Research Article Rate of Convergence of Modified Baskakov …downloads.hindawi.com/archive/2014/235480.pdf · 2019. 7. 31. · Research Article Rate of Convergence of Modified Baskakov-Durrmeyer

Research ArticleRate of Convergence of Modified Baskakov-Durrmeyer TypeOperators for Functions of Bounded Variation

Prashantkumar Patel12 and Vishnu Narayan Mishra13

1 Department of Applied Mathematics amp Humanities Sardar Vallabhbhai National Institute of TechnologySurat Gujarat 395 007 India

2Department of Mathematics St Xavierrsquos College Ahmedabad Gujarat 380 009 India3 L 1627 Awadh Puri Colony Beniganj Phase-III Opposite-Industrial Training Institute (ITI) Ayodhya Main RoadFaizabad Uttar Pradesh 224 001 India

Correspondence should be addressed to Prashantkumar Patel prashant225gmailcom

Received 19 April 2014 Accepted 11 June 2014 Published 2 July 2014

Academic Editor Abdelalim A Elsadany

Copyright copy 2014 P Patel and V N Mishra 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 study a certain integral modification of well-known Baskakov operators with weight function of beta basis function Weestablish rate of convergence for these operators for functions having derivative of bounded variation Also we discuss Stancutype generalization of these operators

1 Introduction

The integral modification of Baskakov operators havingweight function of some beta basis function are defined asthe following for 119909 isin [0infin) 120574 gt 0

119861119899120574(119891 119909) =

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) 119891 (119905) 119889119905

+ (1 + 120574119909)minus119899

119891 (0)

= int

infin

0

119882119899120574(119909 119905) 119891 (119905) 119889119905

(1)

where

119901119899119896120574

(119909) =

Γ (119899120574 + 119896)

Γ (119896 + 1) Γ (119899120574)

sdot

(120574119909)119896

(1 + 120574119909)(119899120574)+119896

119887119899119896120574

(119905) =

120574Γ (119899120574 + 119896 + 1)

Γ (119896) Γ (119899120574 + 1)

sdot

(120574119905)119896minus1

(1 + 120574119905)(119899120574)+119896+1

119882119899120574(119909 119905) =

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) 119889119905 + (1 + 120574119909)minus119899120574

120575 (119905)

(2)

120575(119905) being the Dirac delta functionThe operators defined by (1) were introduced by Gupta

[1] these operators are different from the usual Baskakov-Durrmeyer operators Actually these operators satisfy condi-tion 119861

119899120574(119886119905 + 119887 119909) = 119886119909 + 119887 where 119886 and 119887 are constants In

[1] the author estimated some direct results in simultaneousapproximation for these operators (1) In particular case 120574 =1 the operators (1) reduce to the operators studied in [2 3]

In recent years a lot of work has been done on suchoperators We refer to some of the important papers on therecent development on similar type of operators [4ndash9] Therate of convergence for certainDurrmeyer type operators andthe generalizations is one of the important areas of researchin recent years In present article we extend the studies andhere we estimate the rate of convergence for functions havingderivative of bounded variation

Hindawi Publishing CorporationJournal of Difference EquationsVolume 2014 Article ID 235480 6 pageshttpdxdoiorg1011552014235480

2 Journal of Difference Equations

We denote 120601119899120574(119909 119905) = int

119905

0

119882119899120574(119909 119904)119889119904 then in particular

we have

120601119899120574(119909infin) = int

infin

0

119882119899120574(119909 119904) 119889119904 = 1 (3)

By 119863119861119903(0infin) 119903 ge 0 we denote the class of absolutely

continuous functions 119891 defined on the interval (0infin) suchthat

(i) 119891(119905) = 119874(119905119903) 119905 rarr infin(ii) having a derivative 1198911015840 on the interval (0infin) coincid-

ing ae with a function which is of bounded variationon every finite subinterval of (0infin)

It can be observed that all function 119891 isin 119861119863119903(0infin) possess

for each 119888 gt 0 a representation

119891 (119909) = 119891 (119888) + int

119909

119888

120595 (119905) 119889119905 119909 ge 119888 (4)

2 Rate of Convergence for 119861119899120574

Lemma 1 (see [1]) Let the function 119879119899119898120574

(119909)119898 isin N cup 0 bedefined as

119879119899119898120574

(119909) = 119861119899120574((119905 minus 119909)

119898

119909)

=

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119905 minus 119909)119898

119889119905

+ (1 + 120574119909)minus119899120574

(minus119909)119898

(5)

Then it is easily verified that for each 119909 isin (0infin) 1198791198990120574

(119909) = 11198791198991120574

(119909) = 0 and 1198791198992120574

(119909) = 2119909(1 + 120574119909)(119899 minus 120574) and also thefollowing recurrence relation holds

(119899 minus 120574119898)119879119899119898+1120574

(119909) = 119909 (1 + 120574119909)

times [119879(1)

119899119898120574(119909) + 2119898119879

119899119898minus1120574(119909)]

+ 119898 (1 + 2120574119909) 119879119899119898120574

(119909)

(6)

From the recurrence relation it can be easily be verified that forall 119909 isin [0infin) we have 119879

119899119898120574(119909) = 119874(119899

minus[(119898+1)2]

)

Remark 2 FromLemma 1 usingCauchy-Schwarz inequalityit follows that

119861119899120574(|119905 minus 119909| 119909) le [119861

119899120574((119905 minus 119909)

2

119909)]

12

le radic

2119909 (1 + 120574119909)

119899 minus 120574

(7)

Lemma 3 Let 119909 isin (0infin) and119882119899120574(119909 119905) be the kernel defined

in (1) Then for 119899 being sufficiently large one has

(a) 120601119899120574(119909 119910) = int

119910

0

119882119899120574(119909 119905)119889119905 le 2119909(1 + 120574119909)(119899 minus

120574)(119909 minus 119910)2

0 le 119910 lt 119909

(b) 1 minus 120601119899120574(119909 119911) = int

infin

119911

119882119899120574(119909 119905)119889119905 le 2119909(1 + 120574119909)(119899 minus

120574)(119911 minus 119909)2

119909 lt 119911 lt infin

Proof First we prove (a) by using Lemma 1 we have

int

119910

0

119882119899120574(119909 119905) 119889119905 le int

119910

0

(119909 minus 119905)2

(119909 minus 119910)2119882119899120574(119909 119905) 119889119905

le (119909 minus 119910)minus2

1198791198992120574

(119909)

le

2119909 (1 + 120574119909)

(119899 minus 120574) (119909 minus 119910)2

(8)

The proof of (b) is similar we omit the details

Theorem 4 Let 119891 isin 119863119861119903(0infin) 119903 isin N and 119909 isin (0infin) Then

for 119899 being sufficiently large we have10038161003816100381610038161003816119861119899120574(119891 119909) minus 119891 (119909)

10038161003816100381610038161003816

le

2 (1 + 120574119909)

119899 minus 120574

[radic119899]

sum

119896=1

119909+119909119896

⋁

119909minus119909119896

((1198911015840

)119909

)

+

119909

radic119899

119909+119909radic119899

⋁

119909minus119909radic119899

((1198911015840

)119909

) +

2 (1 + 120574119909)

(119899 minus 120574) 119909

times [

10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891

1015840

(119909+

)

10038161003816100381610038161003816+1003816100381610038161003816119891 (119909)

1003816100381610038161003816]

+ radic

2119909 (1 + 120574119909)

119899 minus 120574

[1198722119903

119874(119899minus1199032

) +

100381610038161003816100381610038161198911015840

(119909+

)

10038161003816100381610038161003816]

+ radic

119909 (1 + 120574119909)

2 (119899 minus 120574)

100381610038161003816100381610038161198911015840

(119909+

) minus 1198911015840

(119909minus

)

10038161003816100381610038161003816

(9)

where the auxiliary function 119891119909is given by

119891119909(119905) =

119891 (119905) minus 119891 (119909minus

) 0 le 119905 lt 119909

0 119905 = 119909

119891 (119905) minus 119891 (119909+

) 119909 lt 119905 lt infin

(10)

⋁119887

119886119891(119909) denotes the total variation of 119891

119909on [119886 119887]

Proof By the application of mean value theorem we have

119861119899120574(119891 119909) minus 119891 (119909) = int

infin

0

119882119899120574(119909 119905) (119891 (119905) minus 119891 (119909)) 119889119905

= int

infin

0

int

119905

119909

119882119899120574(119909 119905) (119891

1015840

(119906) 119889119906) 119889119905

(11)

Also using the identity

1198911015840

(119906) =

1198911015840

(119909+

) + 1198911015840

(119909minus

)

2

+ (1198911015840

)119909

(119906)

+

1198911015840

(119909+

) minus 1198911015840

(119909minus

)

2

sgn (119906 minus 119909)

+ [1198911015840

(119909) minus

1198911015840

(119909+

) + 1198911015840

(119909minus

)

2

] 120594119909(119906)

(12)

Journal of Difference Equations 3

where

120594119909(119906) =

1 119906 = 119909

0 119906 = 119909

(13)

we can see that

int

infin

0

(int

119905

119909

(1198911015840

(119909) minus

1198911015840

(119909+

) + 1198911015840

(119909minus

)

2

)120594119909(119906) 119889119906)

times119882119899120574(119905 119909) 119889119905 = 0

(14)

Also

int

infin

0

(int

119905

119909

(

1198911015840

(119909+

) minus 1198911015840

(119909minus

)

2

) sgn (119906 minus 119909) 119889119906)119882119899120574(119905 119909) 119889119905

=

1198911015840

(119909+

) minus 1198911015840

(119909minus

)

2

119861119899120574(|119905 minus 119909| 119909)

int

infin

0

(int

119905

119909

(

1198911015840

(119909+

) + 1198911015840

(119909minus

)

2

) 119889119906)119882119899120574(119905 119909) 119889119905

=

1198911015840

(119909+

) + 1198911015840

(119909minus

)

2

119861119899120574(119905 minus 119909 119909)

(15)

Substitute value of 1198911015840(119906) from (12) in (11) and using (14) and(15) we get

10038161003816100381610038161003816119861119899120574(119891 119909) minus 119891 (119909)

10038161003816100381610038161003816le

10038161003816100381610038161003816100381610038161003816

int

infin

119909

(int

119905

119909

(1198911015840

)119909

(119906) 119889119906)119882119899120574(119909 119905) 119889119905

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

int

119909

0

(int

119905

119909

(1198911015840

)119909

(119906) 119889119906)119882119899120574(119909 119905) 119889119905

10038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161198911015840

(119909+

) minus 1198911015840

(119909minus

)

10038161003816100381610038161003816

2

119861119899120574(|119905 minus 119909| 119909)

+

100381610038161003816100381610038161198911015840

(119909+

) + 1198911015840

(119909minus

)

10038161003816100381610038161003816

2

119861119899120574(119905 minus 119909 119909)

(16)

Using Lemma 1 and Remark 2 we obtain

10038161003816100381610038161003816119861119899120574(119891 119909) minus 119891 (119909)

10038161003816100381610038161003816le

10038161003816100381610038161003816100381610038161003816

int

infin

119909

(int

119905

119909

(1198911015840

)119909

(119906) 119889119906)119882119899120574(119909 119905) 119889119905

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

int

119909

0

(int

119905

119909

(1198911015840

)119909

(119906) 119889119906)119882119899120574(119909 119905) 119889119905

10038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161198911015840

(119909+

) minus 1198911015840

(119909minus

)

10038161003816100381610038161003816

2

radic

2119909 (1 + 120574119909)

119899 minus 120574

=

10038161003816100381610038161003816119875119899120574(119891 119909)

10038161003816100381610038161003816+

10038161003816100381610038161003816119876119899120574(119891 119909)

10038161003816100381610038161003816

+

100381610038161003816100381610038161198911015840

(119909+

) minus 1198911015840

(119909minus

)

10038161003816100381610038161003816

2

radic

2119909 (1 + 120574119909)

119899 minus 120574

(17)

On applying Lemma 3 with 119910 = 119909 minus 119909radic119899 and integrating byparts we have

10038161003816100381610038161003816119876119899120574(119891 119909)

10038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816

int

119909

0

int

119905

119909

(1198911015840

)119909

(119906) 119889119906 119889119905 (120601119899120574(119909 119910))

10038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816

int

119909

0

120601119899120574(119909 119910) (119891

1015840

)119909

(119905) 119889119905

10038161003816100381610038161003816100381610038161003816

le (int

119910

0

+int

119909

119910

)

10038161003816100381610038161003816(1198911015840

)119909

(119905)

10038161003816100381610038161003816

10038161003816100381610038161003816120601119899120574(119909 119905)

10038161003816100381610038161003816119889119905

le

2119909 (1 + 120574119909)

119899 minus 120574

int

119910

0

119909

⋁

119905

((1198911015840

)119909

)

1

(119909 minus 119905)2119889119905

+ int

119909

119910

119909

⋁

119905

((1198911015840

)119909

) 119889119905

le

2119909 (1 + 120574119909)

119899 minus 120574

int

119910

0

119909

⋁

119905

((1198911015840

)119909

)

1

(119909 minus 119905)2119889119905

+

119909

radic119899

119909

⋁

119909minus119909radic119899

((1198911015840

)119909

)

=

2119909 (1 + 120574119909)

119899 minus 120574

int

radic119899

1

119909

⋁

119909minus119909119906

((1198911015840

)119909

) 119889119906

+

119909

radic119899

119909

⋁

119909minus119909radic119899

((1198911015840

)119909

)

le

2119909 (1 + 120574119909)

119899 minus 120574

[radic119899]

sum

119896=1

119909

⋁

119909minus119909119896

((1198911015840

)119909

)

+

119909

radic119899

119909

⋁

119909minus119909radic119899

((1198911015840

)119909

)

(18)

where 119906 = 119909(119909 minus 119905)On the other hand we have10038161003816100381610038161003816119875119899120574(119891 119909)

10038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816

int

infin

119909

(int

119905

119909

(1198911015840

)119909

(119906) 119889119906)119882119899120574(119909 119905) 119889119905

10038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816

int

infin

2119909

(int

119905

119909

(1198911015840

)119909

(119906) 119889119906)119882119899120574(119909 119905) 119889119905

+int

2119909

119909

(int

119905

119909

(1198911015840

)119909

(119906) 119889119906)119889119905 (1 minus 120601119899120574(119909 119905))

100381610038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

int

infin

2119909

(119891 (119905) minus 119891 (119909))119882119899120574(119909 119905) 119889119905

10038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161198911015840

(119909+

)

10038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

infin

2119909

(119905 minus 119909)119882119899120574(119909 119905) 119889119905

10038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

2119909

119909

(1198911015840

)119909

(119906) 119889119906

100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381610038161 minus 120601119909120574(119909 2119909)

10038161003816100381610038161003816


+ int

2119909

119909

10038161003816100381610038161003816(1198911015840

)119909

(119905)

10038161003816100381610038161003816

100381610038161003816100381610038161 minus 120601119909120574(119909 119905)

10038161003816100381610038161003816119889119905

le [

119872

119909

int

infin

2119909

119882119899120574(119909 119905) 119905

119903

|119905 minus 119909| 119889119905

+

1003816100381610038161003816119891 (119909)

1003816100381610038161003816

1199092

int

infin

2119909

119882119899120574(119909 119905) (119905 minus 119909)

2

119889119905]

+ [

100381610038161003816100381610038161198911015840

(119909+

)

10038161003816100381610038161003816int

infin

2119909

119882119899120574(119909 119905) |119905 minus 119909| 119889119905]

+

2 (1 + 120574119909)

(119899 minus 120574) 119909

10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891

1015840

(119909+

)

10038161003816100381610038161003816

+

2 (1 + 120574119909)

119899 minus 120574

[radic119899]

sum

119896=1

119909+119909119896

⋁

119909

((1198911015840

)119909

)

+

119909

radic119899

119909+119909radic119899

⋁

119909

((1198911015840

)119909

)

= 119877119899120574(119891 119909) + 119878

119899120574(119891 119909)

+

2 (1 + 120574119909)

(119899 minus 120574) 119909

10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891

1015840

(119909+

)

10038161003816100381610038161003816

+

2 (1 + 120574119909)

119899 minus 120574

[radic119899]

sum

119896=1

119909+119909119896

⋁

119909

((1198911015840

)119909

)

+

119909

radic119899

119909+119909radic119899

⋁

119909

((1198911015840

)119909

)

(19)

Applying Holderrsquos inequality Remark 2 and Lemma 1 wehave

119877119899120574(119891 119909) le

119872

119909

(int

infin

2119909

119882119899120574(119909 119905) 119905

2119903

119889119905)

12

times(int

infin

0

119882119899120574(119909 119905) (119905 minus 119909)

2

119889119905)

12

+

1003816100381610038161003816119891 (119909)

1003816100381610038161003816

1199092

int

infin

2119909

119882119899120574(119909 119905) (119905 minus 119909)

2

119889119905

le 1198722119903

119874(119899minus1199032

)radic

2119909 (1 + 120574119909)

119899 minus 120574

+1003816100381610038161003816119891 (119909)

1003816100381610038161003816

2 (1 + 120574119909)

(119899 minus 120574) 119909

(20)

Also

119878119899120574(119891 119909) le

1003816100381610038161003816119891 (119909+

)1003816100381610038161003816int

infin

0

119882119899(119905 119909) |119905 minus 119909| 119889119905

le1003816100381610038161003816119891 (119909+

)1003816100381610038161003816radic

2119909 (1 + 120574119909)

119899 minus 120574

(21)

Combining the estimates (17)ndash(21) we get the desired resultsThis completes the proof of Theorem

3 Rate of Convergence for Stancu TypeGeneralization of 119861

119899120574

In 1968 Stancu introduces Bernstein-Stancu operators in[10] a sequence of the linear positive operators depending ontwo parameters 120572 and 120573 satisfying the condition 0 le 120572 le 120573Recently many researchers applied this approach to manyoperators for details see [11ndash17] For 119891 isin 119862[0infin) Stancugeneralization of operators (1) is as follows

119861120572120573

119899120574(119891 (119905) 119909) =

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) 119891(

119899119905 + 120572

119899 + 120573

)119889119905

+ 1199011198990120574

(119909) 119891(

120572

119899 + 120573

)

= int

infin

0

119882119899120574(119909 119905) 119891(

119899119905 + 120572

119899 + 120573

)119889119905

(22)

where 119901119899119896120574

(119909) 119887119899119896120574

(119909) and119882119899120574(119909 119905)are as defined in (1)

Lemma5 (see [18]) If we define the centralmoments for every119898 isin N as

120583120572120573

119899119898120574(119909) = 119861

120572120573

119899120574((119905 minus 119909)

119898

119909)

=

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (

119899119905 + 120572

119899 + 120573

minus 119909)

119898

119889119905

+ 1199011198990120574

(119909) (

120572

119899 + 120573

minus 119909)

119898

(23)

then 1205831205721205731198990120574

(119909) = 1 1205831205721205731198991120574

(119909) = (120572 minus 120573119909)(119899 + 120573) and

120583120572120573

1198992120574(119909) =

1205722

(119899 + 120573)2+ 119909

(21198992

minus 2119899120572120573 + 2120572120573120574)

(119899 + 120573)2

(119899 minus 120574)

+ 1199092(1198991205732

+ 21198992

120574 minus 1205732

120574)

(119899 + 120573)2

(119899 minus 120574)

(24)

For 119899 gt 119898 we have the following recurrence relation

(119899 minus 120574119898) (119899 + 120573) 120583120572120573

119899119898+1120574(119909)

= 119899119909 (1 + 120574119909) [(120583120572120573

119899119898120574)

(1)

(119909) + 119898120583120572120573

119899119898minus1120574(119909)]

+ [119898119899 + 1198992

119909 minus (2120574119898 minus 119899) (120572 minus (119899 + 120573) 119909)] 120583120572120573

119899119898120574(119909)

+ [119898120574 (119899 + 120573) (

120572

119899 + 120573

minus 119909)

2

minus 119898119899(

120572

119899 + 120573

minus 119909)]

times 120583120572120573

119899119898minus1120574(119909)

(25)


From the recurrence relation it can be easily verified that forall 119909 isin [0infin) we have 120583120572120573

119899119898120574(119909) = 119874(119899

minus[(119898+1)2]

)

Remark 6 Observe that 119861120572120573119899120574

preserve constant functions butnot linear functions If 120572 = 120573 = 0 these operators reduce tothe operators defined in (1) Notice that

120583120572120573

1198992120574(119909) = [

1205732

120574(119899 + 120573)2+

21198992

(119899 + 120573)2

(119899 minus 120574)

] 119909 (1 + 120574119909)

+

120573 [(2120572 + 120573) 120574 minus 119899 (120573 + 2120572120574)]

120574(119899 + 120573)2

(119899 minus 120574)

119909 +

1205722

(119899 + 120573)2

(26)

Remark 7 From Lemma 3 taking 119899 to be sufficiently largeand 119909 isin (0infin) we observe that

120583120572120573

1198992120574(119909) le

119862119909 (1 + 120574119909)

119899 minus 120574

(27)

where 119862 is positive constant

Remark 8 Applying the Cauchy-Schwarz inequality andkeeping the same condition as in Remark 7 for 119909 119899 and 119862we derive from Lemma 5 that

119861120572120573

119899120574(|119905 minus 119909| 119909) le [119861

120572120573

119899120574((119905 minus 119909)

2

119909)]

12

le radic

119862119909 (1 + 120574119909)

119899 minus 120574

(28)


in (1) Then for 119899 being sufficiently large we have

(c) 120601119899120574(119909 119910) = int

119910

0

119882119899120574(119909 119905)119889119905 = 119862119909(1 + 120574119909)(119899 minus

120574)(119909 minus 119910)2

0 le 119910 lt 119909

(d) 1 minus 120601119899120574(119909 119911) = int

infin

119911

119882119899120574(119909 119905)119889119905 = 119862119909(1 + 120574119909)(119899 minus

120574)(119911 minus 119909)2

119909 lt 119911 lt infin

The proof is the same as Lemma 3 thus we omit the details

Theorem 10 Let 119891 isin 119863119861119903(0infin) 119903 isin N and 119909 isin (0infin)

Then for 119899 being sufficiently large we have

10038161003816100381610038161003816119861120572120573

119899120574(119891 119909) minus 119891 (119909)

10038161003816100381610038161003816

le

119862 (1 + 120574119909)

119899 minus 120574

[radic119899]

sum

119896=1

119909+119909119896

⋁

119909minus119909119896

((1198911015840

)119909

)

+

119909

radic119899

119909+119909radic119899

⋁

119909minus119909radic119899

((1198911015840

)119909

) +

119862 (1 + 120574119909)

(119899 minus 120574) 119909

times [

10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891

1015840

(119909+

)

10038161003816100381610038161003816+1003816100381610038161003816119891 (119909)

1003816100381610038161003816]

+ radic

119862119909 (1 + 120574119909)

119899 minus 120574

times [11987212119903

119874(119899minus1199032

) +

100381610038161003816100381610038161198911015840

(119909+

)

10038161003816100381610038161003816]

+

1

2

radic

119862119909 (1 + 120574119909)

(119899 minus 120574)

100381610038161003816100381610038161198911015840

(119909+

) minus 1198911015840

(119909minus

)

10038161003816100381610038161003816

+

1

2

100381610038161003816100381610038161198911015840

(119909+

) + 1198911015840

(119909minus

)

10038161003816100381610038161003816

1003816100381610038161003816120572 minus 120573119909

1003816100381610038161003816

119899 + 120573

(29)

where the auxiliary functions 119891119909and ⋁119887

119886119891(119909) were defined in

Theorem 4The proof of the above theorem follows along the lines of

Theorem 4 thus we omit the details

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are thankful to the anonymous referee formaking valuable comments leading to the better presen-tation of the paper Special thanks are due to ProfessorDr Abdelalim A Elsadany Editor of Journal of DifferenceEquations for kind cooperation and smooth behavior duringcommunication and for his efforts to send the reports of thepaper timely

References

[1] V Gupta ldquoApproximation for modified Baskakov Durrmeyertype operatorsrdquo The Rocky Mountain Journal of Mathematicsvol 39 no 3 pp 825ndash841 2009

[2] V Gupta M A Noor M S Beniwal and M K Gupta ldquoOnsimultaneous approximation for certain Baskakov Durrmeyertype operatorsrdquo Journal of Inequalities in Pure and AppliedMathematics vol 7 pp 1ndash15 2006

[3] V Gupta and P N Agrawal ldquoRate of convergence for certainBaskakov Durrmeyer type operatorsrdquoAnnals of Oradea Univer-sity Fascicola Matematica vol 14 pp 33ndash39 2007

[4] HKarsli andE Ibikli ldquoConvergence rate of a newBezier variantof Chlodowsky operators to bounded variation functionsrdquoJournal of Computational and AppliedMathematics vol 212 no2 pp 431ndash443 2008

[5] V N Mishra H H Khan K Khatri and L N Mishra ldquoHyper-geometric representation for Baskakov-Durrmeyer-Stancu typeoperatorsrdquo Bulletin of Mathematical Analysis and Applicationsvol 5 no 3 pp 18ndash26 2013

[6] V N Mishra K Khatri and L N Mishra ldquoOn simultaneousapproximation for Baskakov-Durrmeyer-Stancu type opera-torsrdquo Journal of Ultra Scientist of Physical Sciences vol 24 no3 pp 567ndash577 2012

[7] H H Khan Approximation of classes of function [PhD thesis]AMU Aligarh India 1974


[8] X M Zeng and F C Cheng ldquoOn the rates of approximation ofBernstein type operatorsrdquo Journal of ApproximationTheory vol109 no 2 pp 242ndash256 2001

[9] M Mursaleen V Karakaya M Erturk and F GursoyldquoWeighted statistical convergence and its application toKorovkin type approximation theoremrdquo Applied Mathematicsand Computation vol 218 no 18 pp 9132ndash9137 2012

[10] D D Stancu ldquoApproximation of function by a new class ofpolynomial operatorsrdquoRevue Roumaine deMathematique Pureset Appliquees vol 3 no 8 pp 1173ndash1194 1968

[11] R Yang J Xiong and F Cao ldquoMultivariate Stancu operatorsdefined on a simplexrdquo Applied Mathematics and Computationvol 138 no 2-3 pp 189ndash198 2003

[12] H M Srivastava M Mursaleen and A Khan ldquoGeneralizedequi-statistical convergence of positive linear operators andassociated approximation theoremsrdquo Mathematical and Com-puter Modelling vol 55 no 9-10 pp 2040ndash2051 2012

[13] M Mursaleen A Khan H M Srivastava and K S NisarldquoOperators constructed by means of q-Lagrange polynomialsand A-statistical approximationrdquo Applied Mathematics andComputation vol 219 no 12 pp 6911ndash6918 2013

[14] V N Mishra and P Patel ldquoApproximation by the Durrmeyer-Baskakov-Stancu operatorsrdquo Lobachevskii Journal ofMathemat-ics vol 34 no 3 pp 272ndash281 2013

[15] V N Mishra K Khatri and L N Mishra ldquoInverse result insimultaneous approximation by Baskakov-Durrmeyer-Stancuoperatorsrdquo Journal of Inequalities and Applications vol 2013 no586 2013

[16] V N Mishra and P Patel ldquoSome approximation properties ofmodified jain-beta operatorsrdquo Journal of Calculus of Variationsvol 2013 Article ID 489249 8 pages 2013

[17] V N Mishra K Khatri and L N Mishra ldquoStatisticalapproximation by Kantorovich-type discrete q-Beta operatorsrdquoAdvances in Difference Equations vol 345 no 1 2013

[18] V N Mishra and P Patel ldquoOn simultaneous approximationfor generalized integral typerdquo Baskakov Operator Submitted forpublication

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

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Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


We denote 120601119899120574(119909 119905) = int

119905

0

119882119899120574(119909 119904)119889119904 then in particular

we have

120601119899120574(119909infin) = int

infin

0

119882119899120574(119909 119904) 119889119904 = 1 (3)

By 119863119861119903(0infin) 119903 ge 0 we denote the class of absolutely

continuous functions 119891 defined on the interval (0infin) suchthat

(i) 119891(119905) = 119874(119905119903) 119905 rarr infin(ii) having a derivative 1198911015840 on the interval (0infin) coincid-

ing ae with a function which is of bounded variationon every finite subinterval of (0infin)

It can be observed that all function 119891 isin 119861119863119903(0infin) possess

for each 119888 gt 0 a representation

119891 (119909) = 119891 (119888) + int

119909

119888

120595 (119905) 119889119905 119909 ge 119888 (4)

2 Rate of Convergence for 119861119899120574

Lemma 1 (see [1]) Let the function 119879119899119898120574

(119909)119898 isin N cup 0 bedefined as

119879119899119898120574

(119909) = 119861119899120574((119905 minus 119909)

119898

119909)

=

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119905 minus 119909)119898

119889119905

+ (1 + 120574119909)minus119899120574

(minus119909)119898

(5)

Then it is easily verified that for each 119909 isin (0infin) 1198791198990120574

(119909) = 11198791198991120574

(119909) = 0 and 1198791198992120574

(119909) = 2119909(1 + 120574119909)(119899 minus 120574) and also thefollowing recurrence relation holds

(119899 minus 120574119898)119879119899119898+1120574

(119909) = 119909 (1 + 120574119909)

times [119879(1)

119899119898120574(119909) + 2119898119879

119899119898minus1120574(119909)]

+ 119898 (1 + 2120574119909) 119879119899119898120574

(119909)

(6)

From the recurrence relation it can be easily be verified that forall 119909 isin [0infin) we have 119879

119899119898120574(119909) = 119874(119899

minus[(119898+1)2]

)

Remark 2 FromLemma 1 usingCauchy-Schwarz inequalityit follows that

119861119899120574(|119905 minus 119909| 119909) le [119861

119899120574((119905 minus 119909)

2

119909)]

12

le radic

2119909 (1 + 120574119909)

119899 minus 120574

(7)


in (1) Then for 119899 being sufficiently large one has

(a) 120601119899120574(119909 119910) = int

119910

0

119882119899120574(119909 119905)119889119905 le 2119909(1 + 120574119909)(119899 minus

120574)(119909 minus 119910)2

0 le 119910 lt 119909

(b) 1 minus 120601119899120574(119909 119911) = int

infin

119911

119882119899120574(119909 119905)119889119905 le 2119909(1 + 120574119909)(119899 minus

120574)(119911 minus 119909)2

119909 lt 119911 lt infin

Proof First we prove (a) by using Lemma 1 we have

int

119910

0

119882119899120574(119909 119905) 119889119905 le int

119910

0

(119909 minus 119905)2

(119909 minus 119910)2119882119899120574(119909 119905) 119889119905

le (119909 minus 119910)minus2

1198791198992120574

(119909)

le

2119909 (1 + 120574119909)

(119899 minus 120574) (119909 minus 119910)2

(8)

The proof of (b) is similar we omit the details

Theorem 4 Let 119891 isin 119863119861119903(0infin) 119903 isin N and 119909 isin (0infin) Then

for 119899 being sufficiently large we have10038161003816100381610038161003816119861119899120574(119891 119909) minus 119891 (119909)

10038161003816100381610038161003816

le

2 (1 + 120574119909)

119899 minus 120574

[radic119899]

sum

119896=1

119909+119909119896

⋁

119909minus119909119896

((1198911015840

)119909

)

+

119909

radic119899

119909+119909radic119899

⋁

119909minus119909radic119899

((1198911015840

)119909

) +

2 (1 + 120574119909)

(119899 minus 120574) 119909

times [

10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891

1015840

(119909+

)

10038161003816100381610038161003816+1003816100381610038161003816119891 (119909)

1003816100381610038161003816]

+ radic

2119909 (1 + 120574119909)

119899 minus 120574

[1198722119903

119874(119899minus1199032

) +

100381610038161003816100381610038161198911015840

(119909+

)

10038161003816100381610038161003816]

+ radic

119909 (1 + 120574119909)

2 (119899 minus 120574)

100381610038161003816100381610038161198911015840

(119909+

) minus 1198911015840

(119909minus

)

10038161003816100381610038161003816

(9)

where the auxiliary function 119891119909is given by

119891119909(119905) =

119891 (119905) minus 119891 (119909minus

) 0 le 119905 lt 119909

0 119905 = 119909

119891 (119905) minus 119891 (119909+

) 119909 lt 119905 lt infin

(10)

⋁119887

119886119891(119909) denotes the total variation of 119891

119909on [119886 119887]

Proof By the application of mean value theorem we have

119861119899120574(119891 119909) minus 119891 (119909) = int

infin

0

119882119899120574(119909 119905) (119891 (119905) minus 119891 (119909)) 119889119905

= int

infin

0

int

119905

119909

119882119899120574(119909 119905) (119891

1015840

(119906) 119889119906) 119889119905

(11)

Also using the identity

1198911015840

(119906) =

1198911015840

(119909+

) + 1198911015840

(119909minus

)

2

+ (1198911015840

)119909

(119906)

+

1198911015840

(119909+

) minus 1198911015840

(119909minus

)

2

sgn (119906 minus 119909)

+ [1198911015840

(119909) minus

1198911015840

(119909+

) + 1198911015840

(119909minus

)

2

] 120594119909(119906)

(12)


where

120594119909(119906) =

1 119906 = 119909

0 119906 = 119909

(13)

we can see that

int

infin

0

(int

119905

119909

(1198911015840

(119909) minus

1198911015840

(119909+

) + 1198911015840

(119909minus

)

2

)120594119909(119906) 119889119906)

times119882119899120574(119905 119909) 119889119905 = 0

(14)

Also

int

infin

0

(int

119905

119909

(

1198911015840

(119909+

) minus 1198911015840

(119909minus

)

2

) sgn (119906 minus 119909) 119889119906)119882119899120574(119905 119909) 119889119905

=

1198911015840

(119909+

) minus 1198911015840

(119909minus

)

2

119861119899120574(|119905 minus 119909| 119909)

int

infin

0

(int

119905

119909

(

1198911015840

(119909+

) + 1198911015840

(119909minus

)

2

) 119889119906)119882119899120574(119905 119909) 119889119905

=

1198911015840

(119909+

) + 1198911015840

(119909minus

)

2

119861119899120574(119905 minus 119909 119909)

(15)


10038161003816100381610038161003816119861119899120574(119891 119909) minus 119891 (119909)

10038161003816100381610038161003816le

10038161003816100381610038161003816100381610038161003816

int

infin

119909

(int

119905

119909

(1198911015840

)119909

(119906) 119889119906)119882119899120574(119909 119905) 119889119905

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

int

119909

0

(int

119905

119909

(1198911015840

)119909

(119906) 119889119906)119882119899120574(119909 119905) 119889119905

10038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161198911015840

(119909+

) minus 1198911015840

(119909minus

)

10038161003816100381610038161003816

2

119861119899120574(|119905 minus 119909| 119909)

+

100381610038161003816100381610038161198911015840

(119909+

) + 1198911015840

(119909minus

)

10038161003816100381610038161003816

2

119861119899120574(119905 minus 119909 119909)

(16)


10038161003816100381610038161003816119861119899120574(119891 119909) minus 119891 (119909)

10038161003816100381610038161003816le

10038161003816100381610038161003816100381610038161003816

int

infin

119909

(int

119905

119909

(1198911015840

)119909

(119906) 119889119906)119882119899120574(119909 119905) 119889119905

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

int

119909

0

(int

119905

119909

(1198911015840

)119909

(119906) 119889119906)119882119899120574(119909 119905) 119889119905

10038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161198911015840

(119909+

) minus 1198911015840

(119909minus

)

10038161003816100381610038161003816

2

radic

2119909 (1 + 120574119909)

119899 minus 120574

=

10038161003816100381610038161003816119875119899120574(119891 119909)

10038161003816100381610038161003816+

10038161003816100381610038161003816119876119899120574(119891 119909)

10038161003816100381610038161003816

+

100381610038161003816100381610038161198911015840

(119909+

) minus 1198911015840

(119909minus

)

10038161003816100381610038161003816

2

radic

2119909 (1 + 120574119909)

119899 minus 120574

(17)


10038161003816100381610038161003816119876119899120574(119891 119909)

10038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816

int

119909

0

int

119905

119909

(1198911015840

)119909

(119906) 119889119906 119889119905 (120601119899120574(119909 119910))

10038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816

int

119909

0

120601119899120574(119909 119910) (119891

1015840

)119909

(119905) 119889119905

10038161003816100381610038161003816100381610038161003816

le (int

119910

0

+int

119909

119910

)

10038161003816100381610038161003816(1198911015840

)119909

(119905)

10038161003816100381610038161003816

10038161003816100381610038161003816120601119899120574(119909 119905)

10038161003816100381610038161003816119889119905

le

2119909 (1 + 120574119909)

119899 minus 120574

int

119910

0

119909

⋁

119905

((1198911015840

)119909

)

1

(119909 minus 119905)2119889119905

+ int

119909

119910

119909

⋁

119905

((1198911015840

)119909

) 119889119905

le

2119909 (1 + 120574119909)

119899 minus 120574

int

119910

0

119909

⋁

119905

((1198911015840

)119909

)

1

(119909 minus 119905)2119889119905

+

119909

radic119899

119909

⋁

119909minus119909radic119899

((1198911015840

)119909

)

=

2119909 (1 + 120574119909)

119899 minus 120574

int

radic119899

1

119909

⋁

119909minus119909119906

((1198911015840

)119909

) 119889119906

+

119909

radic119899

119909

⋁

119909minus119909radic119899

((1198911015840

)119909

)

le

2119909 (1 + 120574119909)

119899 minus 120574

[radic119899]

sum

119896=1

119909

⋁

119909minus119909119896

((1198911015840

)119909

)

+

119909

radic119899

119909

⋁

119909minus119909radic119899

((1198911015840

)119909

)

(18)


10038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816

int

infin

119909

(int

119905

119909

(1198911015840

)119909

(119906) 119889119906)119882119899120574(119909 119905) 119889119905

10038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816

int

infin

2119909

(int

119905

119909

(1198911015840

)119909

(119906) 119889119906)119882119899120574(119909 119905) 119889119905

+int

2119909

119909

(int

119905

119909

(1198911015840

)119909

(119906) 119889119906)119889119905 (1 minus 120601119899120574(119909 119905))

100381610038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

int

infin

2119909

(119891 (119905) minus 119891 (119909))119882119899120574(119909 119905) 119889119905

10038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161198911015840

(119909+

)

10038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

infin

2119909

(119905 minus 119909)119882119899120574(119909 119905) 119889119905

10038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

2119909

119909

(1198911015840

)119909

(119906) 119889119906

100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381610038161 minus 120601119909120574(119909 2119909)

10038161003816100381610038161003816


+ int

2119909

119909

10038161003816100381610038161003816(1198911015840

)119909

(119905)

10038161003816100381610038161003816

100381610038161003816100381610038161 minus 120601119909120574(119909 119905)

10038161003816100381610038161003816119889119905

le [

119872

119909

int

infin

2119909

119882119899120574(119909 119905) 119905

119903

|119905 minus 119909| 119889119905

+

1003816100381610038161003816119891 (119909)

1003816100381610038161003816

1199092

int

infin

2119909

119882119899120574(119909 119905) (119905 minus 119909)

2

119889119905]

+ [

100381610038161003816100381610038161198911015840

(119909+

)

10038161003816100381610038161003816int

infin

2119909

119882119899120574(119909 119905) |119905 minus 119909| 119889119905]

+

2 (1 + 120574119909)

(119899 minus 120574) 119909

10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891

1015840

(119909+

)

10038161003816100381610038161003816

+

2 (1 + 120574119909)

119899 minus 120574

[radic119899]

sum

119896=1

119909+119909119896

⋁

119909

((1198911015840

)119909

)

+

119909

radic119899

119909+119909radic119899

⋁

119909

((1198911015840

)119909

)

= 119877119899120574(119891 119909) + 119878

119899120574(119891 119909)

+

2 (1 + 120574119909)

(119899 minus 120574) 119909

10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891

1015840

(119909+

)

10038161003816100381610038161003816

+

2 (1 + 120574119909)

119899 minus 120574

[radic119899]

sum

119896=1

119909+119909119896

⋁

119909

((1198911015840

)119909

)

+

119909

radic119899

119909+119909radic119899

⋁

119909

((1198911015840

)119909

)

(19)


119877119899120574(119891 119909) le

119872

119909

(int

infin

2119909

119882119899120574(119909 119905) 119905

2119903

119889119905)

12

times(int

infin

0

119882119899120574(119909 119905) (119905 minus 119909)

2

119889119905)

12

+

1003816100381610038161003816119891 (119909)

1003816100381610038161003816

1199092

int

infin

2119909

119882119899120574(119909 119905) (119905 minus 119909)

2

119889119905

le 1198722119903

119874(119899minus1199032

)radic

2119909 (1 + 120574119909)

119899 minus 120574

+1003816100381610038161003816119891 (119909)

1003816100381610038161003816

2 (1 + 120574119909)

(119899 minus 120574) 119909

(20)

Also

119878119899120574(119891 119909) le

1003816100381610038161003816119891 (119909+

)1003816100381610038161003816int

infin

0

119882119899(119905 119909) |119905 minus 119909| 119889119905

le1003816100381610038161003816119891 (119909+

)1003816100381610038161003816radic

2119909 (1 + 120574119909)

119899 minus 120574

(21)



119899120574


119861120572120573

119899120574(119891 (119905) 119909) =

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) 119891(

119899119905 + 120572

119899 + 120573

)119889119905

+ 1199011198990120574

(119909) 119891(

120572

119899 + 120573

)

= int

infin

0

119882119899120574(119909 119905) 119891(

119899119905 + 120572

119899 + 120573

)119889119905

(22)

where 119901119899119896120574

(119909) 119887119899119896120574



120583120572120573

119899119898120574(119909) = 119861

120572120573

119899120574((119905 minus 119909)

119898

119909)

=

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (

119899119905 + 120572

119899 + 120573

minus 119909)

119898

119889119905

+ 1199011198990120574

(119909) (

120572

119899 + 120573

minus 119909)

119898

(23)

then 1205831205721205731198990120574

(119909) = 1 1205831205721205731198991120574

(119909) = (120572 minus 120573119909)(119899 + 120573) and

120583120572120573

1198992120574(119909) =

1205722

(119899 + 120573)2+ 119909

(21198992

minus 2119899120572120573 + 2120572120573120574)

(119899 + 120573)2

(119899 minus 120574)

+ 1199092(1198991205732

+ 21198992

120574 minus 1205732

120574)

(119899 + 120573)2

(119899 minus 120574)

(24)


(119899 minus 120574119898) (119899 + 120573) 120583120572120573

119899119898+1120574(119909)

= 119899119909 (1 + 120574119909) [(120583120572120573

119899119898120574)

(1)

(119909) + 119898120583120572120573

119899119898minus1120574(119909)]

+ [119898119899 + 1198992


119899119898120574(119909)

+ [119898120574 (119899 + 120573) (

120572

119899 + 120573

minus 119909)

2

minus 119898119899(

120572

119899 + 120573

minus 119909)]

times 120583120572120573

119899119898minus1120574(119909)

(25)



119899119898120574(119909) = 119874(119899

minus[(119898+1)2]

)



120583120572120573

1198992120574(119909) = [

1205732

120574(119899 + 120573)2+

21198992

(119899 + 120573)2

(119899 minus 120574)

] 119909 (1 + 120574119909)

+

120573 [(2120572 + 120573) 120574 minus 119899 (120573 + 2120572120574)]

120574(119899 + 120573)2

(119899 minus 120574)

119909 +

1205722

(119899 + 120573)2

(26)


120583120572120573

1198992120574(119909) le

119862119909 (1 + 120574119909)

119899 minus 120574

(27)



119861120572120573

119899120574(|119905 minus 119909| 119909) le [119861

120572120573

119899120574((119905 minus 119909)

2

119909)]

12

le radic

119862119909 (1 + 120574119909)

119899 minus 120574

(28)



(c) 120601119899120574(119909 119910) = int

119910

0

119882119899120574(119909 119905)119889119905 = 119862119909(1 + 120574119909)(119899 minus

120574)(119909 minus 119910)2

0 le 119910 lt 119909

(d) 1 minus 120601119899120574(119909 119911) = int

infin

119911

119882119899120574(119909 119905)119889119905 = 119862119909(1 + 120574119909)(119899 minus

120574)(119911 minus 119909)2

119909 lt 119911 lt infin




10038161003816100381610038161003816119861120572120573

119899120574(119891 119909) minus 119891 (119909)

10038161003816100381610038161003816

le

119862 (1 + 120574119909)

119899 minus 120574

[radic119899]

sum

119896=1

119909+119909119896

⋁

119909minus119909119896

((1198911015840

)119909

)

+

119909

radic119899

119909+119909radic119899

⋁

119909minus119909radic119899

((1198911015840

)119909

) +

119862 (1 + 120574119909)

(119899 minus 120574) 119909

times [

10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891

1015840

(119909+

)

10038161003816100381610038161003816+1003816100381610038161003816119891 (119909)

1003816100381610038161003816]

+ radic

119862119909 (1 + 120574119909)

119899 minus 120574

times [11987212119903

119874(119899minus1199032

) +

100381610038161003816100381610038161198911015840

(119909+

)

10038161003816100381610038161003816]

+

1

2

radic

119862119909 (1 + 120574119909)

(119899 minus 120574)

100381610038161003816100381610038161198911015840

(119909+

) minus 1198911015840

(119909minus

)

10038161003816100381610038161003816

+

1

2

100381610038161003816100381610038161198911015840

(119909+

) + 1198911015840

(119909minus

)

10038161003816100381610038161003816

1003816100381610038161003816120572 minus 120573119909

1003816100381610038161003816

119899 + 120573

(29)


119886119891(119909) were defined in





Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

120594119909(119906) =

1 119906 = 119909

0 119906 = 119909

(13)

we can see that

int

infin

0

(int

119905

119909

(1198911015840

(119909) minus

1198911015840

(119909+

) + 1198911015840

(119909minus

)

2

)120594119909(119906) 119889119906)

times119882119899120574(119905 119909) 119889119905 = 0

(14)

Also

int

infin

0

(int

119905

119909

(

1198911015840

(119909+

) minus 1198911015840

(119909minus

)

2

) sgn (119906 minus 119909) 119889119906)119882119899120574(119905 119909) 119889119905

=

1198911015840

(119909+

) minus 1198911015840

(119909minus

)

2

119861119899120574(|119905 minus 119909| 119909)

int

infin

0

(int

119905

119909

(

1198911015840

(119909+

) + 1198911015840

(119909minus

)

2

) 119889119906)119882119899120574(119905 119909) 119889119905

=

1198911015840

(119909+

) + 1198911015840

(119909minus

)

2

119861119899120574(119905 minus 119909 119909)

(15)


10038161003816100381610038161003816119861119899120574(119891 119909) minus 119891 (119909)

10038161003816100381610038161003816le

10038161003816100381610038161003816100381610038161003816

int

infin

119909

(int

119905

119909

(1198911015840

)119909

(119906) 119889119906)119882119899120574(119909 119905) 119889119905

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

int

119909

0

(int

119905

119909

(1198911015840

)119909

(119906) 119889119906)119882119899120574(119909 119905) 119889119905

10038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161198911015840

(119909+

) minus 1198911015840

(119909minus

)

10038161003816100381610038161003816

2

119861119899120574(|119905 minus 119909| 119909)

+

100381610038161003816100381610038161198911015840

(119909+

) + 1198911015840

(119909minus

)

10038161003816100381610038161003816

2

119861119899120574(119905 minus 119909 119909)

(16)


10038161003816100381610038161003816119861119899120574(119891 119909) minus 119891 (119909)

10038161003816100381610038161003816le

10038161003816100381610038161003816100381610038161003816

int

infin

119909

(int

119905

119909

(1198911015840

)119909

(119906) 119889119906)119882119899120574(119909 119905) 119889119905

10038161003816100381610038161003816100381610038161003816

+

10038161003816100381610038161003816100381610038161003816

int

119909

0

(int

119905

119909

(1198911015840

)119909

(119906) 119889119906)119882119899120574(119909 119905) 119889119905

10038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161198911015840

(119909+

) minus 1198911015840

(119909minus

)

10038161003816100381610038161003816

2

radic

2119909 (1 + 120574119909)

119899 minus 120574

=

10038161003816100381610038161003816119875119899120574(119891 119909)

10038161003816100381610038161003816+

10038161003816100381610038161003816119876119899120574(119891 119909)

10038161003816100381610038161003816

+

100381610038161003816100381610038161198911015840

(119909+

) minus 1198911015840

(119909minus

)

10038161003816100381610038161003816

2

radic

2119909 (1 + 120574119909)

119899 minus 120574

(17)


10038161003816100381610038161003816119876119899120574(119891 119909)

10038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161003816

int

119909

0

int

119905

119909

(1198911015840

)119909

(119906) 119889119906 119889119905 (120601119899120574(119909 119910))

10038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816

int

119909

0

120601119899120574(119909 119910) (119891

1015840

)119909

(119905) 119889119905

10038161003816100381610038161003816100381610038161003816

le (int

119910

0

+int

119909

119910

)

10038161003816100381610038161003816(1198911015840

)119909

(119905)

10038161003816100381610038161003816

10038161003816100381610038161003816120601119899120574(119909 119905)

10038161003816100381610038161003816119889119905

le

2119909 (1 + 120574119909)

119899 minus 120574

int

119910

0

119909

⋁

119905

((1198911015840

)119909

)

1

(119909 minus 119905)2119889119905

+ int

119909

119910

119909

⋁

119905

((1198911015840

)119909

) 119889119905

le

2119909 (1 + 120574119909)

119899 minus 120574

int

119910

0

119909

⋁

119905

((1198911015840

)119909

)

1

(119909 minus 119905)2119889119905

+

119909

radic119899

119909

⋁

119909minus119909radic119899

((1198911015840

)119909

)

=

2119909 (1 + 120574119909)

119899 minus 120574

int

radic119899

1

119909

⋁

119909minus119909119906

((1198911015840

)119909

) 119889119906

+

119909

radic119899

119909

⋁

119909minus119909radic119899

((1198911015840

)119909

)

le

2119909 (1 + 120574119909)

119899 minus 120574

[radic119899]

sum

119896=1

119909

⋁

119909minus119909119896

((1198911015840

)119909

)

+

119909

radic119899

119909

⋁

119909minus119909radic119899

((1198911015840

)119909

)

(18)


10038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816

int

infin

119909

(int

119905

119909

(1198911015840

)119909

(119906) 119889119906)119882119899120574(119909 119905) 119889119905

10038161003816100381610038161003816100381610038161003816

=

10038161003816100381610038161003816100381610038161003816

int

infin

2119909

(int

119905

119909

(1198911015840

)119909

(119906) 119889119906)119882119899120574(119909 119905) 119889119905

+int

2119909

119909

(int

119905

119909

(1198911015840

)119909

(119906) 119889119906)119889119905 (1 minus 120601119899120574(119909 119905))

100381610038161003816100381610038161003816100381610038161003816

le

10038161003816100381610038161003816100381610038161003816

int

infin

2119909

(119891 (119905) minus 119891 (119909))119882119899120574(119909 119905) 119889119905

10038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161198911015840

(119909+

)

10038161003816100381610038161003816

10038161003816100381610038161003816100381610038161003816

int

infin

2119909

(119905 minus 119909)119882119899120574(119909 119905) 119889119905

10038161003816100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816100381610038161003816

int

2119909

119909

(1198911015840

)119909

(119906) 119889119906

100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381610038161 minus 120601119909120574(119909 2119909)

10038161003816100381610038161003816


+ int

2119909

119909

10038161003816100381610038161003816(1198911015840

)119909

(119905)

10038161003816100381610038161003816

100381610038161003816100381610038161 minus 120601119909120574(119909 119905)

10038161003816100381610038161003816119889119905

le [

119872

119909

int

infin

2119909

119882119899120574(119909 119905) 119905

119903

|119905 minus 119909| 119889119905

+

1003816100381610038161003816119891 (119909)

1003816100381610038161003816

1199092

int

infin

2119909

119882119899120574(119909 119905) (119905 minus 119909)

2

119889119905]

+ [

100381610038161003816100381610038161198911015840

(119909+

)

10038161003816100381610038161003816int

infin

2119909

119882119899120574(119909 119905) |119905 minus 119909| 119889119905]

+

2 (1 + 120574119909)

(119899 minus 120574) 119909

10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891

1015840

(119909+

)

10038161003816100381610038161003816

+

2 (1 + 120574119909)

119899 minus 120574

[radic119899]

sum

119896=1

119909+119909119896

⋁

119909

((1198911015840

)119909

)

+

119909

radic119899

119909+119909radic119899

⋁

119909

((1198911015840

)119909

)

= 119877119899120574(119891 119909) + 119878

119899120574(119891 119909)

+

2 (1 + 120574119909)

(119899 minus 120574) 119909

10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891

1015840

(119909+

)

10038161003816100381610038161003816

+

2 (1 + 120574119909)

119899 minus 120574

[radic119899]

sum

119896=1

119909+119909119896

⋁

119909

((1198911015840

)119909

)

+

119909

radic119899

119909+119909radic119899

⋁

119909

((1198911015840

)119909

)

(19)


119877119899120574(119891 119909) le

119872

119909

(int

infin

2119909

119882119899120574(119909 119905) 119905

2119903

119889119905)

12

times(int

infin

0

119882119899120574(119909 119905) (119905 minus 119909)

2

119889119905)

12

+

1003816100381610038161003816119891 (119909)

1003816100381610038161003816

1199092

int

infin

2119909

119882119899120574(119909 119905) (119905 minus 119909)

2

119889119905

le 1198722119903

119874(119899minus1199032

)radic

2119909 (1 + 120574119909)

119899 minus 120574

+1003816100381610038161003816119891 (119909)

1003816100381610038161003816

2 (1 + 120574119909)

(119899 minus 120574) 119909

(20)

Also

119878119899120574(119891 119909) le

1003816100381610038161003816119891 (119909+

)1003816100381610038161003816int

infin

0

119882119899(119905 119909) |119905 minus 119909| 119889119905

le1003816100381610038161003816119891 (119909+

)1003816100381610038161003816radic

2119909 (1 + 120574119909)

119899 minus 120574

(21)



119899120574


119861120572120573

119899120574(119891 (119905) 119909) =

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) 119891(

119899119905 + 120572

119899 + 120573

)119889119905

+ 1199011198990120574

(119909) 119891(

120572

119899 + 120573

)

= int

infin

0

119882119899120574(119909 119905) 119891(

119899119905 + 120572

119899 + 120573

)119889119905

(22)

where 119901119899119896120574

(119909) 119887119899119896120574



120583120572120573

119899119898120574(119909) = 119861

120572120573

119899120574((119905 minus 119909)

119898

119909)

=

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (

119899119905 + 120572

119899 + 120573

minus 119909)

119898

119889119905

+ 1199011198990120574

(119909) (

120572

119899 + 120573

minus 119909)

119898

(23)

then 1205831205721205731198990120574

(119909) = 1 1205831205721205731198991120574

(119909) = (120572 minus 120573119909)(119899 + 120573) and

120583120572120573

1198992120574(119909) =

1205722

(119899 + 120573)2+ 119909

(21198992

minus 2119899120572120573 + 2120572120573120574)

(119899 + 120573)2

(119899 minus 120574)

+ 1199092(1198991205732

+ 21198992

120574 minus 1205732

120574)

(119899 + 120573)2

(119899 minus 120574)

(24)


(119899 minus 120574119898) (119899 + 120573) 120583120572120573

119899119898+1120574(119909)

= 119899119909 (1 + 120574119909) [(120583120572120573

119899119898120574)

(1)

(119909) + 119898120583120572120573

119899119898minus1120574(119909)]

+ [119898119899 + 1198992


119899119898120574(119909)

+ [119898120574 (119899 + 120573) (

120572

119899 + 120573

minus 119909)

2

minus 119898119899(

120572

119899 + 120573

minus 119909)]

times 120583120572120573

119899119898minus1120574(119909)

(25)



119899119898120574(119909) = 119874(119899

minus[(119898+1)2]

)



120583120572120573

1198992120574(119909) = [

1205732

120574(119899 + 120573)2+

21198992

(119899 + 120573)2

(119899 minus 120574)

] 119909 (1 + 120574119909)

+

120573 [(2120572 + 120573) 120574 minus 119899 (120573 + 2120572120574)]

120574(119899 + 120573)2

(119899 minus 120574)

119909 +

1205722

(119899 + 120573)2

(26)


120583120572120573

1198992120574(119909) le

119862119909 (1 + 120574119909)

119899 minus 120574

(27)



119861120572120573

119899120574(|119905 minus 119909| 119909) le [119861

120572120573

119899120574((119905 minus 119909)

2

119909)]

12

le radic

119862119909 (1 + 120574119909)

119899 minus 120574

(28)



(c) 120601119899120574(119909 119910) = int

119910

0

119882119899120574(119909 119905)119889119905 = 119862119909(1 + 120574119909)(119899 minus

120574)(119909 minus 119910)2

0 le 119910 lt 119909

(d) 1 minus 120601119899120574(119909 119911) = int

infin

119911

119882119899120574(119909 119905)119889119905 = 119862119909(1 + 120574119909)(119899 minus

120574)(119911 minus 119909)2

119909 lt 119911 lt infin




10038161003816100381610038161003816119861120572120573

119899120574(119891 119909) minus 119891 (119909)

10038161003816100381610038161003816

le

119862 (1 + 120574119909)

119899 minus 120574

[radic119899]

sum

119896=1

119909+119909119896

⋁

119909minus119909119896

((1198911015840

)119909

)

+

119909

radic119899

119909+119909radic119899

⋁

119909minus119909radic119899

((1198911015840

)119909

) +

119862 (1 + 120574119909)

(119899 minus 120574) 119909

times [

10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891

1015840

(119909+

)

10038161003816100381610038161003816+1003816100381610038161003816119891 (119909)

1003816100381610038161003816]

+ radic

119862119909 (1 + 120574119909)

119899 minus 120574

times [11987212119903

119874(119899minus1199032

) +

100381610038161003816100381610038161198911015840

(119909+

)

10038161003816100381610038161003816]

+

1

2

radic

119862119909 (1 + 120574119909)

(119899 minus 120574)

100381610038161003816100381610038161198911015840

(119909+

) minus 1198911015840

(119909minus

)

10038161003816100381610038161003816

+

1

2

100381610038161003816100381610038161198911015840

(119909+

) + 1198911015840

(119909minus

)

10038161003816100381610038161003816

1003816100381610038161003816120572 minus 120573119909

1003816100381610038161003816

119899 + 120573

(29)


119886119891(119909) were defined in





Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+ int

2119909

119909

10038161003816100381610038161003816(1198911015840

)119909

(119905)

10038161003816100381610038161003816

100381610038161003816100381610038161 minus 120601119909120574(119909 119905)

10038161003816100381610038161003816119889119905

le [

119872

119909

int

infin

2119909

119882119899120574(119909 119905) 119905

119903

|119905 minus 119909| 119889119905

+

1003816100381610038161003816119891 (119909)

1003816100381610038161003816

1199092

int

infin

2119909

119882119899120574(119909 119905) (119905 minus 119909)

2

119889119905]

+ [

100381610038161003816100381610038161198911015840

(119909+

)

10038161003816100381610038161003816int

infin

2119909

119882119899120574(119909 119905) |119905 minus 119909| 119889119905]

+

2 (1 + 120574119909)

(119899 minus 120574) 119909

10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891

1015840

(119909+

)

10038161003816100381610038161003816

+

2 (1 + 120574119909)

119899 minus 120574

[radic119899]

sum

119896=1

119909+119909119896

⋁

119909

((1198911015840

)119909

)

+

119909

radic119899

119909+119909radic119899

⋁

119909

((1198911015840

)119909

)

= 119877119899120574(119891 119909) + 119878

119899120574(119891 119909)

+

2 (1 + 120574119909)

(119899 minus 120574) 119909

10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891

1015840

(119909+

)

10038161003816100381610038161003816

+

2 (1 + 120574119909)

119899 minus 120574

[radic119899]

sum

119896=1

119909+119909119896

⋁

119909

((1198911015840

)119909

)

+

119909

radic119899

119909+119909radic119899

⋁

119909

((1198911015840

)119909

)

(19)


119877119899120574(119891 119909) le

119872

119909

(int

infin

2119909

119882119899120574(119909 119905) 119905

2119903

119889119905)

12

times(int

infin

0

119882119899120574(119909 119905) (119905 minus 119909)

2

119889119905)

12

+

1003816100381610038161003816119891 (119909)

1003816100381610038161003816

1199092

int

infin

2119909

119882119899120574(119909 119905) (119905 minus 119909)

2

119889119905

le 1198722119903

119874(119899minus1199032

)radic

2119909 (1 + 120574119909)

119899 minus 120574

+1003816100381610038161003816119891 (119909)

1003816100381610038161003816

2 (1 + 120574119909)

(119899 minus 120574) 119909

(20)

Also

119878119899120574(119891 119909) le

1003816100381610038161003816119891 (119909+

)1003816100381610038161003816int

infin

0

119882119899(119905 119909) |119905 minus 119909| 119889119905

le1003816100381610038161003816119891 (119909+

)1003816100381610038161003816radic

2119909 (1 + 120574119909)

119899 minus 120574

(21)



119899120574


119861120572120573

119899120574(119891 (119905) 119909) =

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) 119891(

119899119905 + 120572

119899 + 120573

)119889119905

+ 1199011198990120574

(119909) 119891(

120572

119899 + 120573

)

= int

infin

0

119882119899120574(119909 119905) 119891(

119899119905 + 120572

119899 + 120573

)119889119905

(22)

where 119901119899119896120574

(119909) 119887119899119896120574



120583120572120573

119899119898120574(119909) = 119861

120572120573

119899120574((119905 minus 119909)

119898

119909)

=

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (

119899119905 + 120572

119899 + 120573

minus 119909)

119898

119889119905

+ 1199011198990120574

(119909) (

120572

119899 + 120573

minus 119909)

119898

(23)

then 1205831205721205731198990120574

(119909) = 1 1205831205721205731198991120574

(119909) = (120572 minus 120573119909)(119899 + 120573) and

120583120572120573

1198992120574(119909) =

1205722

(119899 + 120573)2+ 119909

(21198992

minus 2119899120572120573 + 2120572120573120574)

(119899 + 120573)2

(119899 minus 120574)

+ 1199092(1198991205732

+ 21198992

120574 minus 1205732

120574)

(119899 + 120573)2

(119899 minus 120574)

(24)


(119899 minus 120574119898) (119899 + 120573) 120583120572120573

119899119898+1120574(119909)

= 119899119909 (1 + 120574119909) [(120583120572120573

119899119898120574)

(1)

(119909) + 119898120583120572120573

119899119898minus1120574(119909)]

+ [119898119899 + 1198992


119899119898120574(119909)

+ [119898120574 (119899 + 120573) (

120572

119899 + 120573

minus 119909)

2

minus 119898119899(

120572

119899 + 120573

minus 119909)]

times 120583120572120573

119899119898minus1120574(119909)

(25)



119899119898120574(119909) = 119874(119899

minus[(119898+1)2]

)



120583120572120573

1198992120574(119909) = [

1205732

120574(119899 + 120573)2+

21198992

(119899 + 120573)2

(119899 minus 120574)

] 119909 (1 + 120574119909)

+

120573 [(2120572 + 120573) 120574 minus 119899 (120573 + 2120572120574)]

120574(119899 + 120573)2

(119899 minus 120574)

119909 +

1205722

(119899 + 120573)2

(26)


120583120572120573

1198992120574(119909) le

119862119909 (1 + 120574119909)

119899 minus 120574

(27)



119861120572120573

119899120574(|119905 minus 119909| 119909) le [119861

120572120573

119899120574((119905 minus 119909)

2

119909)]

12

le radic

119862119909 (1 + 120574119909)

119899 minus 120574

(28)



(c) 120601119899120574(119909 119910) = int

119910

0

119882119899120574(119909 119905)119889119905 = 119862119909(1 + 120574119909)(119899 minus

120574)(119909 minus 119910)2

0 le 119910 lt 119909

(d) 1 minus 120601119899120574(119909 119911) = int

infin

119911

119882119899120574(119909 119905)119889119905 = 119862119909(1 + 120574119909)(119899 minus

120574)(119911 minus 119909)2

119909 lt 119911 lt infin




10038161003816100381610038161003816119861120572120573

119899120574(119891 119909) minus 119891 (119909)

10038161003816100381610038161003816

le

119862 (1 + 120574119909)

119899 minus 120574

[radic119899]

sum

119896=1

119909+119909119896

⋁

119909minus119909119896

((1198911015840

)119909

)

+

119909

radic119899

119909+119909radic119899

⋁

119909minus119909radic119899

((1198911015840

)119909

) +

119862 (1 + 120574119909)

(119899 minus 120574) 119909

times [

10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891

1015840

(119909+

)

10038161003816100381610038161003816+1003816100381610038161003816119891 (119909)

1003816100381610038161003816]

+ radic

119862119909 (1 + 120574119909)

119899 minus 120574

times [11987212119903

119874(119899minus1199032

) +

100381610038161003816100381610038161198911015840

(119909+

)

10038161003816100381610038161003816]

+

1

2

radic

119862119909 (1 + 120574119909)

(119899 minus 120574)

100381610038161003816100381610038161198911015840

(119909+

) minus 1198911015840

(119909minus

)

10038161003816100381610038161003816

+

1

2

100381610038161003816100381610038161198911015840

(119909+

) + 1198911015840

(119909minus

)

10038161003816100381610038161003816

1003816100381610038161003816120572 minus 120573119909

1003816100381610038161003816

119899 + 120573

(29)


119886119891(119909) were defined in





Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119899119898120574(119909) = 119874(119899

minus[(119898+1)2]

)



120583120572120573

1198992120574(119909) = [

1205732

120574(119899 + 120573)2+

21198992

(119899 + 120573)2

(119899 minus 120574)

] 119909 (1 + 120574119909)

+

120573 [(2120572 + 120573) 120574 minus 119899 (120573 + 2120572120574)]

120574(119899 + 120573)2

(119899 minus 120574)

119909 +

1205722

(119899 + 120573)2

(26)


120583120572120573

1198992120574(119909) le

119862119909 (1 + 120574119909)

119899 minus 120574

(27)



119861120572120573

119899120574(|119905 minus 119909| 119909) le [119861

120572120573

119899120574((119905 minus 119909)

2

119909)]

12

le radic

119862119909 (1 + 120574119909)

119899 minus 120574

(28)



(c) 120601119899120574(119909 119910) = int

119910

0

119882119899120574(119909 119905)119889119905 = 119862119909(1 + 120574119909)(119899 minus

120574)(119909 minus 119910)2

0 le 119910 lt 119909

(d) 1 minus 120601119899120574(119909 119911) = int

infin

119911

119882119899120574(119909 119905)119889119905 = 119862119909(1 + 120574119909)(119899 minus

120574)(119911 minus 119909)2

119909 lt 119911 lt infin




10038161003816100381610038161003816119861120572120573

119899120574(119891 119909) minus 119891 (119909)

10038161003816100381610038161003816

le

119862 (1 + 120574119909)

119899 minus 120574

[radic119899]

sum

119896=1

119909+119909119896

⋁

119909minus119909119896

((1198911015840

)119909

)

+

119909

radic119899

119909+119909radic119899

⋁

119909minus119909radic119899

((1198911015840

)119909

) +

119862 (1 + 120574119909)

(119899 minus 120574) 119909

times [

10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891

1015840

(119909+

)

10038161003816100381610038161003816+1003816100381610038161003816119891 (119909)

1003816100381610038161003816]

+ radic

119862119909 (1 + 120574119909)

119899 minus 120574

times [11987212119903

119874(119899minus1199032

) +

100381610038161003816100381610038161198911015840

(119909+

)

10038161003816100381610038161003816]

+

1

2

radic

119862119909 (1 + 120574119909)

(119899 minus 120574)

100381610038161003816100381610038161198911015840

(119909+

) minus 1198911015840

(119909minus

)

10038161003816100381610038161003816

+

1

2

100381610038161003816100381610038161198911015840

(119909+

) + 1198911015840

(119909minus

)

10038161003816100381610038161003816

1003816100381610038161003816120572 minus 120573119909

1003816100381610038161003816

119899 + 120573

(29)


119886119891(119909) were defined in





Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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Research Article Rate of Convergence of Modified Baskakov …downloads.hindawi.com/archive/2014/235480.pdf · 2019. 7. 31. · Research Article Rate of Convergence of Modified Baskakov-Durrmeyer