119890120574

(119909) =1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

+1

Γ (120574)int

119909

minus1

1198701(119909 120579) int

120579

minus1

(120579 minus 120591)minus120583

119890120574

(120591) 119889120591 119889120579

+1

Γ (120574)int

119902119909+119902minus1

minus1

1198702(119909 120578) int

120578

minus1

(120578 minus 119911)minus120583

times 119890120574

(119911) 119889119911 119889120578 + 119867 (119909)

+ 1198691(119909) + 119869

2(119909) + 119869

3(119909) + 119869

4(119909)

+ 1198695(119909) + 119869

6(119909) + 119869

7(119909)

(67)

where

119867(119909) = int

119909

minus1

1198701(119909 120579) int

120579

minus1

(1198694(120579) + 119869

8(120579)) 119889120579

+1

Γ (120574)int

119902119909+119902minus1

minus1

1198702(119909 120578) int

120578

minus1

(1198694(120578) + 119869

8(120578)) 119889120578

+ 1198694(119909) + 119869

8(119909)

(68)


int

119909

minus1

int

120579

minus1

Φ (120579 120591) 119889120591 119889120579 = int

119909

minus1

int

119909

120591

Φ (120579 120591) 119889120579 119889120591 (69)


1

Γ (120574)int

119909

minus1

1198701(119909 120579) int

120579

minus1

(120579 minus 120591)minus120583

119890120574

(120591) 119889120591 119889120579

=1

Γ (120574)int

119909

minus1

(int

119909

120579

1198701(119909 120579) (120579 minus 120591)

minus120583

119889120579) 119890120574

(120591) 119889120591

le1

Γ (120574)21minus120583 max

119909120591isin[minus11]

100381610038161003816100381610038161(119909 120591)

10038161003816100381610038161003816int

119909

minus1

1003816100381610038161003816119890120574

(120591)1003816100381610038161003816 119889120591

≜ 1198721int

119909

minus1

1003816100381610038161003816119890120574

(120591)1003816100381610038161003816 119889120591

(70)


1

Γ (120574)int

119902119909+119902minus1

minus1

1198702(119909 120578) int

120578

minus1

(120578 minus 119911)minus120583

119890120574

(119911) 119889119911 119889120578

=1

Γ (120574)119902int

119909

minus1

1198702(119909 119902120591 + 119902 minus 1)

times (int

119902120591+119902minus1

minus1

(119909 minus 119911)minus120583

119890120574

(119911) 119889119911) 119889120591

=1

Γ (120574)1199021+120574

int

119909

minus1

1198702(119909 119902120591 + 119902 minus 1)

times (int

120591

minus1

(1 minus 120579)minus120583

119890120574

(119902120579 + 119902 minus 1) 119889120579)119889120591

=1

Γ (120574)1199021+120574

int

119909

minus1

(int

119909

120579

1198702(119909 119902120591 + 119902 minus 1)


minus120583

119889120591) 119890120574

times (119902120579 + 119902 minus 1) 119889120579

le1

Γ (120574)21minus120583

1199021+120574 max

119909120591isin[minus11]

10038161003816100381610038161198702(119909 120591)

1003816100381610038161003816

times int

119909

minus1

1003816100381610038161003816119890120574

(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579

≜ 1198722int

119909

minus1

1003816100381610038161003816119890120574

(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579

(71)

Then (67) gives

119890120574

(119909) le1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

+1198721int

119909

minus1

1003816100381610038161003816119890120574

(120591)1003816100381610038161003816 119889120591

+1198722int

119909

minus1

1003816100381610038161003816119890120574

(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579

+ 119867 (119909) + 1198691(119909) + 119869

2(119909) + 119869

3(119909)

+ 1198694(119909) + 119869

5(119909) + 119869

6(119909) + 119869

7(119909)

(72)


1003817100381710038171003817119890120574

(119909)1003817100381710038171003817119871infin(119868)

le

8

sum

119894=1

1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)

(73)

From (65) we have

119890 (119909)119871infin(119868)

le

8

sum

119894=1

1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)

(74)



100381710038171003817100381711986911003817100381710038171003817119871infin(119868)

le

119862119873(12)minus120583max

0le119894le119873

100381610038161003816100381611986811989411003816100381610038161003816 0 lt 120583 lt

1

2

119862 log119873max0le119894le119873

100381610038161003816100381611986811989411003816100381610038161003816

1

2le 120583 lt 1

le

119862119873(12)minus120583minus119898

119870lowast

1(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 0 lt 120583 lt1

2

119862119873minus119898 log119873119870

lowast

1(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 1

2le 120583 lt 1

100381710038171003817100381711986921003817100381710038171003817119871infin(119868)

le

119862119873(12)minus120583max

0le119894le119873

100381610038161003816100381611986811989421003816100381610038161003816 0 lt 120583 lt

1

2

119862 log119873max0le119894le119873

100381610038161003816100381611986811989411003816100381610038161003816

1

2le 120583 lt 1

le

119862119873(12)minus120583minus119898

119870lowast

2(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 0 lt 120583 lt1

2

119862119873minus119898 log119873119870

lowast

2(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 1

2le 120583 lt 1

(75)

Due to Lemma 2

100381710038171003817100381711986931003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus1198981003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868) 0 lt 120583 lt

1

2

119862119873(12)minus119898 log1198731003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

1

2le 120583 lt 1

100381710038171003817100381711986941003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus119898

|119906|119867119898119873

120596119888(119868) 0 lt 120583 lt

1

2

119862119873(12)minus119898 log119873|119906|

119867119898119873

120596119888(119868)

1

2le 120583 lt 1

100381710038171003817100381711986951003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus119898

|119906|119867119898119873

120596119888(119868) 0 lt 120583 lt

1

2

119862119873(12)minus119898 log119873|119906|

119867119898119873

120596119888(119868)

1

2le 120583 lt 1

(76)


100381710038171003817100381711986961003817100381710038171003817119871infin(119868)

le

119862119873minus120583

119890119871infin(119868) 0 lt 120583 lt

1

2

119862119873minus(12)

119890119871infin(119868)

1

2le 120583 lt 1

100381710038171003817100381711986971003817100381710038171003817119871infin(119868)

le

119862119873minus120583

119890119871infin(119868) 0 lt 120583 lt

1

2

119862119873minus(12)

119890119871infin(119868)

1

2le 120583 lt 1

(77)


5 and 6 with 119870(119909 120591) = 1Γ(120574) that

100381710038171003817100381711986951003817100381710038171003817119871infin(119868)

=10038171003817100381710038171003817(119868


119873minus 119868)M119890

12057410038171003817100381710038171003817119871infin(119868)

=10038171003817100381710038171003817(119868


119873minus 119868) (M119890

120574

minusT119873M119890

120574

)10038171003817100381710038171003817119871infin(119868)

le (1 +10038171003817100381710038171003817119868minus120583minus120583

119873

10038171003817100381710038171003817119871infin(119868))119862119873

minus1198961003817100381710038171003817M11989012057410038171003817100381710038170120581

le

119862119873(12)minus120583minus1205811003817100381710038171003817119890

1205741003817100381710038171003817119871infin(119868) 0 lt 120583 lt

1

2

119862119873minus120581 log1198731003817100381710038171003817119890

1205741003817100381710038171003817119871infin(119868)

1

2le 120583 lt 1

(78)


1


1

2

0 lt 120581 lt 1 minus 120583 when 1

2le 120583 lt 1

(79)


120574

(119909) minus 119906120574

(119909)1003817100381710038171003817119871infin(119868)

le

119862119873(12)minus120583minus119898

(119870lowast

1199061198712(119868)

+ 119873(12)U) 0 lt 120583 lt

1

2

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 119873(12)U)

1

2le 120583 lt 1

119880 (119909) minus 119906 (119909)119871infin(119868)

le

119862119873(12)minus120583minus119898

(119870lowast

1199061198712(119868)

+ 119873(12)U) 0 lt 120583 lt

1

2

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 119873(12)U)

1

2le 120583 lt 1

(80)





120574

(119909) minus 119906120574

(119909)10038171003817100381710038171198712120596minus120583minus120583

(119868)

le

119862119873minus119898

(1198811+ 119873

120574minus(12)minus120581

1198812+ 119873

120574minus120581U) 1

2lt 120574 lt 1

119862119873minus119898

(1198811+ 119873

minus120581 log1198731198812+ 119873

(12)minus120581 log119873U) 0 lt 120574 le1

2

119880(119909) minus 119906(119909)1198712

120596minus120583minus120583

(119868)

le

119862119873minus119898

(1198811+ 119873

120574minus(12)minus120581

1198812+ 119873

120574minus120581U) 1

2lt 120574 lt 1

119862119873minus119898

(1198811+ 119873

minus120581 log1198731198812+ 119873

(12)minus120581 log119873U) 0 lt 120574 le1

2

(81)



1198811= 119870

lowast

(1199061198712(119868)

+1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868))

1198812= 119870

lowast

1199061198712(119868)

(82)


100381710038171003817100381711989012057410038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862

8

sum

119894=1

100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583

(119868)

1198901198712

120596minus120583minus120583

(119868)le 119862

8

sum

119894=1

100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583

(119868)

(83)


10038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862 max

119909isin[minus11]

|119868 (119909)|

le 119862119873minus119898

119870lowast

1(119906

1198712(119868)

+ 119890119871infin(119868))

1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862 max

119909isin[minus11]

|119868 (119909)|

le 119862119873minus119898

119870lowast

2(119906

1198712(119868)

+ 119890119871infin(119868))

(84)


119890119871infin(119868)

le 119862(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868)) (85)

So that10038171003817100381710038171198691

10038171003817100381710038171198712120596minus120583minus120583

(119868)

le 119862119873minus119898

119870lowast

1(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868))

1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583

(119868)

le 119862119873minus119898

119870lowast

2(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868))

(86)

Due to Lemma 2 (33a)10038171003817100381710038171198693

10038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1198981003816100381610038161003816119863120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

1003817100381710038171003817119869410038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus119898

|119906|119867119898119873

120596119888(119868)

1003817100381710038171003817119869510038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus119898

|119906|119867119898119873

120596119888(119868)

(87)


1003817100381710038171003817119869610038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1

10038161003816100381610038161003816100381610038161003816

int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583

(119868)

le 119862119873minus1

1198901198712

120596minus120583minus120583

(119868)

1003817100381710038171003817119869710038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1

100381610038161003816100381610038161003816100381610038161003816

int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

1003816100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583

(119868)

le 119862119873minus1

1198901198712

120596minus120583minus120583

(119868)

(88)


1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583

(119868)=10038171003817100381710038171003817(119868


119873minus 119868)M119890

120574100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

=10038171003817100381710038171003817(119868


119873minus 119868) (M119890

120574

minusT119873119890120574

)100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

le10038171003817100381710038171003817119868minus120583minus120583

119873(M119890

120574

minusT119873119890120574

)100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

+1003817100381710038171003817M119890

120574


(119868)

le 1198621003817100381710038171003817M119890

120574

minusT1198731198901205741003817100381710038171003817119871infin(119868)

le 119862119873minus1205811003817100381710038171003817M119890

12057410038171003817100381710038170120581

le 119862119873minus1205811003817100381710038171003817119890

1205741003817100381710038171003817119871infin(119868)

(89)


1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583

(119868)

le


(119870lowast

1199061198712(119868)

+ 11987312U) 0 lt 120583 lt

1

2

119862119873minus119898minus120581 log119873(119870

lowast

1199061198712(119868)

+ 11987312U)

1

2le 120583 lt 1

(90)





119863075

119910 (119905) =6119905225

Γ (325)minus1199052

119890119905

5119910 (119905) minus

(119902119905)2

5119910 (119902119905)

+ int

119905

0

119890119905

119904119910 (119904) 119889119904 + int

119902119905

0

120591119910 (120591) 119889120591 119905 isin [0 1]

119910 (0) = 0

(91)




0 02 04 06 08 10

02

04

06

08

1

12

14


(a)

0 02 04 06 08 10

05

1

15

2

25


(b)


119910(119905) (b)

2 4 6 8 10 12 14 16

100

10minus5

10minus10

2 le N le 16

Linfin

L2

120596minus120583minus120583

-error-error

(a)

2 4 6 8 10 12 14 16

100

10minus5

10minus10

2 le N le 16

Linfin

L2

120596minus120583minus120583

-error-error

(b)



infin and 1198712

120596norms


infin and 1198712




119863120572

119910 (119905) = 2119905 minus 119910 (119905) + 119910 (119902119905) + 119905 (1 + 2119905) int

119905

0

119890119904(119905minus119904)

119910 (119904) 119889119904

+ int

119902119905

0

119902119905119890120591(119902119905minus120591)

119910 (120591) 119889120591 119905 isin [0 1]

119910 (0) = 1

(92)



0 02 04 06 08 105

1

15

2

25

3

35

4


(a)

0 02 04 06 08 10

1

2

3

4

5

6


(b)


2 4 6 8 10 12 14 16 18 20

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

2 le N le 20

Linfin

L2

120596

(a)

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

2 4 6 8 10 12 14 16 18 202 le N le 20

Linfin

L2

120596

(b)



infin and 1198712

120596norms



1199052








Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119896119873



int

1

minus1

1198701(119909


120579 (119909minus120583minus120583

119894 ])) 119906 (120579 (119909minus120583minus120583

119894 ])) 119889]

=

119873

sum

119896=0

1198701(119909

119894 120579 (119909

119894 ]119896)) 119906 (120579 (119909

119894 ]119896)) 120596

00

119896

int

1

minus1

1198702(119909


119894 120578 (119909


119894 120585)) 119906 (120578 (119909


119894 120585)) 119889120585

=

119873

sum

119896=0

1198702(119909

119894 120578 (119909

119894 120585119896)) 119906 (120578 (119909

119894 120585119896)) 120596

00

119896

(23)

The sets ]119896119873

119896=0and 120585

119896119873



119896=0 that is ]

119896119873

119896=0

and 120585119896119873



int

1

minus1

(1 minus ])120583119863120574

119906 (120579 (119909minus120583minus120583

119894 ])) 119889120579

=

119873

sum

119896=0

119863120574

119906 (120579 (119909minus120583minus120583

119894 ]119896)) 120596

minus1205830

119896

(24)




119894 119906120574


119894)

119863120574

119906(119909minus120583minus120583



119880 (119909) =

119873

sum

119895=0

119906119895119865119895(119909) 119880

120574

(119909) =

119873

sum

119895=0

119906120574

119895119865119895(119909) (25)



119894119873



120574

119894119873

119894=0and 119906

119894119873


equations hold

119906120574

119894= 119906

119894+ (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1))

+ 119892 (119909minus120583minus120583

119894)

+1 + 119909


119894

2

119873

sum

119895=0

119906119895(

119873

sum

119896=0

1198701(119909


119894 120579 (119909

119894 ]119896))

times119865119895(120579 (119909


119894 ]119896)) 120596

00

119896)

+

119902 (1 + 119909minus120583minus120583

119894)

2

119873

sum

119895=0

119906119895(

119873

sum

119896=0

1198702(119909


119894 120578 (119909

119894 ]119896))

times 119865119895(120578 (119909


119894 ]119896)) 120596

00

119896)

(26)

119906119894=

1

Γ (120574)(

(1 + 119909minus120583minus120583

119894) 119879

4)

120574

119873

sum

119895=0

119906120574

119895

times (

119873

sum

119896=0

119865119895(119904 (119909


119894 ]119896)) 120596

minus1205830

119896) + 119906 (minus1)

(27)

Writing 119880119873

= (1199060 119906

1 119906

119873)119879 and 119880

120574

119873= (119906

120574

0 119906

120574

1

119906120574


(27)

119880120574

119873= (119864 + 119871 + 119862 + 119863)119880

119873+ 119866

119880119873= 119880

minus1+ 119861119880

120574

119873

(28)

where


119866 = (119892 (119909minus120583minus120583

0) 119892 (119909


1) 119892 (119909


119873))

119879

119871119894119895= 119865

119895(119902119909


119894+ 119902 minus 1)

119862119894119895=1 + 119909


119894

2

119873

sum

119896=0

1(119909


119894 120579 (119909


119894 ]119896)) 119865

119895

times (120579 (119909119894 ]119896)) 120596

00

119896

119863119894119895=

119902 (1 + 119909minus120583minus120583

119894)

2

119873

sum

119896=0

2(119909


119894 120578 (119909


119894 ]119896)) 119865

119895


119894 ]119896)) 120596

00

119896

119861119894119895=

1

Γ (120574)(

(1 + 119909minus120583minus120583

119894) 119879

4)

120574

times

119873

sum

119896=0

119865119895(119904 (119909


119894 ]119896)) 120596

minus1205830

119896

119880minus1


(29)


119894=0and 119906

120574

119894119873



120574






119898

(119868)with 119868 for some119898 ge


N such that100381610038161003816100381610038161003816100381610038161003816

int

1

minus1

119906 (119909) 120593 (119909) 119889119909 minus (119906 120593)119873

100381610038161003816100381610038161003816100381610038161003816

le 119862119873minus119898

|119906|119867119898119873

120596120572120573(119868)

100381710038171003817100381712059310038171003817100381710038171198712120596120572120573(119868)

(30)

where

|119906|119867119898119873

120596120572120573(119868)

= (

119898

sum

119895=min(119898119873+1)

10038171003817100381710038171003817119906(119895)10038171003817100381710038171003817

2

1198712

120596120572120573(119868)

)

12

(119906 120593)119873=

119873

sum

119895=0

119906 (119909119895) 120593 (119909

119895) 120596

119895

(31)


120596minus120583minus120583




119895=0 namely

119868minus120583minus120583

119873119906 =

119873

sum

119894=0

119906 (119909119894) 119865 (119909

119894) (32)


10038171003817100381710038171003817119906 minus 119868


119873119906100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

le 119862119873minus119898

|119906|119867119898119873

120596120572120573(119868) (33a)

10038171003817100381710038171003817119906 minus 119868


11987311990610038171003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus119898

|119906|119867119898119873

120596119888(119868) 0 le 120583 lt

1

2

11986211987312minus119898 log119873|119906|

119867119898119873

120596119888(119868)

1

2le 120583 lt 1

(33b)



119873

119895=0are the 119873-th


100381710038171003817100381710038171003817119868120572120573

119873

100381710038171003817100381710038171003817119871infin(119868)le max

119909isin[minus11]

119873

sum

119895=0

10038161003816100381610038161003816119865119895(119909)

10038161003816100381610038161003816

=


2

O (119873120574+12

) 120574 = max (120572 120573) 119900119905ℎ119890119903119908119894119904119890(34)


119906 (119909) le V (119909) + 119871int

119909

minus1

(119909 minus 120591)minus120583

119906 (120591) 119889120591 (35)


119906 (119909) le V (119909) + 119862119871int

119909

minus1


(36)




119903120581


P119873such that


119903120581119873minus(119903+120581)

V119903120581 (37)

where sdot 119903120581



V119903120581

= max0le120581le119903


1003816100381610038161003816120597120581

119909V (119909)1003816100381610038161003816

+ max0le120581le119903

sup119909119910isin[minus11]119909 = 119910

1003816100381610038161003816120597120581

119909V (119909) minus 120597

120581

119909V (119909)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

120581

(38)


119903120581



(MV) (119909) = int

119909

minus1

(119909 minus 120591)minus120583

119870 (119909 120591) V (120591) 119889120591 (39)


10038161003816100381610038161003816MV (1199091015840) minusMV (11990910158401015840)

1003816100381610038161003816100381610038161003816100381610038161199091015840 minus 11990910158401015840

1003816100381610038161003816


|V (119909)| (40)


11990910158401015840

isin


MV0120581


|V (119909)| 0 lt 120581 lt 1 minus 120583 (41)


sup119873

10038171003817100381710038171003817100381710038171003817100381710038171003817

119873

sum

119895=0

V (119909119895) 119865

119895(119909)

100381710038171003817100381710038171003817100381710038171003817100381710038171198712120596120572120573(119868)


|V (119909)| (42)



119895119873

119895=0


(int

119887

119886

1003816100381610038161003816(119879119891) (119909)1003816100381610038161003816

119902

119906 (119909) 119889119909)

1119902

le (int

119887

119886

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901

V (119909) 119889119909)1119901

(43)



sup119886lt119909lt119887

(int

119887

119909

119906 (119905) 119889119905)

1119902

(int

119909

119886

V1minus1199011015840

(119905) 119889119905)

11199011015840

lt infin 1199011015840

=119901

119901 minus 1

(44)


(119879119865) (119909) = int

119909

119886

119896 (119909 119905) 119891 (119905) 119889119905 (45)




infin space


120574


119898+1

120596minus120583minus120583(119868) then

1003817100381710038171003817119880120574

minus 119863120574

1199061003817100381710038171003817119871infin(119868)

le

119862119873120574minus12minus119898

(119870lowast

1199061198712(119868)

+ 11987312U)

1

2lt 120574 lt 1

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 11987312U) 0 lt 120574 le

1

2

(46)

119880 minus 119906119871infin(119868)

le

119862119873120574minus12minus119898

(119870lowast

1199061198712(119868)

+ 11987312U)

1

2lt 120574 lt 1

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 11987312U) 0 lt 120574 le

1

2

(47)


119870lowast


10038161003816100381610038161198701(119909 120579 (119909 sdot))

100381610038161003816100381611989811987312059600(119868)


10038161003816100381610038161198702(119909 120578 (119909 sdot))

100381610038161003816100381611989811987312059600(119868)

(48)

U =1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868) (49)

Proof We let

(119870 (119904 (119909119894 ])) 120601 (119904 (119909

119894 ])))

119873

=

119873

sum

119896=0

119870(119904 (119909119894 ]119896)) 120601 (119904 (119909

119894 ]119896)) 120596

00

119896

(50)


119906120574

119894= 119906

119894+ (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1)) + 119892 (119909


119894)

+1 + 119909


119894

2(119870

1(119909


119894 120579 (119909


119894 ]))

119880 (120579 (119909minus120583minus120583

119894 ])))

119873

+

119902 (1 + 119909minus120583minus120583

119894)

2(119870

2(119909


119894 120578 (119909


119894 ]))

119880 (120578 (119909minus120583minus120583

119894 ])))

119873

(51)

119906119894=

1

Γ (120574)(119879

2)

120574

int

119909119894

minus1

(119909119894minus 120579)

minus120583

119880120574

(120579) 119889120579 + 119906minus1 (52)

which gives

119906120574

119894= 119906

119894+ (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1)) + 119892 (119909


119894)

+1 + 119909


119894

2int

1

minus1

1198701(119909


119894 120579 (119909


119894 ]))


119894 ])) 119889] + 119868

1(119909


119894)

+

119902 (1 + 119909minus120583minus120583

119894)

2int

1

minus1

1198702(119909


119894 120578 (119909


119894 120585))


119894 120585)) 119889120585 + 119868

2(119909


119894)

(53)

where

1198681(119909) =

1 + 119909

2int

1

minus1

1198701(119909 120579 (119909)) 119880 (120579 (119909 ])) 119889]

minus1 + 119909

2(119870

1(119909 120579 (119909 ])) 119880 (120579 (119909 ])))

119873

1198682(119909) =

119902 (1 + 119909)

2int

1

minus1

1198702(119909 120578 (119909 120585)) 119880 (120578 (119909 120585)) 119889120585

minus119902 (1 + 119909)

2(119870

2(119909 120578 (119909 ])) 119880 (120578 (119909 ])))

119873

(54)



10038161003816100381610038161198681 (119909)1003816100381610038161003816 le 119862119873

minus119898100381610038161003816100381610038161(119909 120579 (119909 sdot))

1003816100381610038161003816100381611986711989811987312059600(119868)

119880 (120579 (119909 sdot))1198712(119868)

le 119862119873minus119898

119870lowast

1119880

1198712(119868)

(55)

10038161003816100381610038161198682 (119909)1003816100381610038161003816 le 119862119873

minus119898100381610038161003816100381610038162(119909 120578 (119909 sdot))

1003816100381610038161003816100381611986711989811987312059600(119868)

1003817100381710038171003817119880 (120578 (119909 sdot))10038171003817100381710038171198712(119868)

le 119862119873minus119898

119870lowast

2119880

1198712(119868)

(56)

119870lowast

1= max

119909isin[minus11]

10038161003816100381610038161198701(119909 120579 (119909 sdot))

100381610038161003816100381611989811987312059600(119868)

119870lowast

2= max

119909isin[minus11]

10038161003816100381610038161198701(119909 120579 (119909 sdot))

100381610038161003816100381611989811987312059600(119868)

(57)


119906120574

119894= 119886 (119909


119894) 119906

119894+ 119887 (119909


119894)

times (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1)) + 119892 (119909


119894)

+ int

119909minus120583minus120583

119894

minus1

1198701(119909


119894 120579)119880 (120579) 119889120579 + 119868

1(119909


119894)

+ int

119902119909minus120583minus120583

119894+119902minus1

minus1

1198702(119909


119894 120578)119880 (120578) 119889120578

+ 1198682(119909


119894)

(58)


119890 (119909) = 119880 (119909) minus 119906 (119909) 119890120574

(119909) = 119880120574

(119909) minus 119906120574

(119909)

(59)

we have

119906120574

119894= 119886 (119909


119894) 119906

119894+ 119887 (119909


119894)

times (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1)) + 119892 (119909


119894)

+ int

119909minus120583minus120583

119894

minus1

1198701(119909


119894 120579) 119906 (120579) 119889120579

+ int

119909minus120583minus120583

119894

minus1

1198701(119909


119894 120579) 119890 (120579) 119889120579

+ int

119902119909minus120583minus120583

119894+119902minus1

minus1

1198702(119909


119894 120578) 119906 (120578) 119889120578

+ int

119902119909minus120583minus120583

119894+119902minus1

minus1

1198702(119909


119894 120578) 119890 (120578) 119889120578

+ 1198681(119909


119894) + 119868

2(119909


119894)

(60)



119880120574

(119909) = 119880 (119909) + 119868minus120583minus120583

119873(

119873

sum

119895=0

119906119895119865119895(119902119909 + 119902 minus 1))

+ 119868minus120583minus120583

119873119892 (119909)

+ 119868minus120583minus120583

119873[119863

120574

119906 (119909) minus 119906 (119909)


+ 119868minus120583minus120583

119873int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

+ 119868minus120583minus120583

119873int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

+ 1198691(119909) + 119869

2(119909)

(61)

119880 (119909) = 119868minus120583minus120583

119873(119906 (119909) minus 119906

minus1)

+ 119868minus120583minus120583

119873(

1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579)

+ 119868minus120583minus120583

119873(119906

minus1)

(62)

where

1198691(119909) = 119868


1198731198681(119909) 119869

2(119909) = 119868


1198731198682(119909) (63)


119890120574

(119909) = 119890 (119909) + int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

+ int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

+ 1198691(119909) + 119869

2(119909) + 119869

3(119909) + 119869

4(119909)

+ 1198695(119909) + 119869

6(119909) + 119869

7(119909)

(64)

119890 (119909) =1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

+ 1198694(119909) + 119869

8(119909)

(65)

where

1198693(119909) = 119868


119873119863120574

119906 (119909) minus 119863120574

119906 (119909)

1198694(119909) = 119868


119873119906 (119909) minus 119906 (119909)

1198695(119909) = 119868



1198696(119909) = 119868


119873int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

minus int

119909

minus1

119870 (119909 119904) 119890 (119904) 119889119904


1198697(119909) = 119868


119873int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

minus int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

1198698(119909) = 119868


119873(

1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579)

minus1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

(66)


119890120574

(119909) =1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

+1

Γ (120574)int

119909

minus1

1198701(119909 120579) int

120579

minus1

(120579 minus 120591)minus120583

119890120574

(120591) 119889120591 119889120579

+1

Γ (120574)int

119902119909+119902minus1

minus1

1198702(119909 120578) int

120578

minus1

(120578 minus 119911)minus120583

times 119890120574

(119911) 119889119911 119889120578 + 119867 (119909)

+ 1198691(119909) + 119869

2(119909) + 119869

3(119909) + 119869

4(119909)

+ 1198695(119909) + 119869

6(119909) + 119869

7(119909)

(67)

where

119867(119909) = int

119909

minus1

1198701(119909 120579) int

120579

minus1

(1198694(120579) + 119869

8(120579)) 119889120579

+1

Γ (120574)int

119902119909+119902minus1

minus1

1198702(119909 120578) int

120578

minus1

(1198694(120578) + 119869

8(120578)) 119889120578

+ 1198694(119909) + 119869

8(119909)

(68)


int

119909

minus1

int

120579

minus1

Φ (120579 120591) 119889120591 119889120579 = int

119909

minus1

int

119909

120591

Φ (120579 120591) 119889120579 119889120591 (69)


1

Γ (120574)int

119909

minus1

1198701(119909 120579) int

120579

minus1

(120579 minus 120591)minus120583

119890120574

(120591) 119889120591 119889120579

=1

Γ (120574)int

119909

minus1

(int

119909

120579

1198701(119909 120579) (120579 minus 120591)

minus120583

119889120579) 119890120574

(120591) 119889120591

le1

Γ (120574)21minus120583 max

119909120591isin[minus11]

100381610038161003816100381610038161(119909 120591)

10038161003816100381610038161003816int

119909

minus1

1003816100381610038161003816119890120574

(120591)1003816100381610038161003816 119889120591

≜ 1198721int

119909

minus1

1003816100381610038161003816119890120574

(120591)1003816100381610038161003816 119889120591

(70)


1

Γ (120574)int

119902119909+119902minus1

minus1

1198702(119909 120578) int

120578

minus1

(120578 minus 119911)minus120583

119890120574

(119911) 119889119911 119889120578

=1

Γ (120574)119902int

119909

minus1

1198702(119909 119902120591 + 119902 minus 1)

times (int

119902120591+119902minus1

minus1

(119909 minus 119911)minus120583

119890120574

(119911) 119889119911) 119889120591

=1

Γ (120574)1199021+120574

int

119909

minus1

1198702(119909 119902120591 + 119902 minus 1)

times (int

120591

minus1

(1 minus 120579)minus120583

119890120574

(119902120579 + 119902 minus 1) 119889120579)119889120591

=1

Γ (120574)1199021+120574

int

119909

minus1

(int

119909

120579

1198702(119909 119902120591 + 119902 minus 1)


minus120583

119889120591) 119890120574

times (119902120579 + 119902 minus 1) 119889120579

le1

Γ (120574)21minus120583

1199021+120574 max

119909120591isin[minus11]

10038161003816100381610038161198702(119909 120591)

1003816100381610038161003816

times int

119909

minus1

1003816100381610038161003816119890120574

(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579

≜ 1198722int

119909

minus1

1003816100381610038161003816119890120574

(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579

(71)

Then (67) gives

119890120574

(119909) le1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

+1198721int

119909

minus1

1003816100381610038161003816119890120574

(120591)1003816100381610038161003816 119889120591

+1198722int

119909

minus1

1003816100381610038161003816119890120574

(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579

+ 119867 (119909) + 1198691(119909) + 119869

2(119909) + 119869

3(119909)

+ 1198694(119909) + 119869

5(119909) + 119869

6(119909) + 119869

7(119909)

(72)


1003817100381710038171003817119890120574

(119909)1003817100381710038171003817119871infin(119868)

le

8

sum

119894=1

1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)

(73)

From (65) we have

119890 (119909)119871infin(119868)

le

8

sum

119894=1

1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)

(74)



100381710038171003817100381711986911003817100381710038171003817119871infin(119868)

le

119862119873(12)minus120583max

0le119894le119873

100381610038161003816100381611986811989411003816100381610038161003816 0 lt 120583 lt

1

2

119862 log119873max0le119894le119873

100381610038161003816100381611986811989411003816100381610038161003816

1

2le 120583 lt 1

le

119862119873(12)minus120583minus119898

119870lowast

1(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 0 lt 120583 lt1

2

119862119873minus119898 log119873119870

lowast

1(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 1

2le 120583 lt 1

100381710038171003817100381711986921003817100381710038171003817119871infin(119868)

le

119862119873(12)minus120583max

0le119894le119873

100381610038161003816100381611986811989421003816100381610038161003816 0 lt 120583 lt

1

2

119862 log119873max0le119894le119873

100381610038161003816100381611986811989411003816100381610038161003816

1

2le 120583 lt 1

le

119862119873(12)minus120583minus119898

119870lowast

2(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 0 lt 120583 lt1

2

119862119873minus119898 log119873119870

lowast

2(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 1

2le 120583 lt 1

(75)

Due to Lemma 2

100381710038171003817100381711986931003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus1198981003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868) 0 lt 120583 lt

1

2

119862119873(12)minus119898 log1198731003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

1

2le 120583 lt 1

100381710038171003817100381711986941003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus119898

|119906|119867119898119873

120596119888(119868) 0 lt 120583 lt

1

2

119862119873(12)minus119898 log119873|119906|

119867119898119873

120596119888(119868)

1

2le 120583 lt 1

100381710038171003817100381711986951003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus119898

|119906|119867119898119873

120596119888(119868) 0 lt 120583 lt

1

2

119862119873(12)minus119898 log119873|119906|

119867119898119873

120596119888(119868)

1

2le 120583 lt 1

(76)


100381710038171003817100381711986961003817100381710038171003817119871infin(119868)

le

119862119873minus120583

119890119871infin(119868) 0 lt 120583 lt

1

2

119862119873minus(12)

119890119871infin(119868)

1

2le 120583 lt 1

100381710038171003817100381711986971003817100381710038171003817119871infin(119868)

le

119862119873minus120583

119890119871infin(119868) 0 lt 120583 lt

1

2

119862119873minus(12)

119890119871infin(119868)

1

2le 120583 lt 1

(77)


5 and 6 with 119870(119909 120591) = 1Γ(120574) that

100381710038171003817100381711986951003817100381710038171003817119871infin(119868)

=10038171003817100381710038171003817(119868


119873minus 119868)M119890

12057410038171003817100381710038171003817119871infin(119868)

=10038171003817100381710038171003817(119868


119873minus 119868) (M119890

120574

minusT119873M119890

120574

)10038171003817100381710038171003817119871infin(119868)

le (1 +10038171003817100381710038171003817119868minus120583minus120583

119873

10038171003817100381710038171003817119871infin(119868))119862119873

minus1198961003817100381710038171003817M11989012057410038171003817100381710038170120581

le

119862119873(12)minus120583minus1205811003817100381710038171003817119890

1205741003817100381710038171003817119871infin(119868) 0 lt 120583 lt

1

2

119862119873minus120581 log1198731003817100381710038171003817119890

1205741003817100381710038171003817119871infin(119868)

1

2le 120583 lt 1

(78)


1


1

2

0 lt 120581 lt 1 minus 120583 when 1

2le 120583 lt 1

(79)


120574

(119909) minus 119906120574

(119909)1003817100381710038171003817119871infin(119868)

le

119862119873(12)minus120583minus119898

(119870lowast

1199061198712(119868)

+ 119873(12)U) 0 lt 120583 lt

1

2

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 119873(12)U)

1

2le 120583 lt 1

119880 (119909) minus 119906 (119909)119871infin(119868)

le

119862119873(12)minus120583minus119898

(119870lowast

1199061198712(119868)

+ 119873(12)U) 0 lt 120583 lt

1

2

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 119873(12)U)

1

2le 120583 lt 1

(80)





120574

(119909) minus 119906120574

(119909)10038171003817100381710038171198712120596minus120583minus120583

(119868)

le

119862119873minus119898

(1198811+ 119873

120574minus(12)minus120581

1198812+ 119873

120574minus120581U) 1

2lt 120574 lt 1

119862119873minus119898

(1198811+ 119873

minus120581 log1198731198812+ 119873

(12)minus120581 log119873U) 0 lt 120574 le1

2

119880(119909) minus 119906(119909)1198712

120596minus120583minus120583

(119868)

le

119862119873minus119898

(1198811+ 119873

120574minus(12)minus120581

1198812+ 119873

120574minus120581U) 1

2lt 120574 lt 1

119862119873minus119898

(1198811+ 119873

minus120581 log1198731198812+ 119873

(12)minus120581 log119873U) 0 lt 120574 le1

2

(81)



1198811= 119870

lowast

(1199061198712(119868)

+1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868))

1198812= 119870

lowast

1199061198712(119868)

(82)


100381710038171003817100381711989012057410038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862

8

sum

119894=1

100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583

(119868)

1198901198712

120596minus120583minus120583

(119868)le 119862

8

sum

119894=1

100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583

(119868)

(83)


10038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862 max

119909isin[minus11]

|119868 (119909)|

le 119862119873minus119898

119870lowast

1(119906

1198712(119868)

+ 119890119871infin(119868))

1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862 max

119909isin[minus11]

|119868 (119909)|

le 119862119873minus119898

119870lowast

2(119906

1198712(119868)

+ 119890119871infin(119868))

(84)


119890119871infin(119868)

le 119862(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868)) (85)

So that10038171003817100381710038171198691

10038171003817100381710038171198712120596minus120583minus120583

(119868)

le 119862119873minus119898

119870lowast

1(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868))

1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583

(119868)

le 119862119873minus119898

119870lowast

2(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868))

(86)

Due to Lemma 2 (33a)10038171003817100381710038171198693

10038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1198981003816100381610038161003816119863120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

1003817100381710038171003817119869410038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus119898

|119906|119867119898119873

120596119888(119868)

1003817100381710038171003817119869510038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus119898

|119906|119867119898119873

120596119888(119868)

(87)


1003817100381710038171003817119869610038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1

10038161003816100381610038161003816100381610038161003816

int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583

(119868)

le 119862119873minus1

1198901198712

120596minus120583minus120583

(119868)

1003817100381710038171003817119869710038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1

100381610038161003816100381610038161003816100381610038161003816

int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

1003816100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583

(119868)

le 119862119873minus1

1198901198712

120596minus120583minus120583

(119868)

(88)


1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583

(119868)=10038171003817100381710038171003817(119868


119873minus 119868)M119890

120574100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

=10038171003817100381710038171003817(119868


119873minus 119868) (M119890

120574

minusT119873119890120574

)100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

le10038171003817100381710038171003817119868minus120583minus120583

119873(M119890

120574

minusT119873119890120574

)100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

+1003817100381710038171003817M119890

120574


(119868)

le 1198621003817100381710038171003817M119890

120574

minusT1198731198901205741003817100381710038171003817119871infin(119868)

le 119862119873minus1205811003817100381710038171003817M119890

12057410038171003817100381710038170120581

le 119862119873minus1205811003817100381710038171003817119890

1205741003817100381710038171003817119871infin(119868)

(89)


1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583

(119868)

le


(119870lowast

1199061198712(119868)

+ 11987312U) 0 lt 120583 lt

1

2

119862119873minus119898minus120581 log119873(119870

lowast

1199061198712(119868)

+ 11987312U)

1

2le 120583 lt 1

(90)





119863075

119910 (119905) =6119905225

Γ (325)minus1199052

119890119905

5119910 (119905) minus

(119902119905)2

5119910 (119902119905)

+ int

119905

0

119890119905

119904119910 (119904) 119889119904 + int

119902119905

0

120591119910 (120591) 119889120591 119905 isin [0 1]

119910 (0) = 0

(91)




0 02 04 06 08 10

02

04

06

08

1

12

14


(a)

0 02 04 06 08 10

05

1

15

2

25


(b)


119910(119905) (b)

2 4 6 8 10 12 14 16

100

10minus5

10minus10

2 le N le 16

Linfin

L2

120596minus120583minus120583

-error-error

(a)

2 4 6 8 10 12 14 16

100

10minus5

10minus10

2 le N le 16

Linfin

L2

120596minus120583minus120583

-error-error

(b)



infin and 1198712

120596norms


infin and 1198712




119863120572

119910 (119905) = 2119905 minus 119910 (119905) + 119910 (119902119905) + 119905 (1 + 2119905) int

119905

0

119890119904(119905minus119904)

119910 (119904) 119889119904

+ int

119902119905

0

119902119905119890120591(119902119905minus120591)

119910 (120591) 119889120591 119905 isin [0 1]

119910 (0) = 1

(92)



0 02 04 06 08 105

1

15

2

25

3

35

4


(a)

0 02 04 06 08 10

1

2

3

4

5

6


(b)


2 4 6 8 10 12 14 16 18 20

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

2 le N le 20

Linfin

L2

120596

(a)

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

2 4 6 8 10 12 14 16 18 202 le N le 20

Linfin

L2

120596

(b)



infin and 1198712

120596norms



1199052








Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119898

(119868)with 119868 for some119898 ge


N such that100381610038161003816100381610038161003816100381610038161003816

int

1

minus1

119906 (119909) 120593 (119909) 119889119909 minus (119906 120593)119873

100381610038161003816100381610038161003816100381610038161003816

le 119862119873minus119898

|119906|119867119898119873

120596120572120573(119868)

100381710038171003817100381712059310038171003817100381710038171198712120596120572120573(119868)

(30)

where

|119906|119867119898119873

120596120572120573(119868)

= (

119898

sum

119895=min(119898119873+1)

10038171003817100381710038171003817119906(119895)10038171003817100381710038171003817

2

1198712

120596120572120573(119868)

)

12

(119906 120593)119873=

119873

sum

119895=0

119906 (119909119895) 120593 (119909

119895) 120596

119895

(31)


120596minus120583minus120583




119895=0 namely

119868minus120583minus120583

119873119906 =

119873

sum

119894=0

119906 (119909119894) 119865 (119909

119894) (32)


10038171003817100381710038171003817119906 minus 119868


119873119906100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

le 119862119873minus119898

|119906|119867119898119873

120596120572120573(119868) (33a)

10038171003817100381710038171003817119906 minus 119868


11987311990610038171003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus119898

|119906|119867119898119873

120596119888(119868) 0 le 120583 lt

1

2

11986211987312minus119898 log119873|119906|

119867119898119873

120596119888(119868)

1

2le 120583 lt 1

(33b)



119873

119895=0are the 119873-th


100381710038171003817100381710038171003817119868120572120573

119873

100381710038171003817100381710038171003817119871infin(119868)le max

119909isin[minus11]

119873

sum

119895=0

10038161003816100381610038161003816119865119895(119909)

10038161003816100381610038161003816

=


2

O (119873120574+12

) 120574 = max (120572 120573) 119900119905ℎ119890119903119908119894119904119890(34)


119906 (119909) le V (119909) + 119871int

119909

minus1

(119909 minus 120591)minus120583

119906 (120591) 119889120591 (35)


119906 (119909) le V (119909) + 119862119871int

119909

minus1


(36)




119903120581


P119873such that


119903120581119873minus(119903+120581)

V119903120581 (37)

where sdot 119903120581



V119903120581

= max0le120581le119903


1003816100381610038161003816120597120581

119909V (119909)1003816100381610038161003816

+ max0le120581le119903

sup119909119910isin[minus11]119909 = 119910

1003816100381610038161003816120597120581

119909V (119909) minus 120597

120581

119909V (119909)1003816100381610038161003816

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

120581

(38)


119903120581



(MV) (119909) = int

119909

minus1

(119909 minus 120591)minus120583

119870 (119909 120591) V (120591) 119889120591 (39)


10038161003816100381610038161003816MV (1199091015840) minusMV (11990910158401015840)

1003816100381610038161003816100381610038161003816100381610038161199091015840 minus 11990910158401015840

1003816100381610038161003816


|V (119909)| (40)


11990910158401015840

isin


MV0120581


|V (119909)| 0 lt 120581 lt 1 minus 120583 (41)


sup119873

10038171003817100381710038171003817100381710038171003817100381710038171003817

119873

sum

119895=0

V (119909119895) 119865

119895(119909)

100381710038171003817100381710038171003817100381710038171003817100381710038171198712120596120572120573(119868)


|V (119909)| (42)



119895119873

119895=0


(int

119887

119886

1003816100381610038161003816(119879119891) (119909)1003816100381610038161003816

119902

119906 (119909) 119889119909)

1119902

le (int

119887

119886

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901

V (119909) 119889119909)1119901

(43)



sup119886lt119909lt119887

(int

119887

119909

119906 (119905) 119889119905)

1119902

(int

119909

119886

V1minus1199011015840

(119905) 119889119905)

11199011015840

lt infin 1199011015840

=119901

119901 minus 1

(44)


(119879119865) (119909) = int

119909

119886

119896 (119909 119905) 119891 (119905) 119889119905 (45)




infin space


120574


119898+1

120596minus120583minus120583(119868) then

1003817100381710038171003817119880120574

minus 119863120574

1199061003817100381710038171003817119871infin(119868)

le

119862119873120574minus12minus119898

(119870lowast

1199061198712(119868)

+ 11987312U)

1

2lt 120574 lt 1

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 11987312U) 0 lt 120574 le

1

2

(46)

119880 minus 119906119871infin(119868)

le

119862119873120574minus12minus119898

(119870lowast

1199061198712(119868)

+ 11987312U)

1

2lt 120574 lt 1

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 11987312U) 0 lt 120574 le

1

2

(47)


119870lowast


10038161003816100381610038161198701(119909 120579 (119909 sdot))

100381610038161003816100381611989811987312059600(119868)


10038161003816100381610038161198702(119909 120578 (119909 sdot))

100381610038161003816100381611989811987312059600(119868)

(48)

U =1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868) (49)

Proof We let

(119870 (119904 (119909119894 ])) 120601 (119904 (119909

119894 ])))

119873

=

119873

sum

119896=0

119870(119904 (119909119894 ]119896)) 120601 (119904 (119909

119894 ]119896)) 120596

00

119896

(50)


119906120574

119894= 119906

119894+ (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1)) + 119892 (119909


119894)

+1 + 119909


119894

2(119870

1(119909


119894 120579 (119909


119894 ]))

119880 (120579 (119909minus120583minus120583

119894 ])))

119873

+

119902 (1 + 119909minus120583minus120583

119894)

2(119870

2(119909


119894 120578 (119909


119894 ]))

119880 (120578 (119909minus120583minus120583

119894 ])))

119873

(51)

119906119894=

1

Γ (120574)(119879

2)

120574

int

119909119894

minus1

(119909119894minus 120579)

minus120583

119880120574

(120579) 119889120579 + 119906minus1 (52)

which gives

119906120574

119894= 119906

119894+ (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1)) + 119892 (119909


119894)

+1 + 119909


119894

2int

1

minus1

1198701(119909


119894 120579 (119909


119894 ]))


119894 ])) 119889] + 119868

1(119909


119894)

+

119902 (1 + 119909minus120583minus120583

119894)

2int

1

minus1

1198702(119909


119894 120578 (119909


119894 120585))


119894 120585)) 119889120585 + 119868

2(119909


119894)

(53)

where

1198681(119909) =

1 + 119909

2int

1

minus1

1198701(119909 120579 (119909)) 119880 (120579 (119909 ])) 119889]

minus1 + 119909

2(119870

1(119909 120579 (119909 ])) 119880 (120579 (119909 ])))

119873

1198682(119909) =

119902 (1 + 119909)

2int

1

minus1

1198702(119909 120578 (119909 120585)) 119880 (120578 (119909 120585)) 119889120585

minus119902 (1 + 119909)

2(119870

2(119909 120578 (119909 ])) 119880 (120578 (119909 ])))

119873

(54)



10038161003816100381610038161198681 (119909)1003816100381610038161003816 le 119862119873

minus119898100381610038161003816100381610038161(119909 120579 (119909 sdot))

1003816100381610038161003816100381611986711989811987312059600(119868)

119880 (120579 (119909 sdot))1198712(119868)

le 119862119873minus119898

119870lowast

1119880

1198712(119868)

(55)

10038161003816100381610038161198682 (119909)1003816100381610038161003816 le 119862119873

minus119898100381610038161003816100381610038162(119909 120578 (119909 sdot))

1003816100381610038161003816100381611986711989811987312059600(119868)

1003817100381710038171003817119880 (120578 (119909 sdot))10038171003817100381710038171198712(119868)

le 119862119873minus119898

119870lowast

2119880

1198712(119868)

(56)

119870lowast

1= max

119909isin[minus11]

10038161003816100381610038161198701(119909 120579 (119909 sdot))

100381610038161003816100381611989811987312059600(119868)

119870lowast

2= max

119909isin[minus11]

10038161003816100381610038161198701(119909 120579 (119909 sdot))

100381610038161003816100381611989811987312059600(119868)

(57)


119906120574

119894= 119886 (119909


119894) 119906

119894+ 119887 (119909


119894)

times (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1)) + 119892 (119909


119894)

+ int

119909minus120583minus120583

119894

minus1

1198701(119909


119894 120579)119880 (120579) 119889120579 + 119868

1(119909


119894)

+ int

119902119909minus120583minus120583

119894+119902minus1

minus1

1198702(119909


119894 120578)119880 (120578) 119889120578

+ 1198682(119909


119894)

(58)


119890 (119909) = 119880 (119909) minus 119906 (119909) 119890120574

(119909) = 119880120574

(119909) minus 119906120574

(119909)

(59)

we have

119906120574

119894= 119886 (119909


119894) 119906

119894+ 119887 (119909


119894)

times (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1)) + 119892 (119909


119894)

+ int

119909minus120583minus120583

119894

minus1

1198701(119909


119894 120579) 119906 (120579) 119889120579

+ int

119909minus120583minus120583

119894

minus1

1198701(119909


119894 120579) 119890 (120579) 119889120579

+ int

119902119909minus120583minus120583

119894+119902minus1

minus1

1198702(119909


119894 120578) 119906 (120578) 119889120578

+ int

119902119909minus120583minus120583

119894+119902minus1

minus1

1198702(119909


119894 120578) 119890 (120578) 119889120578

+ 1198681(119909


119894) + 119868

2(119909


119894)

(60)



119880120574

(119909) = 119880 (119909) + 119868minus120583minus120583

119873(

119873

sum

119895=0

119906119895119865119895(119902119909 + 119902 minus 1))

+ 119868minus120583minus120583

119873119892 (119909)

+ 119868minus120583minus120583

119873[119863

120574

119906 (119909) minus 119906 (119909)


+ 119868minus120583minus120583

119873int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

+ 119868minus120583minus120583

119873int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

+ 1198691(119909) + 119869

2(119909)

(61)

119880 (119909) = 119868minus120583minus120583

119873(119906 (119909) minus 119906

minus1)

+ 119868minus120583minus120583

119873(

1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579)

+ 119868minus120583minus120583

119873(119906

minus1)

(62)

where

1198691(119909) = 119868


1198731198681(119909) 119869

2(119909) = 119868


1198731198682(119909) (63)


119890120574

(119909) = 119890 (119909) + int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

+ int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

+ 1198691(119909) + 119869

2(119909) + 119869

3(119909) + 119869

4(119909)

+ 1198695(119909) + 119869

6(119909) + 119869

7(119909)

(64)

119890 (119909) =1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

+ 1198694(119909) + 119869

8(119909)

(65)

where

1198693(119909) = 119868


119873119863120574

119906 (119909) minus 119863120574

119906 (119909)

1198694(119909) = 119868


119873119906 (119909) minus 119906 (119909)

1198695(119909) = 119868



1198696(119909) = 119868


119873int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

minus int

119909

minus1

119870 (119909 119904) 119890 (119904) 119889119904


1198697(119909) = 119868


119873int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

minus int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

1198698(119909) = 119868


119873(

1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579)

minus1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

(66)


119890120574

(119909) =1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

+1

Γ (120574)int

119909

minus1

1198701(119909 120579) int

120579

minus1

(120579 minus 120591)minus120583

119890120574

(120591) 119889120591 119889120579

+1

Γ (120574)int

119902119909+119902minus1

minus1

1198702(119909 120578) int

120578

minus1

(120578 minus 119911)minus120583

times 119890120574

(119911) 119889119911 119889120578 + 119867 (119909)

+ 1198691(119909) + 119869

2(119909) + 119869

3(119909) + 119869

4(119909)

+ 1198695(119909) + 119869

6(119909) + 119869

7(119909)

(67)

where

119867(119909) = int

119909

minus1

1198701(119909 120579) int

120579

minus1

(1198694(120579) + 119869

8(120579)) 119889120579

+1

Γ (120574)int

119902119909+119902minus1

minus1

1198702(119909 120578) int

120578

minus1

(1198694(120578) + 119869

8(120578)) 119889120578

+ 1198694(119909) + 119869

8(119909)

(68)


int

119909

minus1

int

120579

minus1

Φ (120579 120591) 119889120591 119889120579 = int

119909

minus1

int

119909

120591

Φ (120579 120591) 119889120579 119889120591 (69)


1

Γ (120574)int

119909

minus1

1198701(119909 120579) int

120579

minus1

(120579 minus 120591)minus120583

119890120574

(120591) 119889120591 119889120579

=1

Γ (120574)int

119909

minus1

(int

119909

120579

1198701(119909 120579) (120579 minus 120591)

minus120583

119889120579) 119890120574

(120591) 119889120591

le1

Γ (120574)21minus120583 max

119909120591isin[minus11]

100381610038161003816100381610038161(119909 120591)

10038161003816100381610038161003816int

119909

minus1

1003816100381610038161003816119890120574

(120591)1003816100381610038161003816 119889120591

≜ 1198721int

119909

minus1

1003816100381610038161003816119890120574

(120591)1003816100381610038161003816 119889120591

(70)


1

Γ (120574)int

119902119909+119902minus1

minus1

1198702(119909 120578) int

120578

minus1

(120578 minus 119911)minus120583

119890120574

(119911) 119889119911 119889120578

=1

Γ (120574)119902int

119909

minus1

1198702(119909 119902120591 + 119902 minus 1)

times (int

119902120591+119902minus1

minus1

(119909 minus 119911)minus120583

119890120574

(119911) 119889119911) 119889120591

=1

Γ (120574)1199021+120574

int

119909

minus1

1198702(119909 119902120591 + 119902 minus 1)

times (int

120591

minus1

(1 minus 120579)minus120583

119890120574

(119902120579 + 119902 minus 1) 119889120579)119889120591

=1

Γ (120574)1199021+120574

int

119909

minus1

(int

119909

120579

1198702(119909 119902120591 + 119902 minus 1)


minus120583

119889120591) 119890120574

times (119902120579 + 119902 minus 1) 119889120579

le1

Γ (120574)21minus120583

1199021+120574 max

119909120591isin[minus11]

10038161003816100381610038161198702(119909 120591)

1003816100381610038161003816

times int

119909

minus1

1003816100381610038161003816119890120574

(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579

≜ 1198722int

119909

minus1

1003816100381610038161003816119890120574

(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579

(71)

Then (67) gives

119890120574

(119909) le1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

+1198721int

119909

minus1

1003816100381610038161003816119890120574

(120591)1003816100381610038161003816 119889120591

+1198722int

119909

minus1

1003816100381610038161003816119890120574

(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579

+ 119867 (119909) + 1198691(119909) + 119869

2(119909) + 119869

3(119909)

+ 1198694(119909) + 119869

5(119909) + 119869

6(119909) + 119869

7(119909)

(72)


1003817100381710038171003817119890120574

(119909)1003817100381710038171003817119871infin(119868)

le

8

sum

119894=1

1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)

(73)

From (65) we have

119890 (119909)119871infin(119868)

le

8

sum

119894=1

1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)

(74)



100381710038171003817100381711986911003817100381710038171003817119871infin(119868)

le

119862119873(12)minus120583max

0le119894le119873

100381610038161003816100381611986811989411003816100381610038161003816 0 lt 120583 lt

1

2

119862 log119873max0le119894le119873

100381610038161003816100381611986811989411003816100381610038161003816

1

2le 120583 lt 1

le

119862119873(12)minus120583minus119898

119870lowast

1(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 0 lt 120583 lt1

2

119862119873minus119898 log119873119870

lowast

1(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 1

2le 120583 lt 1

100381710038171003817100381711986921003817100381710038171003817119871infin(119868)

le

119862119873(12)minus120583max

0le119894le119873

100381610038161003816100381611986811989421003816100381610038161003816 0 lt 120583 lt

1

2

119862 log119873max0le119894le119873

100381610038161003816100381611986811989411003816100381610038161003816

1

2le 120583 lt 1

le

119862119873(12)minus120583minus119898

119870lowast

2(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 0 lt 120583 lt1

2

119862119873minus119898 log119873119870

lowast

2(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 1

2le 120583 lt 1

(75)

Due to Lemma 2

100381710038171003817100381711986931003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus1198981003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868) 0 lt 120583 lt

1

2

119862119873(12)minus119898 log1198731003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

1

2le 120583 lt 1

100381710038171003817100381711986941003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus119898

|119906|119867119898119873

120596119888(119868) 0 lt 120583 lt

1

2

119862119873(12)minus119898 log119873|119906|

119867119898119873

120596119888(119868)

1

2le 120583 lt 1

100381710038171003817100381711986951003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus119898

|119906|119867119898119873

120596119888(119868) 0 lt 120583 lt

1

2

119862119873(12)minus119898 log119873|119906|

119867119898119873

120596119888(119868)

1

2le 120583 lt 1

(76)


100381710038171003817100381711986961003817100381710038171003817119871infin(119868)

le

119862119873minus120583

119890119871infin(119868) 0 lt 120583 lt

1

2

119862119873minus(12)

119890119871infin(119868)

1

2le 120583 lt 1

100381710038171003817100381711986971003817100381710038171003817119871infin(119868)

le

119862119873minus120583

119890119871infin(119868) 0 lt 120583 lt

1

2

119862119873minus(12)

119890119871infin(119868)

1

2le 120583 lt 1

(77)


5 and 6 with 119870(119909 120591) = 1Γ(120574) that

100381710038171003817100381711986951003817100381710038171003817119871infin(119868)

=10038171003817100381710038171003817(119868


119873minus 119868)M119890

12057410038171003817100381710038171003817119871infin(119868)

=10038171003817100381710038171003817(119868


119873minus 119868) (M119890

120574

minusT119873M119890

120574

)10038171003817100381710038171003817119871infin(119868)

le (1 +10038171003817100381710038171003817119868minus120583minus120583

119873

10038171003817100381710038171003817119871infin(119868))119862119873

minus1198961003817100381710038171003817M11989012057410038171003817100381710038170120581

le

119862119873(12)minus120583minus1205811003817100381710038171003817119890

1205741003817100381710038171003817119871infin(119868) 0 lt 120583 lt

1

2

119862119873minus120581 log1198731003817100381710038171003817119890

1205741003817100381710038171003817119871infin(119868)

1

2le 120583 lt 1

(78)


1


1

2

0 lt 120581 lt 1 minus 120583 when 1

2le 120583 lt 1

(79)


120574

(119909) minus 119906120574

(119909)1003817100381710038171003817119871infin(119868)

le

119862119873(12)minus120583minus119898

(119870lowast

1199061198712(119868)

+ 119873(12)U) 0 lt 120583 lt

1

2

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 119873(12)U)

1

2le 120583 lt 1

119880 (119909) minus 119906 (119909)119871infin(119868)

le

119862119873(12)minus120583minus119898

(119870lowast

1199061198712(119868)

+ 119873(12)U) 0 lt 120583 lt

1

2

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 119873(12)U)

1

2le 120583 lt 1

(80)





120574

(119909) minus 119906120574

(119909)10038171003817100381710038171198712120596minus120583minus120583

(119868)

le

119862119873minus119898

(1198811+ 119873

120574minus(12)minus120581

1198812+ 119873

120574minus120581U) 1

2lt 120574 lt 1

119862119873minus119898

(1198811+ 119873

minus120581 log1198731198812+ 119873

(12)minus120581 log119873U) 0 lt 120574 le1

2

119880(119909) minus 119906(119909)1198712

120596minus120583minus120583

(119868)

le

119862119873minus119898

(1198811+ 119873

120574minus(12)minus120581

1198812+ 119873

120574minus120581U) 1

2lt 120574 lt 1

119862119873minus119898

(1198811+ 119873

minus120581 log1198731198812+ 119873

(12)minus120581 log119873U) 0 lt 120574 le1

2

(81)



1198811= 119870

lowast

(1199061198712(119868)

+1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868))

1198812= 119870

lowast

1199061198712(119868)

(82)


100381710038171003817100381711989012057410038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862

8

sum

119894=1

100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583

(119868)

1198901198712

120596minus120583minus120583

(119868)le 119862

8

sum

119894=1

100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583

(119868)

(83)


10038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862 max

119909isin[minus11]

|119868 (119909)|

le 119862119873minus119898

119870lowast

1(119906

1198712(119868)

+ 119890119871infin(119868))

1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862 max

119909isin[minus11]

|119868 (119909)|

le 119862119873minus119898

119870lowast

2(119906

1198712(119868)

+ 119890119871infin(119868))

(84)


119890119871infin(119868)

le 119862(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868)) (85)

So that10038171003817100381710038171198691

10038171003817100381710038171198712120596minus120583minus120583

(119868)

le 119862119873minus119898

119870lowast

1(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868))

1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583

(119868)

le 119862119873minus119898

119870lowast

2(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868))

(86)

Due to Lemma 2 (33a)10038171003817100381710038171198693

10038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1198981003816100381610038161003816119863120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

1003817100381710038171003817119869410038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus119898

|119906|119867119898119873

120596119888(119868)

1003817100381710038171003817119869510038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus119898

|119906|119867119898119873

120596119888(119868)

(87)


1003817100381710038171003817119869610038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1

10038161003816100381610038161003816100381610038161003816

int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583

(119868)

le 119862119873minus1

1198901198712

120596minus120583minus120583

(119868)

1003817100381710038171003817119869710038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1

100381610038161003816100381610038161003816100381610038161003816

int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

1003816100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583

(119868)

le 119862119873minus1

1198901198712

120596minus120583minus120583

(119868)

(88)


1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583

(119868)=10038171003817100381710038171003817(119868


119873minus 119868)M119890

120574100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

=10038171003817100381710038171003817(119868


119873minus 119868) (M119890

120574

minusT119873119890120574

)100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

le10038171003817100381710038171003817119868minus120583minus120583

119873(M119890

120574

minusT119873119890120574

)100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

+1003817100381710038171003817M119890

120574


(119868)

le 1198621003817100381710038171003817M119890

120574

minusT1198731198901205741003817100381710038171003817119871infin(119868)

le 119862119873minus1205811003817100381710038171003817M119890

12057410038171003817100381710038170120581

le 119862119873minus1205811003817100381710038171003817119890

1205741003817100381710038171003817119871infin(119868)

(89)


1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583

(119868)

le


(119870lowast

1199061198712(119868)

+ 11987312U) 0 lt 120583 lt

1

2

119862119873minus119898minus120581 log119873(119870

lowast

1199061198712(119868)

+ 11987312U)

1

2le 120583 lt 1

(90)





119863075

119910 (119905) =6119905225

Γ (325)minus1199052

119890119905

5119910 (119905) minus

(119902119905)2

5119910 (119902119905)

+ int

119905

0

119890119905

119904119910 (119904) 119889119904 + int

119902119905

0

120591119910 (120591) 119889120591 119905 isin [0 1]

119910 (0) = 0

(91)




0 02 04 06 08 10

02

04

06

08

1

12

14


(a)

0 02 04 06 08 10

05

1

15

2

25


(b)


119910(119905) (b)

2 4 6 8 10 12 14 16

100

10minus5

10minus10

2 le N le 16

Linfin

L2

120596minus120583minus120583

-error-error

(a)

2 4 6 8 10 12 14 16

100

10minus5

10minus10

2 le N le 16

Linfin

L2

120596minus120583minus120583

-error-error

(b)



infin and 1198712

120596norms


infin and 1198712




119863120572

119910 (119905) = 2119905 minus 119910 (119905) + 119910 (119902119905) + 119905 (1 + 2119905) int

119905

0

119890119904(119905minus119904)

119910 (119904) 119889119904

+ int

119902119905

0

119902119905119890120591(119902119905minus120591)

119910 (120591) 119889120591 119905 isin [0 1]

119910 (0) = 1

(92)



0 02 04 06 08 105

1

15

2

25

3

35

4


(a)

0 02 04 06 08 10

1

2

3

4

5

6


(b)


2 4 6 8 10 12 14 16 18 20

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

2 le N le 20

Linfin

L2

120596

(a)

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

2 4 6 8 10 12 14 16 18 202 le N le 20

Linfin

L2

120596

(b)



infin and 1198712

120596norms



1199052








Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











sup119886lt119909lt119887

(int

119887

119909

119906 (119905) 119889119905)

1119902

(int

119909

119886

V1minus1199011015840

(119905) 119889119905)

11199011015840

lt infin 1199011015840

=119901

119901 minus 1

(44)


(119879119865) (119909) = int

119909

119886

119896 (119909 119905) 119891 (119905) 119889119905 (45)




infin space


120574


119898+1

120596minus120583minus120583(119868) then

1003817100381710038171003817119880120574

minus 119863120574

1199061003817100381710038171003817119871infin(119868)

le

119862119873120574minus12minus119898

(119870lowast

1199061198712(119868)

+ 11987312U)

1

2lt 120574 lt 1

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 11987312U) 0 lt 120574 le

1

2

(46)

119880 minus 119906119871infin(119868)

le

119862119873120574minus12minus119898

(119870lowast

1199061198712(119868)

+ 11987312U)

1

2lt 120574 lt 1

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 11987312U) 0 lt 120574 le

1

2

(47)


119870lowast


10038161003816100381610038161198701(119909 120579 (119909 sdot))

100381610038161003816100381611989811987312059600(119868)


10038161003816100381610038161198702(119909 120578 (119909 sdot))

100381610038161003816100381611989811987312059600(119868)

(48)

U =1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868) (49)

Proof We let

(119870 (119904 (119909119894 ])) 120601 (119904 (119909

119894 ])))

119873

=

119873

sum

119896=0

119870(119904 (119909119894 ]119896)) 120601 (119904 (119909

119894 ]119896)) 120596

00

119896

(50)


119906120574

119894= 119906

119894+ (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1)) + 119892 (119909


119894)

+1 + 119909


119894

2(119870

1(119909


119894 120579 (119909


119894 ]))

119880 (120579 (119909minus120583minus120583

119894 ])))

119873

+

119902 (1 + 119909minus120583minus120583

119894)

2(119870

2(119909


119894 120578 (119909


119894 ]))

119880 (120578 (119909minus120583minus120583

119894 ])))

119873

(51)

119906119894=

1

Γ (120574)(119879

2)

120574

int

119909119894

minus1

(119909119894minus 120579)

minus120583

119880120574

(120579) 119889120579 + 119906minus1 (52)

which gives

119906120574

119894= 119906

119894+ (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1)) + 119892 (119909


119894)

+1 + 119909


119894

2int

1

minus1

1198701(119909


119894 120579 (119909


119894 ]))


119894 ])) 119889] + 119868

1(119909


119894)

+

119902 (1 + 119909minus120583minus120583

119894)

2int

1

minus1

1198702(119909


119894 120578 (119909


119894 120585))


119894 120585)) 119889120585 + 119868

2(119909


119894)

(53)

where

1198681(119909) =

1 + 119909

2int

1

minus1

1198701(119909 120579 (119909)) 119880 (120579 (119909 ])) 119889]

minus1 + 119909

2(119870

1(119909 120579 (119909 ])) 119880 (120579 (119909 ])))

119873

1198682(119909) =

119902 (1 + 119909)

2int

1

minus1

1198702(119909 120578 (119909 120585)) 119880 (120578 (119909 120585)) 119889120585

minus119902 (1 + 119909)

2(119870

2(119909 120578 (119909 ])) 119880 (120578 (119909 ])))

119873

(54)



10038161003816100381610038161198681 (119909)1003816100381610038161003816 le 119862119873

minus119898100381610038161003816100381610038161(119909 120579 (119909 sdot))

1003816100381610038161003816100381611986711989811987312059600(119868)

119880 (120579 (119909 sdot))1198712(119868)

le 119862119873minus119898

119870lowast

1119880

1198712(119868)

(55)

10038161003816100381610038161198682 (119909)1003816100381610038161003816 le 119862119873

minus119898100381610038161003816100381610038162(119909 120578 (119909 sdot))

1003816100381610038161003816100381611986711989811987312059600(119868)

1003817100381710038171003817119880 (120578 (119909 sdot))10038171003817100381710038171198712(119868)

le 119862119873minus119898

119870lowast

2119880

1198712(119868)

(56)

119870lowast

1= max

119909isin[minus11]

10038161003816100381610038161198701(119909 120579 (119909 sdot))

100381610038161003816100381611989811987312059600(119868)

119870lowast

2= max

119909isin[minus11]

10038161003816100381610038161198701(119909 120579 (119909 sdot))

100381610038161003816100381611989811987312059600(119868)

(57)


119906120574

119894= 119886 (119909


119894) 119906

119894+ 119887 (119909


119894)

times (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1)) + 119892 (119909


119894)

+ int

119909minus120583minus120583

119894

minus1

1198701(119909


119894 120579)119880 (120579) 119889120579 + 119868

1(119909


119894)

+ int

119902119909minus120583minus120583

119894+119902minus1

minus1

1198702(119909


119894 120578)119880 (120578) 119889120578

+ 1198682(119909


119894)

(58)


119890 (119909) = 119880 (119909) minus 119906 (119909) 119890120574

(119909) = 119880120574

(119909) minus 119906120574

(119909)

(59)

we have

119906120574

119894= 119886 (119909


119894) 119906

119894+ 119887 (119909


119894)

times (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1)) + 119892 (119909


119894)

+ int

119909minus120583minus120583

119894

minus1

1198701(119909


119894 120579) 119906 (120579) 119889120579

+ int

119909minus120583minus120583

119894

minus1

1198701(119909


119894 120579) 119890 (120579) 119889120579

+ int

119902119909minus120583minus120583

119894+119902minus1

minus1

1198702(119909


119894 120578) 119906 (120578) 119889120578

+ int

119902119909minus120583minus120583

119894+119902minus1

minus1

1198702(119909


119894 120578) 119890 (120578) 119889120578

+ 1198681(119909


119894) + 119868

2(119909


119894)

(60)



119880120574

(119909) = 119880 (119909) + 119868minus120583minus120583

119873(

119873

sum

119895=0

119906119895119865119895(119902119909 + 119902 minus 1))

+ 119868minus120583minus120583

119873119892 (119909)

+ 119868minus120583minus120583

119873[119863

120574

119906 (119909) minus 119906 (119909)


+ 119868minus120583minus120583

119873int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

+ 119868minus120583minus120583

119873int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

+ 1198691(119909) + 119869

2(119909)

(61)

119880 (119909) = 119868minus120583minus120583

119873(119906 (119909) minus 119906

minus1)

+ 119868minus120583minus120583

119873(

1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579)

+ 119868minus120583minus120583

119873(119906

minus1)

(62)

where

1198691(119909) = 119868


1198731198681(119909) 119869

2(119909) = 119868


1198731198682(119909) (63)


119890120574

(119909) = 119890 (119909) + int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

+ int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

+ 1198691(119909) + 119869

2(119909) + 119869

3(119909) + 119869

4(119909)

+ 1198695(119909) + 119869

6(119909) + 119869

7(119909)

(64)

119890 (119909) =1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

+ 1198694(119909) + 119869

8(119909)

(65)

where

1198693(119909) = 119868


119873119863120574

119906 (119909) minus 119863120574

119906 (119909)

1198694(119909) = 119868


119873119906 (119909) minus 119906 (119909)

1198695(119909) = 119868



1198696(119909) = 119868


119873int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

minus int

119909

minus1

119870 (119909 119904) 119890 (119904) 119889119904


1198697(119909) = 119868


119873int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

minus int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

1198698(119909) = 119868


119873(

1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579)

minus1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

(66)


119890120574

(119909) =1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

+1

Γ (120574)int

119909

minus1

1198701(119909 120579) int

120579

minus1

(120579 minus 120591)minus120583

119890120574

(120591) 119889120591 119889120579

+1

Γ (120574)int

119902119909+119902minus1

minus1

1198702(119909 120578) int

120578

minus1

(120578 minus 119911)minus120583

times 119890120574

(119911) 119889119911 119889120578 + 119867 (119909)

+ 1198691(119909) + 119869

2(119909) + 119869

3(119909) + 119869

4(119909)

+ 1198695(119909) + 119869

6(119909) + 119869

7(119909)

(67)

where

119867(119909) = int

119909

minus1

1198701(119909 120579) int

120579

minus1

(1198694(120579) + 119869

8(120579)) 119889120579

+1

Γ (120574)int

119902119909+119902minus1

minus1

1198702(119909 120578) int

120578

minus1

(1198694(120578) + 119869

8(120578)) 119889120578

+ 1198694(119909) + 119869

8(119909)

(68)


int

119909

minus1

int

120579

minus1

Φ (120579 120591) 119889120591 119889120579 = int

119909

minus1

int

119909

120591

Φ (120579 120591) 119889120579 119889120591 (69)


1

Γ (120574)int

119909

minus1

1198701(119909 120579) int

120579

minus1

(120579 minus 120591)minus120583

119890120574

(120591) 119889120591 119889120579

=1

Γ (120574)int

119909

minus1

(int

119909

120579

1198701(119909 120579) (120579 minus 120591)

minus120583

119889120579) 119890120574

(120591) 119889120591

le1

Γ (120574)21minus120583 max

119909120591isin[minus11]

100381610038161003816100381610038161(119909 120591)

10038161003816100381610038161003816int

119909

minus1

1003816100381610038161003816119890120574

(120591)1003816100381610038161003816 119889120591

≜ 1198721int

119909

minus1

1003816100381610038161003816119890120574

(120591)1003816100381610038161003816 119889120591

(70)


1

Γ (120574)int

119902119909+119902minus1

minus1

1198702(119909 120578) int

120578

minus1

(120578 minus 119911)minus120583

119890120574

(119911) 119889119911 119889120578

=1

Γ (120574)119902int

119909

minus1

1198702(119909 119902120591 + 119902 minus 1)

times (int

119902120591+119902minus1

minus1

(119909 minus 119911)minus120583

119890120574

(119911) 119889119911) 119889120591

=1

Γ (120574)1199021+120574

int

119909

minus1

1198702(119909 119902120591 + 119902 minus 1)

times (int

120591

minus1

(1 minus 120579)minus120583

119890120574

(119902120579 + 119902 minus 1) 119889120579)119889120591

=1

Γ (120574)1199021+120574

int

119909

minus1

(int

119909

120579

1198702(119909 119902120591 + 119902 minus 1)


minus120583

119889120591) 119890120574

times (119902120579 + 119902 minus 1) 119889120579

le1

Γ (120574)21minus120583

1199021+120574 max

119909120591isin[minus11]

10038161003816100381610038161198702(119909 120591)

1003816100381610038161003816

times int

119909

minus1

1003816100381610038161003816119890120574

(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579

≜ 1198722int

119909

minus1

1003816100381610038161003816119890120574

(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579

(71)

Then (67) gives

119890120574

(119909) le1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

+1198721int

119909

minus1

1003816100381610038161003816119890120574

(120591)1003816100381610038161003816 119889120591

+1198722int

119909

minus1

1003816100381610038161003816119890120574

(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579

+ 119867 (119909) + 1198691(119909) + 119869

2(119909) + 119869

3(119909)

+ 1198694(119909) + 119869

5(119909) + 119869

6(119909) + 119869

7(119909)

(72)


1003817100381710038171003817119890120574

(119909)1003817100381710038171003817119871infin(119868)

le

8

sum

119894=1

1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)

(73)

From (65) we have

119890 (119909)119871infin(119868)

le

8

sum

119894=1

1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)

(74)



100381710038171003817100381711986911003817100381710038171003817119871infin(119868)

le

119862119873(12)minus120583max

0le119894le119873

100381610038161003816100381611986811989411003816100381610038161003816 0 lt 120583 lt

1

2

119862 log119873max0le119894le119873

100381610038161003816100381611986811989411003816100381610038161003816

1

2le 120583 lt 1

le

119862119873(12)minus120583minus119898

119870lowast

1(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 0 lt 120583 lt1

2

119862119873minus119898 log119873119870

lowast

1(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 1

2le 120583 lt 1

100381710038171003817100381711986921003817100381710038171003817119871infin(119868)

le

119862119873(12)minus120583max

0le119894le119873

100381610038161003816100381611986811989421003816100381610038161003816 0 lt 120583 lt

1

2

119862 log119873max0le119894le119873

100381610038161003816100381611986811989411003816100381610038161003816

1

2le 120583 lt 1

le

119862119873(12)minus120583minus119898

119870lowast

2(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 0 lt 120583 lt1

2

119862119873minus119898 log119873119870

lowast

2(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 1

2le 120583 lt 1

(75)

Due to Lemma 2

100381710038171003817100381711986931003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus1198981003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868) 0 lt 120583 lt

1

2

119862119873(12)minus119898 log1198731003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

1

2le 120583 lt 1

100381710038171003817100381711986941003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus119898

|119906|119867119898119873

120596119888(119868) 0 lt 120583 lt

1

2

119862119873(12)minus119898 log119873|119906|

119867119898119873

120596119888(119868)

1

2le 120583 lt 1

100381710038171003817100381711986951003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus119898

|119906|119867119898119873

120596119888(119868) 0 lt 120583 lt

1

2

119862119873(12)minus119898 log119873|119906|

119867119898119873

120596119888(119868)

1

2le 120583 lt 1

(76)


100381710038171003817100381711986961003817100381710038171003817119871infin(119868)

le

119862119873minus120583

119890119871infin(119868) 0 lt 120583 lt

1

2

119862119873minus(12)

119890119871infin(119868)

1

2le 120583 lt 1

100381710038171003817100381711986971003817100381710038171003817119871infin(119868)

le

119862119873minus120583

119890119871infin(119868) 0 lt 120583 lt

1

2

119862119873minus(12)

119890119871infin(119868)

1

2le 120583 lt 1

(77)


5 and 6 with 119870(119909 120591) = 1Γ(120574) that

100381710038171003817100381711986951003817100381710038171003817119871infin(119868)

=10038171003817100381710038171003817(119868


119873minus 119868)M119890

12057410038171003817100381710038171003817119871infin(119868)

=10038171003817100381710038171003817(119868


119873minus 119868) (M119890

120574

minusT119873M119890

120574

)10038171003817100381710038171003817119871infin(119868)

le (1 +10038171003817100381710038171003817119868minus120583minus120583

119873

10038171003817100381710038171003817119871infin(119868))119862119873

minus1198961003817100381710038171003817M11989012057410038171003817100381710038170120581

le

119862119873(12)minus120583minus1205811003817100381710038171003817119890

1205741003817100381710038171003817119871infin(119868) 0 lt 120583 lt

1

2

119862119873minus120581 log1198731003817100381710038171003817119890

1205741003817100381710038171003817119871infin(119868)

1

2le 120583 lt 1

(78)


1


1

2

0 lt 120581 lt 1 minus 120583 when 1

2le 120583 lt 1

(79)


120574

(119909) minus 119906120574

(119909)1003817100381710038171003817119871infin(119868)

le

119862119873(12)minus120583minus119898

(119870lowast

1199061198712(119868)

+ 119873(12)U) 0 lt 120583 lt

1

2

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 119873(12)U)

1

2le 120583 lt 1

119880 (119909) minus 119906 (119909)119871infin(119868)

le

119862119873(12)minus120583minus119898

(119870lowast

1199061198712(119868)

+ 119873(12)U) 0 lt 120583 lt

1

2

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 119873(12)U)

1

2le 120583 lt 1

(80)





120574

(119909) minus 119906120574

(119909)10038171003817100381710038171198712120596minus120583minus120583

(119868)

le

119862119873minus119898

(1198811+ 119873

120574minus(12)minus120581

1198812+ 119873

120574minus120581U) 1

2lt 120574 lt 1

119862119873minus119898

(1198811+ 119873

minus120581 log1198731198812+ 119873

(12)minus120581 log119873U) 0 lt 120574 le1

2

119880(119909) minus 119906(119909)1198712

120596minus120583minus120583

(119868)

le

119862119873minus119898

(1198811+ 119873

120574minus(12)minus120581

1198812+ 119873

120574minus120581U) 1

2lt 120574 lt 1

119862119873minus119898

(1198811+ 119873

minus120581 log1198731198812+ 119873

(12)minus120581 log119873U) 0 lt 120574 le1

2

(81)



1198811= 119870

lowast

(1199061198712(119868)

+1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868))

1198812= 119870

lowast

1199061198712(119868)

(82)


100381710038171003817100381711989012057410038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862

8

sum

119894=1

100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583

(119868)

1198901198712

120596minus120583minus120583

(119868)le 119862

8

sum

119894=1

100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583

(119868)

(83)


10038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862 max

119909isin[minus11]

|119868 (119909)|

le 119862119873minus119898

119870lowast

1(119906

1198712(119868)

+ 119890119871infin(119868))

1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862 max

119909isin[minus11]

|119868 (119909)|

le 119862119873minus119898

119870lowast

2(119906

1198712(119868)

+ 119890119871infin(119868))

(84)


119890119871infin(119868)

le 119862(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868)) (85)

So that10038171003817100381710038171198691

10038171003817100381710038171198712120596minus120583minus120583

(119868)

le 119862119873minus119898

119870lowast

1(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868))

1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583

(119868)

le 119862119873minus119898

119870lowast

2(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868))

(86)

Due to Lemma 2 (33a)10038171003817100381710038171198693

10038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1198981003816100381610038161003816119863120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

1003817100381710038171003817119869410038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus119898

|119906|119867119898119873

120596119888(119868)

1003817100381710038171003817119869510038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus119898

|119906|119867119898119873

120596119888(119868)

(87)


1003817100381710038171003817119869610038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1

10038161003816100381610038161003816100381610038161003816

int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583

(119868)

le 119862119873minus1

1198901198712

120596minus120583minus120583

(119868)

1003817100381710038171003817119869710038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1

100381610038161003816100381610038161003816100381610038161003816

int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

1003816100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583

(119868)

le 119862119873minus1

1198901198712

120596minus120583minus120583

(119868)

(88)


1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583

(119868)=10038171003817100381710038171003817(119868


119873minus 119868)M119890

120574100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

=10038171003817100381710038171003817(119868


119873minus 119868) (M119890

120574

minusT119873119890120574

)100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

le10038171003817100381710038171003817119868minus120583minus120583

119873(M119890

120574

minusT119873119890120574

)100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

+1003817100381710038171003817M119890

120574


(119868)

le 1198621003817100381710038171003817M119890

120574

minusT1198731198901205741003817100381710038171003817119871infin(119868)

le 119862119873minus1205811003817100381710038171003817M119890

12057410038171003817100381710038170120581

le 119862119873minus1205811003817100381710038171003817119890

1205741003817100381710038171003817119871infin(119868)

(89)


1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583

(119868)

le


(119870lowast

1199061198712(119868)

+ 11987312U) 0 lt 120583 lt

1

2

119862119873minus119898minus120581 log119873(119870

lowast

1199061198712(119868)

+ 11987312U)

1

2le 120583 lt 1

(90)





119863075

119910 (119905) =6119905225

Γ (325)minus1199052

119890119905

5119910 (119905) minus

(119902119905)2

5119910 (119902119905)

+ int

119905

0

119890119905

119904119910 (119904) 119889119904 + int

119902119905

0

120591119910 (120591) 119889120591 119905 isin [0 1]

119910 (0) = 0

(91)




0 02 04 06 08 10

02

04

06

08

1

12

14


(a)

0 02 04 06 08 10

05

1

15

2

25


(b)


119910(119905) (b)

2 4 6 8 10 12 14 16

100

10minus5

10minus10

2 le N le 16

Linfin

L2

120596minus120583minus120583

-error-error

(a)

2 4 6 8 10 12 14 16

100

10minus5

10minus10

2 le N le 16

Linfin

L2

120596minus120583minus120583

-error-error

(b)



infin and 1198712

120596norms


infin and 1198712




119863120572

119910 (119905) = 2119905 minus 119910 (119905) + 119910 (119902119905) + 119905 (1 + 2119905) int

119905

0

119890119904(119905minus119904)

119910 (119904) 119889119904

+ int

119902119905

0

119902119905119890120591(119902119905minus120591)

119910 (120591) 119889120591 119905 isin [0 1]

119910 (0) = 1

(92)



0 02 04 06 08 105

1

15

2

25

3

35

4


(a)

0 02 04 06 08 10

1

2

3

4

5

6


(b)


2 4 6 8 10 12 14 16 18 20

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

2 le N le 20

Linfin

L2

120596

(a)

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

2 4 6 8 10 12 14 16 18 202 le N le 20

Linfin

L2

120596

(b)



infin and 1198712

120596norms



1199052








Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











10038161003816100381610038161198681 (119909)1003816100381610038161003816 le 119862119873

minus119898100381610038161003816100381610038161(119909 120579 (119909 sdot))

1003816100381610038161003816100381611986711989811987312059600(119868)

119880 (120579 (119909 sdot))1198712(119868)

le 119862119873minus119898

119870lowast

1119880

1198712(119868)

(55)

10038161003816100381610038161198682 (119909)1003816100381610038161003816 le 119862119873

minus119898100381610038161003816100381610038162(119909 120578 (119909 sdot))

1003816100381610038161003816100381611986711989811987312059600(119868)

1003817100381710038171003817119880 (120578 (119909 sdot))10038171003817100381710038171198712(119868)

le 119862119873minus119898

119870lowast

2119880

1198712(119868)

(56)

119870lowast

1= max

119909isin[minus11]

10038161003816100381610038161198701(119909 120579 (119909 sdot))

100381610038161003816100381611989811987312059600(119868)

119870lowast

2= max

119909isin[minus11]

10038161003816100381610038161198701(119909 120579 (119909 sdot))

100381610038161003816100381611989811987312059600(119868)

(57)


119906120574

119894= 119886 (119909


119894) 119906

119894+ 119887 (119909


119894)

times (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1)) + 119892 (119909


119894)

+ int

119909minus120583minus120583

119894

minus1

1198701(119909


119894 120579)119880 (120579) 119889120579 + 119868

1(119909


119894)

+ int

119902119909minus120583minus120583

119894+119902minus1

minus1

1198702(119909


119894 120578)119880 (120578) 119889120578

+ 1198682(119909


119894)

(58)


119890 (119909) = 119880 (119909) minus 119906 (119909) 119890120574

(119909) = 119880120574

(119909) minus 119906120574

(119909)

(59)

we have

119906120574

119894= 119886 (119909


119894) 119906

119894+ 119887 (119909


119894)

times (

119873

sum

119895=0

119906119895119865119895(119902119909


119894+ 119902 minus 1)) + 119892 (119909


119894)

+ int

119909minus120583minus120583

119894

minus1

1198701(119909


119894 120579) 119906 (120579) 119889120579

+ int

119909minus120583minus120583

119894

minus1

1198701(119909


119894 120579) 119890 (120579) 119889120579

+ int

119902119909minus120583minus120583

119894+119902minus1

minus1

1198702(119909


119894 120578) 119906 (120578) 119889120578

+ int

119902119909minus120583minus120583

119894+119902minus1

minus1

1198702(119909


119894 120578) 119890 (120578) 119889120578

+ 1198681(119909


119894) + 119868

2(119909


119894)

(60)



119880120574

(119909) = 119880 (119909) + 119868minus120583minus120583

119873(

119873

sum

119895=0

119906119895119865119895(119902119909 + 119902 minus 1))

+ 119868minus120583minus120583

119873119892 (119909)

+ 119868minus120583minus120583

119873[119863

120574

119906 (119909) minus 119906 (119909)


+ 119868minus120583minus120583

119873int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

+ 119868minus120583minus120583

119873int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

+ 1198691(119909) + 119869

2(119909)

(61)

119880 (119909) = 119868minus120583minus120583

119873(119906 (119909) minus 119906

minus1)

+ 119868minus120583minus120583

119873(

1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579)

+ 119868minus120583minus120583

119873(119906

minus1)

(62)

where

1198691(119909) = 119868


1198731198681(119909) 119869

2(119909) = 119868


1198731198682(119909) (63)


119890120574

(119909) = 119890 (119909) + int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

+ int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

+ 1198691(119909) + 119869

2(119909) + 119869

3(119909) + 119869

4(119909)

+ 1198695(119909) + 119869

6(119909) + 119869

7(119909)

(64)

119890 (119909) =1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

+ 1198694(119909) + 119869

8(119909)

(65)

where

1198693(119909) = 119868


119873119863120574

119906 (119909) minus 119863120574

119906 (119909)

1198694(119909) = 119868


119873119906 (119909) minus 119906 (119909)

1198695(119909) = 119868



1198696(119909) = 119868


119873int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

minus int

119909

minus1

119870 (119909 119904) 119890 (119904) 119889119904


1198697(119909) = 119868


119873int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

minus int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

1198698(119909) = 119868


119873(

1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579)

minus1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

(66)


119890120574

(119909) =1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

+1

Γ (120574)int

119909

minus1

1198701(119909 120579) int

120579

minus1

(120579 minus 120591)minus120583

119890120574

(120591) 119889120591 119889120579

+1

Γ (120574)int

119902119909+119902minus1

minus1

1198702(119909 120578) int

120578

minus1

(120578 minus 119911)minus120583

times 119890120574

(119911) 119889119911 119889120578 + 119867 (119909)

+ 1198691(119909) + 119869

2(119909) + 119869

3(119909) + 119869

4(119909)

+ 1198695(119909) + 119869

6(119909) + 119869

7(119909)

(67)

where

119867(119909) = int

119909

minus1

1198701(119909 120579) int

120579

minus1

(1198694(120579) + 119869

8(120579)) 119889120579

+1

Γ (120574)int

119902119909+119902minus1

minus1

1198702(119909 120578) int

120578

minus1

(1198694(120578) + 119869

8(120578)) 119889120578

+ 1198694(119909) + 119869

8(119909)

(68)


int

119909

minus1

int

120579

minus1

Φ (120579 120591) 119889120591 119889120579 = int

119909

minus1

int

119909

120591

Φ (120579 120591) 119889120579 119889120591 (69)


1

Γ (120574)int

119909

minus1

1198701(119909 120579) int

120579

minus1

(120579 minus 120591)minus120583

119890120574

(120591) 119889120591 119889120579

=1

Γ (120574)int

119909

minus1

(int

119909

120579

1198701(119909 120579) (120579 minus 120591)

minus120583

119889120579) 119890120574

(120591) 119889120591

le1

Γ (120574)21minus120583 max

119909120591isin[minus11]

100381610038161003816100381610038161(119909 120591)

10038161003816100381610038161003816int

119909

minus1

1003816100381610038161003816119890120574

(120591)1003816100381610038161003816 119889120591

≜ 1198721int

119909

minus1

1003816100381610038161003816119890120574

(120591)1003816100381610038161003816 119889120591

(70)


1

Γ (120574)int

119902119909+119902minus1

minus1

1198702(119909 120578) int

120578

minus1

(120578 minus 119911)minus120583

119890120574

(119911) 119889119911 119889120578

=1

Γ (120574)119902int

119909

minus1

1198702(119909 119902120591 + 119902 minus 1)

times (int

119902120591+119902minus1

minus1

(119909 minus 119911)minus120583

119890120574

(119911) 119889119911) 119889120591

=1

Γ (120574)1199021+120574

int

119909

minus1

1198702(119909 119902120591 + 119902 minus 1)

times (int

120591

minus1

(1 minus 120579)minus120583

119890120574

(119902120579 + 119902 minus 1) 119889120579)119889120591

=1

Γ (120574)1199021+120574

int

119909

minus1

(int

119909

120579

1198702(119909 119902120591 + 119902 minus 1)


minus120583

119889120591) 119890120574

times (119902120579 + 119902 minus 1) 119889120579

le1

Γ (120574)21minus120583

1199021+120574 max

119909120591isin[minus11]

10038161003816100381610038161198702(119909 120591)

1003816100381610038161003816

times int

119909

minus1

1003816100381610038161003816119890120574

(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579

≜ 1198722int

119909

minus1

1003816100381610038161003816119890120574

(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579

(71)

Then (67) gives

119890120574

(119909) le1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

+1198721int

119909

minus1

1003816100381610038161003816119890120574

(120591)1003816100381610038161003816 119889120591

+1198722int

119909

minus1

1003816100381610038161003816119890120574

(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579

+ 119867 (119909) + 1198691(119909) + 119869

2(119909) + 119869

3(119909)

+ 1198694(119909) + 119869

5(119909) + 119869

6(119909) + 119869

7(119909)

(72)


1003817100381710038171003817119890120574

(119909)1003817100381710038171003817119871infin(119868)

le

8

sum

119894=1

1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)

(73)

From (65) we have

119890 (119909)119871infin(119868)

le

8

sum

119894=1

1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)

(74)



100381710038171003817100381711986911003817100381710038171003817119871infin(119868)

le

119862119873(12)minus120583max

0le119894le119873

100381610038161003816100381611986811989411003816100381610038161003816 0 lt 120583 lt

1

2

119862 log119873max0le119894le119873

100381610038161003816100381611986811989411003816100381610038161003816

1

2le 120583 lt 1

le

119862119873(12)minus120583minus119898

119870lowast

1(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 0 lt 120583 lt1

2

119862119873minus119898 log119873119870

lowast

1(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 1

2le 120583 lt 1

100381710038171003817100381711986921003817100381710038171003817119871infin(119868)

le

119862119873(12)minus120583max

0le119894le119873

100381610038161003816100381611986811989421003816100381610038161003816 0 lt 120583 lt

1

2

119862 log119873max0le119894le119873

100381610038161003816100381611986811989411003816100381610038161003816

1

2le 120583 lt 1

le

119862119873(12)minus120583minus119898

119870lowast

2(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 0 lt 120583 lt1

2

119862119873minus119898 log119873119870

lowast

2(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 1

2le 120583 lt 1

(75)

Due to Lemma 2

100381710038171003817100381711986931003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus1198981003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868) 0 lt 120583 lt

1

2

119862119873(12)minus119898 log1198731003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

1

2le 120583 lt 1

100381710038171003817100381711986941003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus119898

|119906|119867119898119873

120596119888(119868) 0 lt 120583 lt

1

2

119862119873(12)minus119898 log119873|119906|

119867119898119873

120596119888(119868)

1

2le 120583 lt 1

100381710038171003817100381711986951003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus119898

|119906|119867119898119873

120596119888(119868) 0 lt 120583 lt

1

2

119862119873(12)minus119898 log119873|119906|

119867119898119873

120596119888(119868)

1

2le 120583 lt 1

(76)


100381710038171003817100381711986961003817100381710038171003817119871infin(119868)

le

119862119873minus120583

119890119871infin(119868) 0 lt 120583 lt

1

2

119862119873minus(12)

119890119871infin(119868)

1

2le 120583 lt 1

100381710038171003817100381711986971003817100381710038171003817119871infin(119868)

le

119862119873minus120583

119890119871infin(119868) 0 lt 120583 lt

1

2

119862119873minus(12)

119890119871infin(119868)

1

2le 120583 lt 1

(77)


5 and 6 with 119870(119909 120591) = 1Γ(120574) that

100381710038171003817100381711986951003817100381710038171003817119871infin(119868)

=10038171003817100381710038171003817(119868


119873minus 119868)M119890

12057410038171003817100381710038171003817119871infin(119868)

=10038171003817100381710038171003817(119868


119873minus 119868) (M119890

120574

minusT119873M119890

120574

)10038171003817100381710038171003817119871infin(119868)

le (1 +10038171003817100381710038171003817119868minus120583minus120583

119873

10038171003817100381710038171003817119871infin(119868))119862119873

minus1198961003817100381710038171003817M11989012057410038171003817100381710038170120581

le

119862119873(12)minus120583minus1205811003817100381710038171003817119890

1205741003817100381710038171003817119871infin(119868) 0 lt 120583 lt

1

2

119862119873minus120581 log1198731003817100381710038171003817119890

1205741003817100381710038171003817119871infin(119868)

1

2le 120583 lt 1

(78)


1


1

2

0 lt 120581 lt 1 minus 120583 when 1

2le 120583 lt 1

(79)


120574

(119909) minus 119906120574

(119909)1003817100381710038171003817119871infin(119868)

le

119862119873(12)minus120583minus119898

(119870lowast

1199061198712(119868)

+ 119873(12)U) 0 lt 120583 lt

1

2

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 119873(12)U)

1

2le 120583 lt 1

119880 (119909) minus 119906 (119909)119871infin(119868)

le

119862119873(12)minus120583minus119898

(119870lowast

1199061198712(119868)

+ 119873(12)U) 0 lt 120583 lt

1

2

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 119873(12)U)

1

2le 120583 lt 1

(80)





120574

(119909) minus 119906120574

(119909)10038171003817100381710038171198712120596minus120583minus120583

(119868)

le

119862119873minus119898

(1198811+ 119873

120574minus(12)minus120581

1198812+ 119873

120574minus120581U) 1

2lt 120574 lt 1

119862119873minus119898

(1198811+ 119873

minus120581 log1198731198812+ 119873

(12)minus120581 log119873U) 0 lt 120574 le1

2

119880(119909) minus 119906(119909)1198712

120596minus120583minus120583

(119868)

le

119862119873minus119898

(1198811+ 119873

120574minus(12)minus120581

1198812+ 119873

120574minus120581U) 1

2lt 120574 lt 1

119862119873minus119898

(1198811+ 119873

minus120581 log1198731198812+ 119873

(12)minus120581 log119873U) 0 lt 120574 le1

2

(81)



1198811= 119870

lowast

(1199061198712(119868)

+1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868))

1198812= 119870

lowast

1199061198712(119868)

(82)


100381710038171003817100381711989012057410038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862

8

sum

119894=1

100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583

(119868)

1198901198712

120596minus120583minus120583

(119868)le 119862

8

sum

119894=1

100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583

(119868)

(83)


10038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862 max

119909isin[minus11]

|119868 (119909)|

le 119862119873minus119898

119870lowast

1(119906

1198712(119868)

+ 119890119871infin(119868))

1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862 max

119909isin[minus11]

|119868 (119909)|

le 119862119873minus119898

119870lowast

2(119906

1198712(119868)

+ 119890119871infin(119868))

(84)


119890119871infin(119868)

le 119862(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868)) (85)

So that10038171003817100381710038171198691

10038171003817100381710038171198712120596minus120583minus120583

(119868)

le 119862119873minus119898

119870lowast

1(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868))

1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583

(119868)

le 119862119873minus119898

119870lowast

2(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868))

(86)

Due to Lemma 2 (33a)10038171003817100381710038171198693

10038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1198981003816100381610038161003816119863120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

1003817100381710038171003817119869410038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus119898

|119906|119867119898119873

120596119888(119868)

1003817100381710038171003817119869510038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus119898

|119906|119867119898119873

120596119888(119868)

(87)


1003817100381710038171003817119869610038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1

10038161003816100381610038161003816100381610038161003816

int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583

(119868)

le 119862119873minus1

1198901198712

120596minus120583minus120583

(119868)

1003817100381710038171003817119869710038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1

100381610038161003816100381610038161003816100381610038161003816

int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

1003816100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583

(119868)

le 119862119873minus1

1198901198712

120596minus120583minus120583

(119868)

(88)


1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583

(119868)=10038171003817100381710038171003817(119868


119873minus 119868)M119890

120574100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

=10038171003817100381710038171003817(119868


119873minus 119868) (M119890

120574

minusT119873119890120574

)100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

le10038171003817100381710038171003817119868minus120583minus120583

119873(M119890

120574

minusT119873119890120574

)100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

+1003817100381710038171003817M119890

120574


(119868)

le 1198621003817100381710038171003817M119890

120574

minusT1198731198901205741003817100381710038171003817119871infin(119868)

le 119862119873minus1205811003817100381710038171003817M119890

12057410038171003817100381710038170120581

le 119862119873minus1205811003817100381710038171003817119890

1205741003817100381710038171003817119871infin(119868)

(89)


1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583

(119868)

le


(119870lowast

1199061198712(119868)

+ 11987312U) 0 lt 120583 lt

1

2

119862119873minus119898minus120581 log119873(119870

lowast

1199061198712(119868)

+ 11987312U)

1

2le 120583 lt 1

(90)





119863075

119910 (119905) =6119905225

Γ (325)minus1199052

119890119905

5119910 (119905) minus

(119902119905)2

5119910 (119902119905)

+ int

119905

0

119890119905

119904119910 (119904) 119889119904 + int

119902119905

0

120591119910 (120591) 119889120591 119905 isin [0 1]

119910 (0) = 0

(91)




0 02 04 06 08 10

02

04

06

08

1

12

14


(a)

0 02 04 06 08 10

05

1

15

2

25


(b)


119910(119905) (b)

2 4 6 8 10 12 14 16

100

10minus5

10minus10

2 le N le 16

Linfin

L2

120596minus120583minus120583

-error-error

(a)

2 4 6 8 10 12 14 16

100

10minus5

10minus10

2 le N le 16

Linfin

L2

120596minus120583minus120583

-error-error

(b)



infin and 1198712

120596norms


infin and 1198712




119863120572

119910 (119905) = 2119905 minus 119910 (119905) + 119910 (119902119905) + 119905 (1 + 2119905) int

119905

0

119890119904(119905minus119904)

119910 (119904) 119889119904

+ int

119902119905

0

119902119905119890120591(119902119905minus120591)

119910 (120591) 119889120591 119905 isin [0 1]

119910 (0) = 1

(92)



0 02 04 06 08 105

1

15

2

25

3

35

4


(a)

0 02 04 06 08 10

1

2

3

4

5

6


(b)


2 4 6 8 10 12 14 16 18 20

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

2 le N le 20

Linfin

L2

120596

(a)

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

2 4 6 8 10 12 14 16 18 202 le N le 20

Linfin

L2

120596

(b)



infin and 1198712

120596norms



1199052








Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1198697(119909) = 119868


119873int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

minus int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

1198698(119909) = 119868


119873(

1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579)

minus1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

(66)


119890120574

(119909) =1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

+1

Γ (120574)int

119909

minus1

1198701(119909 120579) int

120579

minus1

(120579 minus 120591)minus120583

119890120574

(120591) 119889120591 119889120579

+1

Γ (120574)int

119902119909+119902minus1

minus1

1198702(119909 120578) int

120578

minus1

(120578 minus 119911)minus120583

times 119890120574

(119911) 119889119911 119889120578 + 119867 (119909)

+ 1198691(119909) + 119869

2(119909) + 119869

3(119909) + 119869

4(119909)

+ 1198695(119909) + 119869

6(119909) + 119869

7(119909)

(67)

where

119867(119909) = int

119909

minus1

1198701(119909 120579) int

120579

minus1

(1198694(120579) + 119869

8(120579)) 119889120579

+1

Γ (120574)int

119902119909+119902minus1

minus1

1198702(119909 120578) int

120578

minus1

(1198694(120578) + 119869

8(120578)) 119889120578

+ 1198694(119909) + 119869

8(119909)

(68)


int

119909

minus1

int

120579

minus1

Φ (120579 120591) 119889120591 119889120579 = int

119909

minus1

int

119909

120591

Φ (120579 120591) 119889120579 119889120591 (69)


1

Γ (120574)int

119909

minus1

1198701(119909 120579) int

120579

minus1

(120579 minus 120591)minus120583

119890120574

(120591) 119889120591 119889120579

=1

Γ (120574)int

119909

minus1

(int

119909

120579

1198701(119909 120579) (120579 minus 120591)

minus120583

119889120579) 119890120574

(120591) 119889120591

le1

Γ (120574)21minus120583 max

119909120591isin[minus11]

100381610038161003816100381610038161(119909 120591)

10038161003816100381610038161003816int

119909

minus1

1003816100381610038161003816119890120574

(120591)1003816100381610038161003816 119889120591

≜ 1198721int

119909

minus1

1003816100381610038161003816119890120574

(120591)1003816100381610038161003816 119889120591

(70)


1

Γ (120574)int

119902119909+119902minus1

minus1

1198702(119909 120578) int

120578

minus1

(120578 minus 119911)minus120583

119890120574

(119911) 119889119911 119889120578

=1

Γ (120574)119902int

119909

minus1

1198702(119909 119902120591 + 119902 minus 1)

times (int

119902120591+119902minus1

minus1

(119909 minus 119911)minus120583

119890120574

(119911) 119889119911) 119889120591

=1

Γ (120574)1199021+120574

int

119909

minus1

1198702(119909 119902120591 + 119902 minus 1)

times (int

120591

minus1

(1 minus 120579)minus120583

119890120574

(119902120579 + 119902 minus 1) 119889120579)119889120591

=1

Γ (120574)1199021+120574

int

119909

minus1

(int

119909

120579

1198702(119909 119902120591 + 119902 minus 1)


minus120583

119889120591) 119890120574

times (119902120579 + 119902 minus 1) 119889120579

le1

Γ (120574)21minus120583

1199021+120574 max

119909120591isin[minus11]

10038161003816100381610038161198702(119909 120591)

1003816100381610038161003816

times int

119909

minus1

1003816100381610038161003816119890120574

(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579

≜ 1198722int

119909

minus1

1003816100381610038161003816119890120574

(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579

(71)

Then (67) gives

119890120574

(119909) le1

Γ (120574)int

119909

minus1

(119909 minus 120579)minus120583

119890120574

(120579) 119889120579

+1198721int

119909

minus1

1003816100381610038161003816119890120574

(120591)1003816100381610038161003816 119889120591

+1198722int

119909

minus1

1003816100381610038161003816119890120574

(119902120579 + 119902 minus 1)1003816100381610038161003816 119889120579

+ 119867 (119909) + 1198691(119909) + 119869

2(119909) + 119869

3(119909)

+ 1198694(119909) + 119869

5(119909) + 119869

6(119909) + 119869

7(119909)

(72)


1003817100381710038171003817119890120574

(119909)1003817100381710038171003817119871infin(119868)

le

8

sum

119894=1

1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)

(73)

From (65) we have

119890 (119909)119871infin(119868)

le

8

sum

119894=1

1003817100381710038171003817119869119894 (119909)1003817100381710038171003817119871infin(119868)

(74)



100381710038171003817100381711986911003817100381710038171003817119871infin(119868)

le

119862119873(12)minus120583max

0le119894le119873

100381610038161003816100381611986811989411003816100381610038161003816 0 lt 120583 lt

1

2

119862 log119873max0le119894le119873

100381610038161003816100381611986811989411003816100381610038161003816

1

2le 120583 lt 1

le

119862119873(12)minus120583minus119898

119870lowast

1(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 0 lt 120583 lt1

2

119862119873minus119898 log119873119870

lowast

1(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 1

2le 120583 lt 1

100381710038171003817100381711986921003817100381710038171003817119871infin(119868)

le

119862119873(12)minus120583max

0le119894le119873

100381610038161003816100381611986811989421003816100381610038161003816 0 lt 120583 lt

1

2

119862 log119873max0le119894le119873

100381610038161003816100381611986811989411003816100381610038161003816

1

2le 120583 lt 1

le

119862119873(12)minus120583minus119898

119870lowast

2(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 0 lt 120583 lt1

2

119862119873minus119898 log119873119870

lowast

2(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 1

2le 120583 lt 1

(75)

Due to Lemma 2

100381710038171003817100381711986931003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus1198981003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868) 0 lt 120583 lt

1

2

119862119873(12)minus119898 log1198731003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

1

2le 120583 lt 1

100381710038171003817100381711986941003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus119898

|119906|119867119898119873

120596119888(119868) 0 lt 120583 lt

1

2

119862119873(12)minus119898 log119873|119906|

119867119898119873

120596119888(119868)

1

2le 120583 lt 1

100381710038171003817100381711986951003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus119898

|119906|119867119898119873

120596119888(119868) 0 lt 120583 lt

1

2

119862119873(12)minus119898 log119873|119906|

119867119898119873

120596119888(119868)

1

2le 120583 lt 1

(76)


100381710038171003817100381711986961003817100381710038171003817119871infin(119868)

le

119862119873minus120583

119890119871infin(119868) 0 lt 120583 lt

1

2

119862119873minus(12)

119890119871infin(119868)

1

2le 120583 lt 1

100381710038171003817100381711986971003817100381710038171003817119871infin(119868)

le

119862119873minus120583

119890119871infin(119868) 0 lt 120583 lt

1

2

119862119873minus(12)

119890119871infin(119868)

1

2le 120583 lt 1

(77)


5 and 6 with 119870(119909 120591) = 1Γ(120574) that

100381710038171003817100381711986951003817100381710038171003817119871infin(119868)

=10038171003817100381710038171003817(119868


119873minus 119868)M119890

12057410038171003817100381710038171003817119871infin(119868)

=10038171003817100381710038171003817(119868


119873minus 119868) (M119890

120574

minusT119873M119890

120574

)10038171003817100381710038171003817119871infin(119868)

le (1 +10038171003817100381710038171003817119868minus120583minus120583

119873

10038171003817100381710038171003817119871infin(119868))119862119873

minus1198961003817100381710038171003817M11989012057410038171003817100381710038170120581

le

119862119873(12)minus120583minus1205811003817100381710038171003817119890

1205741003817100381710038171003817119871infin(119868) 0 lt 120583 lt

1

2

119862119873minus120581 log1198731003817100381710038171003817119890

1205741003817100381710038171003817119871infin(119868)

1

2le 120583 lt 1

(78)


1


1

2

0 lt 120581 lt 1 minus 120583 when 1

2le 120583 lt 1

(79)


120574

(119909) minus 119906120574

(119909)1003817100381710038171003817119871infin(119868)

le

119862119873(12)minus120583minus119898

(119870lowast

1199061198712(119868)

+ 119873(12)U) 0 lt 120583 lt

1

2

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 119873(12)U)

1

2le 120583 lt 1

119880 (119909) minus 119906 (119909)119871infin(119868)

le

119862119873(12)minus120583minus119898

(119870lowast

1199061198712(119868)

+ 119873(12)U) 0 lt 120583 lt

1

2

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 119873(12)U)

1

2le 120583 lt 1

(80)





120574

(119909) minus 119906120574

(119909)10038171003817100381710038171198712120596minus120583minus120583

(119868)

le

119862119873minus119898

(1198811+ 119873

120574minus(12)minus120581

1198812+ 119873

120574minus120581U) 1

2lt 120574 lt 1

119862119873minus119898

(1198811+ 119873

minus120581 log1198731198812+ 119873

(12)minus120581 log119873U) 0 lt 120574 le1

2

119880(119909) minus 119906(119909)1198712

120596minus120583minus120583

(119868)

le

119862119873minus119898

(1198811+ 119873

120574minus(12)minus120581

1198812+ 119873

120574minus120581U) 1

2lt 120574 lt 1

119862119873minus119898

(1198811+ 119873

minus120581 log1198731198812+ 119873

(12)minus120581 log119873U) 0 lt 120574 le1

2

(81)



1198811= 119870

lowast

(1199061198712(119868)

+1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868))

1198812= 119870

lowast

1199061198712(119868)

(82)


100381710038171003817100381711989012057410038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862

8

sum

119894=1

100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583

(119868)

1198901198712

120596minus120583minus120583

(119868)le 119862

8

sum

119894=1

100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583

(119868)

(83)


10038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862 max

119909isin[minus11]

|119868 (119909)|

le 119862119873minus119898

119870lowast

1(119906

1198712(119868)

+ 119890119871infin(119868))

1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862 max

119909isin[minus11]

|119868 (119909)|

le 119862119873minus119898

119870lowast

2(119906

1198712(119868)

+ 119890119871infin(119868))

(84)


119890119871infin(119868)

le 119862(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868)) (85)

So that10038171003817100381710038171198691

10038171003817100381710038171198712120596minus120583minus120583

(119868)

le 119862119873minus119898

119870lowast

1(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868))

1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583

(119868)

le 119862119873minus119898

119870lowast

2(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868))

(86)

Due to Lemma 2 (33a)10038171003817100381710038171198693

10038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1198981003816100381610038161003816119863120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

1003817100381710038171003817119869410038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus119898

|119906|119867119898119873

120596119888(119868)

1003817100381710038171003817119869510038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus119898

|119906|119867119898119873

120596119888(119868)

(87)


1003817100381710038171003817119869610038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1

10038161003816100381610038161003816100381610038161003816

int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583

(119868)

le 119862119873minus1

1198901198712

120596minus120583minus120583

(119868)

1003817100381710038171003817119869710038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1

100381610038161003816100381610038161003816100381610038161003816

int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

1003816100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583

(119868)

le 119862119873minus1

1198901198712

120596minus120583minus120583

(119868)

(88)


1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583

(119868)=10038171003817100381710038171003817(119868


119873minus 119868)M119890

120574100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

=10038171003817100381710038171003817(119868


119873minus 119868) (M119890

120574

minusT119873119890120574

)100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

le10038171003817100381710038171003817119868minus120583minus120583

119873(M119890

120574

minusT119873119890120574

)100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

+1003817100381710038171003817M119890

120574


(119868)

le 1198621003817100381710038171003817M119890

120574

minusT1198731198901205741003817100381710038171003817119871infin(119868)

le 119862119873minus1205811003817100381710038171003817M119890

12057410038171003817100381710038170120581

le 119862119873minus1205811003817100381710038171003817119890

1205741003817100381710038171003817119871infin(119868)

(89)


1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583

(119868)

le


(119870lowast

1199061198712(119868)

+ 11987312U) 0 lt 120583 lt

1

2

119862119873minus119898minus120581 log119873(119870

lowast

1199061198712(119868)

+ 11987312U)

1

2le 120583 lt 1

(90)





119863075

119910 (119905) =6119905225

Γ (325)minus1199052

119890119905

5119910 (119905) minus

(119902119905)2

5119910 (119902119905)

+ int

119905

0

119890119905

119904119910 (119904) 119889119904 + int

119902119905

0

120591119910 (120591) 119889120591 119905 isin [0 1]

119910 (0) = 0

(91)




0 02 04 06 08 10

02

04

06

08

1

12

14


(a)

0 02 04 06 08 10

05

1

15

2

25


(b)


119910(119905) (b)

2 4 6 8 10 12 14 16

100

10minus5

10minus10

2 le N le 16

Linfin

L2

120596minus120583minus120583

-error-error

(a)

2 4 6 8 10 12 14 16

100

10minus5

10minus10

2 le N le 16

Linfin

L2

120596minus120583minus120583

-error-error

(b)



infin and 1198712

120596norms


infin and 1198712




119863120572

119910 (119905) = 2119905 minus 119910 (119905) + 119910 (119902119905) + 119905 (1 + 2119905) int

119905

0

119890119904(119905minus119904)

119910 (119904) 119889119904

+ int

119902119905

0

119902119905119890120591(119902119905minus120591)

119910 (120591) 119889120591 119905 isin [0 1]

119910 (0) = 1

(92)



0 02 04 06 08 105

1

15

2

25

3

35

4


(a)

0 02 04 06 08 10

1

2

3

4

5

6


(b)


2 4 6 8 10 12 14 16 18 20

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

2 le N le 20

Linfin

L2

120596

(a)

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

2 4 6 8 10 12 14 16 18 202 le N le 20

Linfin

L2

120596

(b)



infin and 1198712

120596norms



1199052








Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











100381710038171003817100381711986911003817100381710038171003817119871infin(119868)

le

119862119873(12)minus120583max

0le119894le119873

100381610038161003816100381611986811989411003816100381610038161003816 0 lt 120583 lt

1

2

119862 log119873max0le119894le119873

100381610038161003816100381611986811989411003816100381610038161003816

1

2le 120583 lt 1

le

119862119873(12)minus120583minus119898

119870lowast

1(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 0 lt 120583 lt1

2

119862119873minus119898 log119873119870

lowast

1(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 1

2le 120583 lt 1

100381710038171003817100381711986921003817100381710038171003817119871infin(119868)

le

119862119873(12)minus120583max

0le119894le119873

100381610038161003816100381611986811989421003816100381610038161003816 0 lt 120583 lt

1

2

119862 log119873max0le119894le119873

100381610038161003816100381611986811989411003816100381610038161003816

1

2le 120583 lt 1

le

119862119873(12)minus120583minus119898

119870lowast

2(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 0 lt 120583 lt1

2

119862119873minus119898 log119873119870

lowast

2(119906

1198712(119868)

+1003817100381710038171003817119890

120574

(119909)1003817100381710038171003817119871infin(119868)

) 1

2le 120583 lt 1

(75)

Due to Lemma 2

100381710038171003817100381711986931003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus1198981003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868) 0 lt 120583 lt

1

2

119862119873(12)minus119898 log1198731003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

1

2le 120583 lt 1

100381710038171003817100381711986941003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus119898

|119906|119867119898119873

120596119888(119868) 0 lt 120583 lt

1

2

119862119873(12)minus119898 log119873|119906|

119867119898119873

120596119888(119868)

1

2le 120583 lt 1

100381710038171003817100381711986951003817100381710038171003817119871infin(119868)

le

1198621198731minus120583minus119898

|119906|119867119898119873

120596119888(119868) 0 lt 120583 lt

1

2

119862119873(12)minus119898 log119873|119906|

119867119898119873

120596119888(119868)

1

2le 120583 lt 1

(76)


100381710038171003817100381711986961003817100381710038171003817119871infin(119868)

le

119862119873minus120583

119890119871infin(119868) 0 lt 120583 lt

1

2

119862119873minus(12)

119890119871infin(119868)

1

2le 120583 lt 1

100381710038171003817100381711986971003817100381710038171003817119871infin(119868)

le

119862119873minus120583

119890119871infin(119868) 0 lt 120583 lt

1

2

119862119873minus(12)

119890119871infin(119868)

1

2le 120583 lt 1

(77)


5 and 6 with 119870(119909 120591) = 1Γ(120574) that

100381710038171003817100381711986951003817100381710038171003817119871infin(119868)

=10038171003817100381710038171003817(119868


119873minus 119868)M119890

12057410038171003817100381710038171003817119871infin(119868)

=10038171003817100381710038171003817(119868


119873minus 119868) (M119890

120574

minusT119873M119890

120574

)10038171003817100381710038171003817119871infin(119868)

le (1 +10038171003817100381710038171003817119868minus120583minus120583

119873

10038171003817100381710038171003817119871infin(119868))119862119873

minus1198961003817100381710038171003817M11989012057410038171003817100381710038170120581

le

119862119873(12)minus120583minus1205811003817100381710038171003817119890

1205741003817100381710038171003817119871infin(119868) 0 lt 120583 lt

1

2

119862119873minus120581 log1198731003817100381710038171003817119890

1205741003817100381710038171003817119871infin(119868)

1

2le 120583 lt 1

(78)


1


1

2

0 lt 120581 lt 1 minus 120583 when 1

2le 120583 lt 1

(79)


120574

(119909) minus 119906120574

(119909)1003817100381710038171003817119871infin(119868)

le

119862119873(12)minus120583minus119898

(119870lowast

1199061198712(119868)

+ 119873(12)U) 0 lt 120583 lt

1

2

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 119873(12)U)

1

2le 120583 lt 1

119880 (119909) minus 119906 (119909)119871infin(119868)

le

119862119873(12)minus120583minus119898

(119870lowast

1199061198712(119868)

+ 119873(12)U) 0 lt 120583 lt

1

2

119862119873minus119898 log119873(119870

lowast

1199061198712(119868)

+ 119873(12)U)

1

2le 120583 lt 1

(80)





120574

(119909) minus 119906120574

(119909)10038171003817100381710038171198712120596minus120583minus120583

(119868)

le

119862119873minus119898

(1198811+ 119873

120574minus(12)minus120581

1198812+ 119873

120574minus120581U) 1

2lt 120574 lt 1

119862119873minus119898

(1198811+ 119873

minus120581 log1198731198812+ 119873

(12)minus120581 log119873U) 0 lt 120574 le1

2

119880(119909) minus 119906(119909)1198712

120596minus120583minus120583

(119868)

le

119862119873minus119898

(1198811+ 119873

120574minus(12)minus120581

1198812+ 119873

120574minus120581U) 1

2lt 120574 lt 1

119862119873minus119898

(1198811+ 119873

minus120581 log1198731198812+ 119873

(12)minus120581 log119873U) 0 lt 120574 le1

2

(81)



1198811= 119870

lowast

(1199061198712(119868)

+1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868))

1198812= 119870

lowast

1199061198712(119868)

(82)


100381710038171003817100381711989012057410038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862

8

sum

119894=1

100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583

(119868)

1198901198712

120596minus120583minus120583

(119868)le 119862

8

sum

119894=1

100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583

(119868)

(83)


10038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862 max

119909isin[minus11]

|119868 (119909)|

le 119862119873minus119898

119870lowast

1(119906

1198712(119868)

+ 119890119871infin(119868))

1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862 max

119909isin[minus11]

|119868 (119909)|

le 119862119873minus119898

119870lowast

2(119906

1198712(119868)

+ 119890119871infin(119868))

(84)


119890119871infin(119868)

le 119862(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868)) (85)

So that10038171003817100381710038171198691

10038171003817100381710038171198712120596minus120583minus120583

(119868)

le 119862119873minus119898

119870lowast

1(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868))

1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583

(119868)

le 119862119873minus119898

119870lowast

2(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868))

(86)

Due to Lemma 2 (33a)10038171003817100381710038171198693

10038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1198981003816100381610038161003816119863120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

1003817100381710038171003817119869410038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus119898

|119906|119867119898119873

120596119888(119868)

1003817100381710038171003817119869510038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus119898

|119906|119867119898119873

120596119888(119868)

(87)


1003817100381710038171003817119869610038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1

10038161003816100381610038161003816100381610038161003816

int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583

(119868)

le 119862119873minus1

1198901198712

120596minus120583minus120583

(119868)

1003817100381710038171003817119869710038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1

100381610038161003816100381610038161003816100381610038161003816

int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

1003816100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583

(119868)

le 119862119873minus1

1198901198712

120596minus120583minus120583

(119868)

(88)


1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583

(119868)=10038171003817100381710038171003817(119868


119873minus 119868)M119890

120574100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

=10038171003817100381710038171003817(119868


119873minus 119868) (M119890

120574

minusT119873119890120574

)100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

le10038171003817100381710038171003817119868minus120583minus120583

119873(M119890

120574

minusT119873119890120574

)100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

+1003817100381710038171003817M119890

120574


(119868)

le 1198621003817100381710038171003817M119890

120574

minusT1198731198901205741003817100381710038171003817119871infin(119868)

le 119862119873minus1205811003817100381710038171003817M119890

12057410038171003817100381710038170120581

le 119862119873minus1205811003817100381710038171003817119890

1205741003817100381710038171003817119871infin(119868)

(89)


1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583

(119868)

le


(119870lowast

1199061198712(119868)

+ 11987312U) 0 lt 120583 lt

1

2

119862119873minus119898minus120581 log119873(119870

lowast

1199061198712(119868)

+ 11987312U)

1

2le 120583 lt 1

(90)





119863075

119910 (119905) =6119905225

Γ (325)minus1199052

119890119905

5119910 (119905) minus

(119902119905)2

5119910 (119902119905)

+ int

119905

0

119890119905

119904119910 (119904) 119889119904 + int

119902119905

0

120591119910 (120591) 119889120591 119905 isin [0 1]

119910 (0) = 0

(91)




0 02 04 06 08 10

02

04

06

08

1

12

14


(a)

0 02 04 06 08 10

05

1

15

2

25


(b)


119910(119905) (b)

2 4 6 8 10 12 14 16

100

10minus5

10minus10

2 le N le 16

Linfin

L2

120596minus120583minus120583

-error-error

(a)

2 4 6 8 10 12 14 16

100

10minus5

10minus10

2 le N le 16

Linfin

L2

120596minus120583minus120583

-error-error

(b)



infin and 1198712

120596norms


infin and 1198712




119863120572

119910 (119905) = 2119905 minus 119910 (119905) + 119910 (119902119905) + 119905 (1 + 2119905) int

119905

0

119890119904(119905minus119904)

119910 (119904) 119889119904

+ int

119902119905

0

119902119905119890120591(119902119905minus120591)

119910 (120591) 119889120591 119905 isin [0 1]

119910 (0) = 1

(92)



0 02 04 06 08 105

1

15

2

25

3

35

4


(a)

0 02 04 06 08 10

1

2

3

4

5

6


(b)


2 4 6 8 10 12 14 16 18 20

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

2 le N le 20

Linfin

L2

120596

(a)

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

2 4 6 8 10 12 14 16 18 202 le N le 20

Linfin

L2

120596

(b)



infin and 1198712

120596norms



1199052








Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198811= 119870

lowast

(1199061198712(119868)

+1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868))

1198812= 119870

lowast

1199061198712(119868)

(82)


100381710038171003817100381711989012057410038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862

8

sum

119894=1

100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583

(119868)

1198901198712

120596minus120583minus120583

(119868)le 119862

8

sum

119894=1

100381710038171003817100381711986911989410038171003817100381710038171198712120596minus120583minus120583

(119868)

(83)


10038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862 max

119909isin[minus11]

|119868 (119909)|

le 119862119873minus119898

119870lowast

1(119906

1198712(119868)

+ 119890119871infin(119868))

1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862 max

119909isin[minus11]

|119868 (119909)|

le 119862119873minus119898

119870lowast

2(119906

1198712(119868)

+ 119890119871infin(119868))

(84)


119890119871infin(119868)

le 119862(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868)) (85)

So that10038171003817100381710038171198691

10038171003817100381710038171198712120596minus120583minus120583

(119868)

le 119862119873minus119898

119870lowast

1(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868))

1003817100381710038171003817119869210038171003817100381710038171198712120596minus120583minus120583

(119868)

le 119862119873minus119898

119870lowast

2(1003816100381610038161003816119863

120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

+ |119906|119867119898119873

120596119888(119868)

+ 1199061198712(119868))

(86)

Due to Lemma 2 (33a)10038171003817100381710038171198693

10038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1198981003816100381610038161003816119863120574

1199061003816100381610038161003816119867119898119873120596119888(119868)

1003817100381710038171003817119869410038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus119898

|119906|119867119898119873

120596119888(119868)

1003817100381710038171003817119869510038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus119898

|119906|119867119898119873

120596119888(119868)

(87)


1003817100381710038171003817119869610038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1

10038161003816100381610038161003816100381610038161003816

int

119909

minus1

1198701(119909 120579) 119890 (120579) 119889120579

100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583

(119868)

le 119862119873minus1

1198901198712

120596minus120583minus120583

(119868)

1003817100381710038171003817119869710038171003817100381710038171198712120596minus120583minus120583

(119868)le 119862119873

minus1

100381610038161003816100381610038161003816100381610038161003816

int

119902119909+119902minus1

minus1

1198702(119909 120578) 119890 (120578) 119889120578

1003816100381610038161003816100381610038161003816100381610038161198671119873120596minus120583minus120583

(119868)

le 119862119873minus1

1198901198712

120596minus120583minus120583

(119868)

(88)


1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583

(119868)=10038171003817100381710038171003817(119868


119873minus 119868)M119890

120574100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

=10038171003817100381710038171003817(119868


119873minus 119868) (M119890

120574

minusT119873119890120574

)100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

le10038171003817100381710038171003817119868minus120583minus120583

119873(M119890

120574

minusT119873119890120574

)100381710038171003817100381710038171198712120596minus120583minus120583

(119868)

+1003817100381710038171003817M119890

120574


(119868)

le 1198621003817100381710038171003817M119890

120574

minusT1198731198901205741003817100381710038171003817119871infin(119868)

le 119862119873minus1205811003817100381710038171003817M119890

12057410038171003817100381710038170120581

le 119862119873minus1205811003817100381710038171003817119890

1205741003817100381710038171003817119871infin(119868)

(89)


1003817100381710038171003817119869810038171003817100381710038171198712120596minus120583minus120583

(119868)

le


(119870lowast

1199061198712(119868)

+ 11987312U) 0 lt 120583 lt

1

2

119862119873minus119898minus120581 log119873(119870

lowast

1199061198712(119868)

+ 11987312U)

1

2le 120583 lt 1

(90)





119863075

119910 (119905) =6119905225

Γ (325)minus1199052

119890119905

5119910 (119905) minus

(119902119905)2

5119910 (119902119905)

+ int

119905

0

119890119905

119904119910 (119904) 119889119904 + int

119902119905

0

120591119910 (120591) 119889120591 119905 isin [0 1]

119910 (0) = 0

(91)




0 02 04 06 08 10

02

04

06

08

1

12

14


(a)

0 02 04 06 08 10

05

1

15

2

25


(b)


119910(119905) (b)

2 4 6 8 10 12 14 16

100

10minus5

10minus10

2 le N le 16

Linfin

L2

120596minus120583minus120583

-error-error

(a)

2 4 6 8 10 12 14 16

100

10minus5

10minus10

2 le N le 16

Linfin

L2

120596minus120583minus120583

-error-error

(b)



infin and 1198712

120596norms


infin and 1198712




119863120572

119910 (119905) = 2119905 minus 119910 (119905) + 119910 (119902119905) + 119905 (1 + 2119905) int

119905

0

119890119904(119905minus119904)

119910 (119904) 119889119904

+ int

119902119905

0

119902119905119890120591(119902119905minus120591)

119910 (120591) 119889120591 119905 isin [0 1]

119910 (0) = 1

(92)



0 02 04 06 08 105

1

15

2

25

3

35

4


(a)

0 02 04 06 08 10

1

2

3

4

5

6


(b)


2 4 6 8 10 12 14 16 18 20

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

2 le N le 20

Linfin

L2

120596

(a)

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

2 4 6 8 10 12 14 16 18 202 le N le 20

Linfin

L2

120596

(b)



infin and 1198712

120596norms



1199052








Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 02 04 06 08 10

02

04

06

08

1

12

14


(a)

0 02 04 06 08 10

05

1

15

2

25


(b)


119910(119905) (b)

2 4 6 8 10 12 14 16

100

10minus5

10minus10

2 le N le 16

Linfin

L2

120596minus120583minus120583

-error-error

(a)

2 4 6 8 10 12 14 16

100

10minus5

10minus10

2 le N le 16

Linfin

L2

120596minus120583minus120583

-error-error

(b)



infin and 1198712

120596norms


infin and 1198712




119863120572

119910 (119905) = 2119905 minus 119910 (119905) + 119910 (119902119905) + 119905 (1 + 2119905) int

119905

0

119890119904(119905minus119904)

119910 (119904) 119889119904

+ int

119902119905

0

119902119905119890120591(119902119905minus120591)

119910 (120591) 119889120591 119905 isin [0 1]

119910 (0) = 1

(92)



0 02 04 06 08 105

1

15

2

25

3

35

4


(a)

0 02 04 06 08 10

1

2

3

4

5

6


(b)


2 4 6 8 10 12 14 16 18 20

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

2 le N le 20

Linfin

L2

120596

(a)

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

2 4 6 8 10 12 14 16 18 202 le N le 20

Linfin

L2

120596

(b)



infin and 1198712

120596norms



1199052








Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 02 04 06 08 105

1

15

2

25

3

35

4


(a)

0 02 04 06 08 10

1

2

3

4

5

6


(b)


2 4 6 8 10 12 14 16 18 20

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

2 le N le 20

Linfin

L2

120596

(a)

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

2 4 6 8 10 12 14 16 18 202 le N le 20

Linfin

L2

120596

(b)



infin and 1198712

120596norms



1199052








Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Spectral-Collocation Methods for...

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