Research ArticleOn Simultaneous Approximation of ModifiedBaskakov-Durrmeyer Operators

Prashantkumar G Patel12 and Vishnu Narayan Mishra13

1Department of Applied Mathematics and Humanities Sardar Vallabhbhai National Institute of TechnologyIchchhanath Mahadev Dumas Road Surat Gujarat 395 007 India2Department of Mathematics St Xavier College Ahmedabad Gujarat 380 009 India3L 1627 Awadh Puri Colony Beniganj Phase-III Opposite-Industrial Training Institute (ITI) Ayodhya Main Road FaizabadUttar Pradesh 224 001 India

Correspondence should be addressed to Prashantkumar G Patel prashant225gmailcom

Received 22 July 2015 Accepted 10 September 2015

Academic Editor Ying Hu

Copyright copy 2015 P G 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 discuss properties of modified Baskakov-Durrmeyer-Stancu (BDS) operators with parameter 120574 gt 0 We compute the momentsof these modified operators Also we establish pointwise convergence Voronovskaja type asymptotic formula and an errorestimation in terms of second order modification of continuity of the function for the operators 119861120572120573

119899120574(119891 119909)

1 Introduction

For 119909 isin [0infin) 120574 gt 0 0 le 120572 le 120573 and 119891 isin 119862[0infin)we consider a certain integral type generalized Baskakovoperators as

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

(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) 119887119899119896120574

(119905) + (1 + 120574119909)minus119899120574

120575 (119905)

(2)

120575(119905) being the Dirac delta functionThe operators defined by (1) are the generalization of

the integral modification of well-known Baskakov operatorshaving weight function of some beta basis function As aspecial case that is 120574 = 1 the operators (1) reduce to theoperators very recently studied in [1 2] Inverse results ofsame type of operatorswere established in [3] Also if120572 = 120573 =

0 the operators (1) reduce to the operators recently studied in[4] and if 120572 = 120573 = 0 and 120574 = 1 the operators (1) reduce to theoperators studied in [5] The 119902-analog of the operators (1) isdiscussed in [6] We refer to some of the important papers onthe recent development on similar type of the operators [7ndash9]

Hindawi Publishing CorporationInternational Journal of AnalysisVolume 2015 Article ID 805395 10 pageshttpdxdoiorg1011552015805395

2 International Journal of Analysis

The present a paper that deals with the study of simultaneousapproximation for the operators 119861120572120573

119899120574

2 Moments and Recurrence Relations

Lemma 1 If one defines the central moments for every119898 isin Nas

120583119899119898120574

(119909) = 119861120572120573

119899120574((119905 minus 119909)

119898

119909)

=

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905

+ 1199011198990120574

(119909) (120572

119899 + 120573minus 119909)

119898

(3)

then 1205831198990120574

(119909) = 1 1205831198991120574

(119909) = (120572minus120573119909)(119899+120573) and for 119899 gt 120574119898one has the following recurrence relation

(119899 minus 120574119898) (119899 + 120573) 120583119899119898+1120574

(119909) = 119899119909 (1 + 120574119909)

sdot 120583(1)

119899119898120574(119909) + 119898120583

119899119898minus1120574(119909)

+ 119898119899 + 1198992

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

sdot 120583119899119898120574

(119909)

+ 119898120574 (119899 + 120573) (120572

119899 + 120573minus 119909)

2

minus 119898119899(120572

119899 + 120573minus 119909)

sdot 120583119899119898minus1120574

(119909)

(4)

From the recurrence relation it can be easily verified that forall 119909 isin [0infin) one has 120583

119899119898120574(119909) = 119874(119899

minus[(119898+1)2]

) where [120572]

denotes the integral part of 120572

Proof Taking derivative of the above

120583(1)

119899119898120574(119909) = minus119898

infin

sum

119896=1

119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898minus1

119889119905 minus 1198981199011198990120574

(119909)

sdot (120572

119899 + 120573minus 119909)

119898minus1

+

infin

sum

119896=1

119901(1)

119899119896120574(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905 + 119901(1)

1198990120574(119909)

sdot (120572

119899 + 120573minus 119909)

119898

= minus119898120583119899119898minus1120574

(119909) +

infin

sum

119896=1

119901(1)

119899119896120574(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905 + 119901(1)

1198990120574(119909)

sdot (120572

119899 + 120573minus 119909)

119898

119909 (1 + 120574119909) 120583(1)

119899119898120574(119909) + 119898120583

119899119898minus1120574(119909)

=

infin

sum

119896=1

119909 (1 + 120574119909) 119901(1)

119899119896120574(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905 + 119909 (1 + 120574119909)

sdot 119901(1)

1198990120574(119909) (

120572

119899 + 120573minus 119909)

119898

(5)

Using 119909(1 + 120574119909)119901(1)

119899119896120574(119909) = (119896 minus 119899119909)119901

119899119896120574(119909) we get

119909 (1 + 120574119909) 120583(1)

119899119898120574(119909) + 119898120583

119899119898minus1120574(119909)

=

infin

sum

119896=1

(119896 minus 119899119909) 119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905 + (minus119899119909) 1199011198990120574

(119909)

sdot (120572

119899 + 120573minus 119909)

119898

=

infin

sum

119896=1

119896119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905 minus 119899119909120583119899119898120574

(119909) = 119868

minus 119899119909120583119899119898120574

(119909)

(6)

We can write 119868 as

119868 =

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

(119896 minus 1) minus (119899 + 2120574) 119905 119887119899119896120574

(119905)

sdot (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905 +

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905)

sdot (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905 + (119899 + 2120574)

infin

sum

119896=1

119901119899119896120574

(119909)

sdot int

infin

0

119905119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905 = 1198681

+ 1198682

(say)

(7)

To estimate 1198682using 119905 = ((119899 + 120573)119899)((119899119905 + 120572)(119899 + 120573) minus 119909) minus

(120572(119899 + 120573) minus 119909) we have

1198682=

(119899 + 2120574) (119899 + 120573)

119899

infin

sum

119896=1

119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898+1

119889119905 minus (120572

119899 + 120573

minus 119909)

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905

International Journal of Analysis 3

=(119899 + 2120574) (119899 + 120573)

119899

infin

sum

119896=1

119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898+1

119889119905 + 1199011198990120574

(119909)

sdot (120572

119899 + 120573minus 119909)

119898+1

minus (120572

119899 + 120573minus 119909)

sdot

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905

+ 1199011198990120574

(119909) (120572

119899 + 120573minus 119909)

119898

=(119899 + 2120574) (119899 + 120573)

119899120583119899119898+1120574

(119909) minus (120572

119899 + 120573minus 119909)

sdot 120583119899119898120574

(119909)

(8)

Next to estimate 1198681using the equality (119896 minus 1) minus (119899 +

2120574)119905119887119899119896120574

(119905) = 119905(1 + 120574119905)119887(1)

119899119896120574(119905) we have

1198681=

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119905119887(1)

119899119896120574(119905) (

119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905

+

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905

+ 120574

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

1199052

119887(1)

119899119896120574(119905) (

119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905

= 1198691+ 1198692

(say)

(9)

Again putting 119905 = ((119899 + 120573)119899)((119899119905 + 120572)(119899 + 120573) minus 119909) minus (120572(119899 +

120573) minus 119909) we get

1198691=

119899 + 120573

119899

infin

sum

119896=1

119901119899119896120574

(119909)

sdot int

infin

0

119887(1)

119899119896120574(119905) (

119899119905 + 120572

119899 + 120573minus 119909)

119898+1

119889119905 + (120572

119899 + 120573minus 119909)

sdot

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887(1)

119899119896120574(119905) (

119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905

+

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905

(10)

Now integrating by parts we get

1198691= minus (119898 + 1)

infin

sum

119896=1

119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905 + 119898(120572

119899 + 120573minus 119909)

sdot

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898minus1

119889119905

+

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905

= minus (119898 + 1)

sdot

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905

+ 1199011198990120574

(119909) (120572

119899 + 120573minus 119909)

119898

+ 119898(120572

119899 + 120573minus 119909)

sdot

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898minus1

119889119905

+ 1199011198990120574

(119909) (120572

119899 + 120573minus 119909)

119898minus1

+

infin

sum

119896=1

119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905 + 1199011198990120574

(119909) (120572

119899 + 120573

minus 119909)

119898

1198691= minus119898120583

119899119898120574(119909) + 119898(

120572

119899 + 120573minus 119909)120583

119899119898minus1120574(119909)

(11)

Proceeding in the similar manner we obtain the estimate 1198692

as

1198692= minus

120574 (119899 + 120573) (119898 + 2)

119899120583119899119898+1120574

(119909)

+ 2120574(119899 + 120573) (119898 + 1)

119899(

120572

119899 + 120573minus 119909)120583

119899119898120574(119909)

minus119898120574 (119899 + 120573)

119899(

120572

119899 + 120573minus 119909)

2

120583119899119898minus1120574

(119909)

(12)

Combining (6)ndash(12) we get

119909 (1 + 120574119909) 120583(1)

119899119898120574(119909) + 119898120583

119899119898minus1120574(119909) = minus119898120583

119899119898120574(119909)

+ 119898(120572

119899 + 120573minus 119909)120583

119899119898minus1120574(119909) minus

120574 (119899 + 120573) (119898 + 2)

119899

sdot 120583119899119898+1120574

(119909) + 2120574(119899 + 120573) (119898 + 1)

119899(

120572

119899 + 120573minus 119909)

4 International Journal of Analysis

sdot 120583119899119898120574

(119909) minus119898120574 (119899 + 120573)

119899(

120572

119899 + 120573minus 119909)

2

sdot 120583119899119898minus1120574

(119909) minus 119899119909120583119899119898120574

(119909)

+(119899 + 2120574) (119899 + 120573)

119899120583119899119898+1120574

(119909)

minus (120572

119899 + 120573minus 119909)120583

119899119898120574(119909)

(13)

Hence

(119899 minus 120574119898) (119899 + 120573) 120583119899119898+1120574

(119909) = 119899119909 (1 + 120574119909) 120583(1)

119899119898120574(119909)

+ 119898120583119899119898minus1120574

(119909) + 119898119899 + 1198992

119909

minus (2120574119898 minus 119899) (120572 minus (119899 + 120573) 119909) 120583119899119898120574

(119909)

+ 119898120574 (119899 + 120573) (120572

119899 + 120573minus 119909)

2

minus 119898119899(120572

119899 + 120573minus 119909)120583

119899119898minus1120574(119909)

(14)

This completes the proof of Lemma 1

Remark 2 (see [10]) For119898 isin Ncup0 if the119898th ordermomentis defined as

119880119899119898120574

(119909) =

infin

sum

119896=0

119901119899119896120574

(119909) (119896

119899minus 119909)

119898

(15)

then 1198801198990120574

(119909) = 1 1198801198991120574

(119909) = 0 and 119899119880119899119898+1120574

(119909) = 119909(1 +

120574119909)(119880(1)

119899119898120574(119909) + 119898119880

119899119898minus1120574(119909))

Consequently for all 119909 isin [0infin) we have 119880119899119898120574

(119909) =

119874(119899minus[(119898+1)2]

)

Remark 3 It is easily verified from Lemma 1 that for each 119909 isin

[0infin)

119861120572120573

119899120574(119905119898

119909) =119899119898

Γ (119899120574 + 119898) Γ (119899120574 minus 119898 + 1)

(119899 + 120573)119898

Γ (119899120574 + 1) Γ (119899120574)119909119898

+119898119899119898minus1

Γ (119899120574 + 119898 minus 1) Γ (119899120574 minus 119898 + 1)

(119899 + 120573)119898

Γ (119899120574 + 1) Γ (119899120574)119899 (119898 minus 1)

+ 120572(119899

120574minus 119898 + 1)119909

119898minus1

+120572119898 (119898 minus 1) 119899

119898minus2

Γ (119899120574 + 119898 minus 2) Γ (119899120574 minus 119898 + 2)

(119899 + 120573)119898

Γ (119899120574 + 1) Γ (119899120574)119899 (119898

minus 2) +120572 (119899120574 minus 119898 + 2)

2119909119898minus2

+ 119874 (119899minus2

)

(16)

Lemma 4 (see [10]) The polynomials119876119894119895119903120574

(119909) exist indepen-dent of 119899 and 119896 such that

119909 (1 + 120574119909)119903

119863119903

[119901119899119896120574

(119909)]

= sum

2119894+119895le119903

119894119895ge0

119899119894

(119896 minus 119899119909)119895

119876119894119895119903120574

(119909) 119901119899119896120574

(119909)

where 119863 equiv119889

119889119909

(17)

Lemma 5 If 119891 is 119903 times differentiable on [0infin) such that119891(119903minus1)

= 119874(119905120592

) 120592 gt 0 as 119905 rarr infin then for 119903 = 1 2 3 and119899 gt 120592 + 120574119903 one has

(119861120572120573

119899120574)(119903)

(119891 119909) =119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)

sdot

infin

sum

119896=0

119901119899+120574119903119896120574

(119909)

sdot int

infin

0

119887119899minus120574119903119896+119903120574

(119905) 119891(119903)

(119899119905 + 120572

119899 + 120573)119889119905

(18)

Proof First

(119861120572120573

119899120574)(1)

(119891 119909)

=

infin

sum

119896=1

119901(1)

119899119896120574(119909) int

infin

0

119887119899119896120574

(119905) 119891(119899119905 + 120572

119899 + 120573)119889119905

minus 119899 (1 + 120574119909)minus119899120574minus1

119891(120572

119899 + 120573)

(19)

Now using the identities

119901(1)

119899119896120574(119909) = 119899 119901

119899+120574119896minus1120574(119909) minus 119901

119899+120574119896120574(119909)

119887(1)

119899119896120574(119909) = (119899 + 120574) 119887

119899+120574119896minus1120574(119909) minus 119887

119899+120574119896120574(119909)

(20)

for 119896 ge 1 we have

(119861120572120573

119899120574)(1)

(119891 119909) =

infin

sum

119896=1

119899 119901119899+120574119896minus1120574

(119909) minus 119901119899+120574119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) 119891(119899119905 + 120572

119899 + 120573)119889119905 minus 119899 (1 + 120574119909)

minus119899120574minus1

sdot 119891 (120572

119899 + 120573) = 119899119901

119899+1205740120574(119909)

sdot int

infin

0

119887119899+1205741120574

(119905) 119891(119899119905 + 120572

119899 + 120573)119889119905 minus 119899 (1 + 120574119909)

minus119899120574minus1

sdot 119891 (120572

119899 + 120573) + 119899

infin

sum

119896=1

119901119899+120574119896120574

(119909)

sdot int

infin

0

119887119899119896+1120574

(119905) minus 119887119899119896120574

(119905) 119891(119899119905 + 120572

119899 + 120573)119889119905

International Journal of Analysis 5

(119861120572120573

119899120574)(1)

(119891 119909) = 119899 (1 + 120574119909)minus119899120574minus1

sdot int

infin

0

(119899 + 120574) (1 + 120574119905)minus119899120574minus2

119891(119899119905 + 120572

119899 + 120573)119889119905

+ 119899

infin

sum

119896=1

119901119899+120574119896120574

(119909)

sdot int

infin

0

(minus1

119899119887(1)

119899minus120574119896+1120574(119905)) 119891(

119899119905 + 120572

119899 + 120573)119889119905

minus 119899 (1 + 120574119909)minus119899120574minus1

119891(120572

119899 + 120573)

(21)

Integrating by parts we get

(119861120572120573

119899120574)(1)

(119891 119909) = 119899 (1 + 120574119909)minus119899120574minus1

119891(120572

119899 + 120573)

+1198992

119899 + 120573(1 + 120574119909)

minus119899120574minus1

sdot int

infin

0

(1 + 120574119905)minus119899120574minus1

119891(1)

(119899119905 + 120572

119899 + 120573)119889119905 +

119899

119899 + 120573

sdot

infin

sum

119896=1

119901119899+120574119896120574

(119909) int

infin

0

119887119899minus120574119896+1120574

(119905) 119891(1)

(119899119905 + 120572

119899 + 120573)119889119905

minus 119899 (1 + 120574119909)minus119899120574minus1

119891(120572

119899 + 120573)

(119861120572120573

119899120574)(1)

(119891 119909) =119899

119899 + 120573

infin

sum

119896=0

119901119899+120574119896120574

(119909)

sdot int

infin

0

119887119899minus120574119896+1120574

(119905) 119891(1)

(119899119905 + 120572

119899 + 120573)119889119905

(22)

Thus the result is true for 119903 = 1 We prove the result byinduction method Suppose that the result is true for 119903 = 119894then

(119861120572120573

119899120574)(119894)

(119891 119909) =119899119894

Γ (119899120574 + 119894) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

sdot

infin

sum

119896=0

119901119899+120574119894119896120574

(119909) int

infin

0

119887119899minus120574119894119896+119894120574

(119905) 119891(119894)

(119899119905 + 120572

119899 + 120573)119889119905

(23)

Thus using the identities (20) we have

(119861120572120573

119899120574)(119894+1)

(119891 119909)

=119899119894

Γ (119899120574 + 119894) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

infin

sum

119896=1

(119899

120574+ 119894)

sdot 119901119899+120574(119894+1)119896minus1120574

(119909) minus 119901119899+120574(119894+1)119896120574

(119909) int

infin

0

119887119899minus120574119894119896+119894120574

(119905)

sdot 119891(119894)

(119899119905 + 120572

119899 + 120573)119889119905 minus (

119899

120574+ 119894) (1 + 120574119909)

minus119899120574minus119894minus1

sdot int

infin

0

119887119899minus120574119894119894120574

(119905) 119891(119894)

(119899119905 + 120572

119899 + 120573)

=119899119894

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

119901119899+120574(119894+1)0120574

(119909)

sdot int

infin

0

119887119899minus1205741198941+119894120574

(119905) 119891(119894)

(119899119905 + 120572

119899 + 120573)119889119905

minus119899119894

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

119901119899+120574(119894+1)0120574

(119909)

sdot int

infin

0

119887119899minus120574119894119894120574

(119905) 119891(119894)

(119899119905 + 120572

119899 + 120573)119889119905

+119899119894

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

infin

sum

119896=1

119901119899+120574(119894+1)119896120574

(119909)

sdot int

infin

0

119887119899minus120574119894119896+119894+1120574

(119905) minus 119887119899minus120574119894119896+119894120574

(119905)

sdot 119891(119894)

(119899119905 + 120572

119899 + 120573)119889119905

=119899119894

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

119901119899+120574(119894+1)0120574

(119909)

sdot int

infin

0

(minus1

119899120574 minus 119894119887(1)

119899minus120574(119894minus1)1+119894120574(119905))119891

(119894)

(119899119905 + 120572

119899 + 120573)119889119905

+119899119894

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

infin

sum

119896=1

119901119899+120574(119894+1)119896120574

(119909)

sdot int

infin

0

(minus1

119899120574 minus 119894119887(1)

119899minus120574(119894minus1)119896+119894+1120574(119905))119891

(119894)

(119899119905 + 120572

119899 + 120573)119889119905

(24)

Integrating by parts we obtain

(119861120572120573

119899120574)(119894+1)

(119891 119909) =119899119894+1

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894+1

Γ (119899120574 + 1) Γ (119899120574)

sdot

infin

sum

119896=0

119901119899+120574(119894+1)119896120574

(119909)

sdot int

infin

0

119887119899minus120574(119894minus1)119896+119894+1120574

(119905) 119891(119894+1)

(119899119905 + 120572

119899 + 120573)119889119905

(25)

This completes the proof of Lemma 5

3 Direct Theorems

This section deals with the direct results we establish herepointwise approximation asymptotic formula and errorestimation in simultaneous approximation

6 International Journal of Analysis

We denote 119862120583[0infin) = 119891 isin 119862[0infin) |119891(119905)| le

119872119905120583 for some 119872 gt 0 120583 gt 0 and the norm sdot

120583on the

class 119862120583[0infin) is defined as 119891

120583= sup

0le119905ltinfin|119891(119905)|119905

minus120583

It canbe easily verified that the operators 119861120572120573

119899120574(119891 119909) are well defined

for 119891 isin 119862120583[0infin)

Theorem 6 Let 119891 isin 119862120583[0infin) and let 119891(119903) exist at a point

119909 isin (0infin) Then one has

lim119899rarrinfin

(119861120572120573

119899120574)(119903)

(119891 119909) = 119891(119903)

(119909) (26)

Proof By Taylorrsquos expansion of 119891 we have

119891 (119905) =

119903

sum

119894=0

119891(119894)

(119909)

119894(119905 minus 119909)

119894

+ 120598 (119905 119909) (119905 minus 119909)119903

(27)

where 120598(119905 119909) rarr 0 as 119905 rarr 119909 Hence

(119861120572120573

119899120574)(119903)

(119891 119909) =

119903

sum

119894=0

119891(119894)

(119909)

119894(119861120572120573

119899120574)(119903)

((119905 minus 119909)119894

119909)

+ (119861120572120573

119899120574)(119903)

(120598 (119905 119909) (119905 minus 119909)119903

119909)

= 1198771+ 1198772

(28)

First to estimate 1198771 using binomial expansion of ((119899119905 +

120572)(119899 + 120573) minus 119909)119894 and Remark 3 we have

1198771=

119903

sum

119894=0

119891(119894)

(119909)

119894

119894

sum

119895=0

(119894

119895) (minus119909)

119894minus119895

(119861120572120573

119899120574)(119903)

(119905119895

119909)

=119891(119903)

(119909)

119903119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)119903

= 119891(119903)

(119909) 119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)

997888rarr 119891(119903)

(119909) as 119899 997888rarr infin

(29)

Next applying Lemma 4 we obtain

1198772= int

infin

0

119882(119903)

119899120574(119905 119909) 120598 (119905 119909) (

119899119905 + 120572

119899 + 120573minus 119909)

119903

119889119905

100381610038161003816100381611987721003816100381610038161003816 le sum

2119894+119895le119903

119894119895ge0

119899119894

10038161003816100381610038161003816119876119894119895119903120574

(119909)10038161003816100381610038161003816

119909 (1 + 120574119909)119903

infin

sum

119896=1

|119896 minus 119899119909|119895

119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) |120598 (119905 119909)|

10038161003816100381610038161003816100381610038161003816

119899119905 + 120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

119903

119889119905

+Γ (119899120574 + 119903 + 2)

Γ (119899120574)(1 + 120574119909)

minus119899120574minus119903

|120598 (0 119909)|

sdot

10038161003816100381610038161003816100381610038161003816

120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

119903

(30)

The second term in the above expression tends to zero as 119899 rarr

infin Since 120598(119905 119909) rarr 0 as 119905 rarr 119909 for given 120576 gt 0 there existsa 120575 isin (0 1) such that |120598(119905 119909)| lt 120576 whenever 0 lt |119905 minus 119909| lt 120575If 120591 gt max120583 119903 where 120591 is any integer then we can find aconstant 119872

3gt 0 such that |120598(119905 119909)((119899119905 + 120572)(119899 + 120573) minus 119909)

119903

| le

1198723|(119899119905 + 120572)(119899 + 120573) minus 119909|

120591 for |119905 minus 119909| ge 120575 Therefore

100381610038161003816100381611987721003816100381610038161003816 le 119872

3sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=0

|119896 minus 119899119909|119895

119901119899119896120574

(119909)

sdot 120576 int|119905minus119909|lt120575

119887119899119896120574

(119909)

10038161003816100381610038161003816100381610038161003816

119899119905 + 120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

119903

119889119905

+ int|119905minus119909|ge120575

119887119899119896120574

(119905)

10038161003816100381610038161003816100381610038161003816

119899119905 + 120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

120591

119889119905 = 1198773+ 1198774

(31)

Applying the Cauchy-Schwarz inequality for integration andsummation respectively we obtain

100381610038161003816100381611987731003816100381610038161003816 le 120576119872

3sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=1

(119896 minus 119899119909)2119895

119901119899119896120574

(119909)

12

sdot

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

2119903

119889119905

12

(32)

Using Remark 2 and Lemma 1 we get 1198773le 120576119874(119899

1199032

)119874(119899minus1199032

)

= 120576 sdot 119874(1)

Again using the Cauchy-Schwarz inequality and Lemma1 we get

100381610038161003816100381611987741003816100381610038161003816 le 119872

4sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=1

|119896 minus 119899119909|119895

119901119899119896120574

(119909)

sdot int|119905minus119909|ge120575

119887119899119896120574

(119905)

10038161003816100381610038161003816100381610038161003816

119899119905 + 120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

120591

119889119905 le 1198724

sum

2119894+119895le119903

119894119895ge0

119899119894

sdot

infin

sum

119896=1

|119896 minus 119899119909|119895

119901119899119896120574

(119909) int|119905minus119909|ge120575

119887119899119896120574

(119905) 119889119905

12

sdot int|119905minus119909|ge120575

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

2120591

119889119905

12

le 1198724

sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=1

(119896 minus 119899119909)2119895

119901119899119896120574

(119909)

12

sdot

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

2120591

119889119905

12

= sum

2119894+119895le119903

119894119895ge0

119899119894

119874(1198991198952

)119874 (119899minus1205912

) = 119874 (119899(119903minus120591)2

) = 119900 (1)

(33)

Collecting the estimation of1198771ndash1198774 we get the required result

International Journal of Analysis 7

Theorem 7 Let 119891 isin 119862120583[0infin) If 119891(119903+2) exists at a point 119909 isin

(0infin) then

lim119899rarrinfin

119899 (119861120572120573

119899120574)(119903)

(119891 119909) minus 119891(119903)

(119909)

= 119903 (120574 (119903 minus 1) minus 120573) 119891(119903)

(119909)

+ 119903120574 (1 + 2119909) + 120572 minus 120573119909119891(119903+1)

(119909)

+ 119909 (1 + 120574119909) 119891(119903+2)

(119909)

(34)

Proof Using Taylorrsquos expansion of 119891 we have

119891 (119905) =

119903+2

sum

119894=0

119891(119894)

(119909)

119894(119905 minus 119909)

119894

+ 120598 (119905 119909) (119905 minus 119909)119903+2

(35)

where 120598(119905 119909) rarr 0 as 119905 rarr 119909 and 120598(119905 119909) = 119874((119905 minus 119909)120583

) 119905 rarr

infin for 120583 gt 0Applying Lemma 1 we have

119899 (119861120572120573

119899120574)(119903)

(119891 119909) minus 119891(119903)

(119909)

= 119899

119903+2

sum

119894=0

119891(119894)

(119909)

119894(119861120572120573

119899120574)(119903)

((119905 minus 119909)119894

119909) minus 119891(119903)

(119909)

+ 119899 (119861120572120573

119899120574)(119903)

(120598 (119905 119909) (119905 minus 119909)119903+2

119909)

= 1198641+ 1198642

(36)

First we have

1198641= 119899

119903+2

sum

119894=0

119891(119894)

(119909)

119894

119894

sum

119895=0

(119894

119895) (minus119909)

119894minus119895

(119861120572120573

119899120574)(119903)

(119905119895

119909)

minus 119899119891(119903)

(119909) =119891(119903)

(119909)

119903119899 (119861

120572120573

119899120574)(119903)

(119905119903

119909) minus 119903

+119891(119903+1)

(119909)

(119903 + 1)119899 (119903 + 1) (minus119909) (119861

120572120573

119899120574)(119903)

(119905119903

119909)

+ (119861120572120573

119899120574)(119903)

(119905119903+1

119909) +119891(119903+2)

(119909)

(119903 + 2)

sdot 119899 (119903 + 2) (119903 + 1)

21199092

(119861120572120573

119899120574)(119903)

(119905119903

119909) + (119903 + 2)

sdot (minus119909) (119861120572120573

119899120574)(119903)

(119905119903+1

119909) + (119861120572120573

119899120574)(119903)

(119905119903+2

119909)

= 119891(119903)

(119909) 119899 119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)minus 1

+119891(119903+1)

(119909)

(119903 + 1)119899(119903 + 1) (minus119909)

sdot119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)119903

+119899119903+1

Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903)

(119899 + 120573)119903+1

Γ (119899120574 + 1) Γ (119899120574)

(119903 + 1)119909

+(119903 + 1) 119899

119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903)

(119899 + 120573)119903+1

Γ (119899120574 + 1) Γ (119899120574)

119899119903 + 120572(119899

120574

minus 119903) 119903 +119891(119903+2)

(119909)

(119903 + 2)119899(

(119903 + 1) (119903 + 2)

21199092

sdot119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)119903 minus 119909 (119903 + 2)

sdot 119899119903+1

Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903)

(119899 + 120573)119903+1

Γ (119899120574 + 1) Γ (119899120574)

(119903 + 1)119909

+(119903 + 1) 119899

119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903)

(119899 + 120573)119903+1

Γ (119899120574 + 1) Γ (119899120574)

119899119903

+ 120572(119899

120574minus 119903) 119903

+119899119903+2

Γ (119899120574 + 119903 + 2) Γ (119899120574 minus 119903 minus 1)

(119899 + 120573)119903+2

Γ (119899120574 + 1) Γ (119899120574)

(119903 + 2)

21199092

+(119903 + 2) 119899

119903+1

Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903 minus 1)

(119899 + 120573)119903+2

Γ (119899120574 + 1) Γ (119899120574)

119899 (119903

+ 1) + 120572(119899

120574minus 119903 minus 1) (119903 + 1)119909

+120572 (119903 + 1) (119903 + 2) 119899

119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)119899119903

+120572 (119899120574 minus 119903)

2 119903)

(37)

Now the coefficients of119891(119903)(119909)119891(119903+1)(119909) and119891(119903+2)

(119909) in theabove expression tend to 119903(120574(119903 minus 1) minus 120573) 119903120574(1 + 2119909) + 120572 minus 120573119909and 119909(1 + 120574119909) respectively which follows by using inductionhypothesis on 119903 and taking the limit as 119899 rarr infin Hence inorder to prove (34) it is sufficient to show that 119864

2rarr 0

as 119899 rarr infin which follows along the lines of the proof ofTheorem 6 and by using Remark 2 and Lemmas 1 and 4

Remark 8 Particular case 120572 = 120573 = 0 was discussed inTheorem 41 in [4] which says that the coefficient of119891(119903+1)(119909)converges to 119903(1 + 2120574119909) but it converges to 119903120574(1 + 2119909) and weget this by putting 120572 = 120573 = 0 in the above theorem

Definition 9 The 119898th order modulus of continuity 120596119898(119891 120575

[119886 119887]) for a function continuous on [119886 119887] is defined by

120596119898(119891 120575 [119886 119887])

= sup 1003816100381610038161003816Δ119898

ℎ119891 (119909)

1003816100381610038161003816 |ℎ| le 120575 119909 119909 + ℎ isin [119886 119887]

(38)

For119898 = 1 120596119898(119891 120575) is usual modulus of continuity

8 International Journal of Analysis

Theorem 10 Let 119891 isin 119862120583[0infin) for some 120583 gt 0 and 0 lt 119886 lt

1198861lt 1198871lt 119887 lt infin Then for 119899 sufficiently large one has

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(119891 sdot) minus 119891(119903)

1003817100381710038171003817100381710038171003817119862[11988611198871]

le 11987211205962(119891(119903)

119899minus12

[1198861 1198871]) + 119872

2119899minus1 1003817100381710038171003817119891

1003817100381710038171003817120583

(39)

where1198721= 1198721(119903) and119872

2= 1198722(119903 119891)

Proof Let us assume that 0 lt 119886 lt 1198861

lt 1198871

lt 119887 lt infinFor sufficiently small 120578 gt 0 we define the function 119891

1205782

corresponding to 119891 isin 119862120583[119886 119887] and 119905 isin [119886

1 1198871] as follows

1198911205782

(119905) = 120578minus2

∬

1205782

minus1205782

(119891 (119905) minus Δ2

ℎ119891 (119905)) 119889119905

11198891199052 (40)

where ℎ = (1199051+ 1199052)2 and Δ

2

ℎis the second order forward

difference operator with step length ℎ For 119891 isin 119862[119886 119887] thefunctions 119891

1205782are known as the Steklov mean of order 2

which satisfy the following properties [11]

(a) 1198911205782

has continuous derivatives up to order 2 over[1198861 1198871]

(b) 119891(119903)1205782

119862[11988611198871]le 1198721120578minus119903

1205962(119891 120578 [119886 119887]) 119903 = 1 2

(c) 119891 minus 1198911205782

119862[11988611198871]le 11987221205962(119891 120578 [119886 119887])

(d) 1198911205782

119862[11988611198871]le 1198723119891120583

where119872119894 119894 = 1 2 3 are certain constants which are different

in each occurrence and are independent of 119891 and 120578We can write by linearity properties of 119861120572120573

119899120574

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(119891 sdot) minus 119891(119903)

1003817100381710038171003817100381710038171003817119862[11988611198871]

le

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(119891 minus 1198911205782

sdot)

1003817100381710038171003817100381710038171003817119862[11988611198871]

+

1003817100381710038171003817100381710038171003817((119861120572120573

119899120574)(119903)

1198911205782

sdot) minus 119891(119903)

1205782

1003817100381710038171003817100381710038171003817119862[11988611198871]

+10038171003817100381710038171003817119891(119903)

minus 119891(119903)

1205782

10038171003817100381710038171003817119862[11988611198871]

= 1198751+ 1198752+ 1198753

(41)

Since 119891(119903)

1205782= (119891(119903)

)1205782

(119905) by property (c) of the function 1198911205782

we get

1198753le 11987241205962(119891(119903)

120578 [119886 119887]) (42)

Next on an application of Theorem 7 it follows that

1198752le 1198725119899minus1

119903+2

sum

119894=119903

10038171003817100381710038171003817119891(119894)

1205782

10038171003817100381710038171003817119862[119886119887] (43)

Using the interpolation property due to Goldberg and Meir[12] for each 119895 = 119903 119903 + 1 119903 + 2 it follows that

10038171003817100381710038171003817119891(119894)

1205782

10038171003817100381710038171003817119862[119886119887]le 1198726100381710038171003817100381710038171198911205782

10038171003817100381710038171003817119862[119886119887]+10038171003817100381710038171003817119891(119903+2)

1205782

10038171003817100381710038171003817119862[119886119887] (44)

Therefore by applying properties (c) and (d) of the function1198911205782 we obtain

1198752le 1198727sdot 119899minus1

1003817100381710038171003817119891

1003817100381710038171003817120583 + 120575minus2

1205962(119891(119903)

120583 [119886 119887]) (45)

Finally we will estimate 1198751 choosing 119886

lowast 119887lowast satisfying theconditions 0 lt 119886 lt 119886

lowast

lt 1198861lt 1198871lt 119887lowast

lt 119887 lt infin Suppose ℏ(119905)denotes the characteristic function of the interval [119886lowast 119887lowast]Then

1198751le

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(ℏ (119905) (119891 (119905) minus 1198911205782

(119905)) sdot)

1003817100381710038171003817100381710038171003817119862[11988611198871]

+

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782

(119905)) sdot)

1003817100381710038171003817100381710038171003817119862[11988611198871]

= 1198754+ 1198755

(46)

By Lemma 5 we have

(119861120572120573

119899120574)(119903)

(ℏ (119905) (119891 (119905) minus 1198911205782

(119905)) 119909)

=119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)

infin

sum

119896=0

119901119899+120574119903119896120574

(119909)

sdot int

infin

0

119887119899minus120574119903119896+119903120574

(119905) ℏ (119905)

sdot (119891(119903)

(119899119905 + 120572

119899 + 120573) minus 119891

(119903)

1205782(119899119905 + 120572

119899 + 120573))119889119905

(47)

Hence100381710038171003817100381710038171003817(119861120572120573

119899120574)119903

(ℏ (119905) (119891 (119905) minus 1198911205782

(119905)) sdot)100381710038171003817100381710038171003817119862[11988611198871]

le 1198728

10038171003817100381710038171003817119891(119903)

minus 119891(119903)

1205782

10038171003817100381710038171003817119862[119886lowast 119887lowast]

(48)

Now for 119909 isin [1198861 1198871] and 119905 isin [0infin) [119886

lowast

119887lowast

] we choose a120575 gt 0 satisfying |(119899119905 + 120572)(119899 + 120573) minus 119909| ge 120575

Therefore by Lemma 4 and the Cauchy-Schwarz inequal-ity we have

119868 equiv (119861120572120573

119899120574)(119903)

((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782

(119905)) 119909)

|119868| le sum

2119894+119895le119903

119894119895ge0

119899119894

10038161003816100381610038161003816119876119894119895119903120574

(119909)10038161003816100381610038161003816

119909 (1 + 120574119909)119903

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|119895

sdot int

infin

0

119887119899119896120574

(119905) (1 minus ℏ (119905))

sdot

10038161003816100381610038161003816100381610038161003816119891 (

119899119905 + 120572

119899 + 120573) minus 1198911205782

(119899119905 + 120572

119899 + 120573)

10038161003816100381610038161003816100381610038161003816119889119905

+Γ (119899120574 + 119903)

Γ (119899120574)(1 + 120574119909)

minus119899120574minus119903

(1 minus ℏ (0))

sdot

10038161003816100381610038161003816100381610038161003816119891 (

120572

119899 + 120573) minus 1198911205782

(120572

119899 + 120573)

10038161003816100381610038161003816100381610038161003816

(49)

International Journal of Analysis 9

For sufficiently large 119899 the second term tends to zero Thus

|119868| le 1198729

10038171003817100381710038171198911003817100381710038171003817120583 sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|119895

sdot int|119905minus119909|ge120575

119887119899119896120574

(119905) 119889119905 le 1198729

10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898

sum

2119894+119895le119903

119894119895ge0

119899119894

sdot

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|119895

(int

infin

0

119887119899119896120574

(119905) 119889119905)

12

sdot (int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

4119898

119889119905)

12

le 1198729

10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898

sum

2119894+119895le119903

119894119895ge0

119899119894

sdot (

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|2119895

)

12

(

infin

sum

119896=1

119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

4119898

119889119905)

12

(50)

Hence by using Remark 2 and Lemma 1 we have

|119868| le 11987210

10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898

119874(119899(119894+(1198952)minus119898)

) le 11987211119899minus119902 1003817100381710038171003817119891

1003817100381710038171003817120583 (51)

where 119902 = 119898 minus (1199032) Now choosing 119898 gt 0 satisfying 119902 ge 1we obtain 119868 le 119872

11119899minus1

119891120583 Therefore by property (c) of the

function 1198911205782

(119905) we get

1198751le 1198728

10038171003817100381710038171003817119891(119903)

minus 119891(119903)

1205782

10038171003817100381710038171003817119862[119886lowast 119887lowast]+ 11987211119899minus1 1003817100381710038171003817119891

1003817100381710038171003817120583

le 119872121205962(119891(119903)

120578 [119886 119887]) + 11987211119899minus1 1003817100381710038171003817119891

1003817100381710038171003817120583

(52)

Choosing 120578 = 119899minus12 the theorem follows

Remark 11 In the last decade the applications of 119902-calculus inapproximation theory are one of themain areas of research In2008 Gupta [13] introduced 119902-Durrmeyer operators whoseapproximation properties were studied in [14] More work inthis direction can be seen in [15ndash17]

A Durrmeyer type 119902-analogue of the 119861120572120573

119899120574(119891 119909) is intro-

duced as follows

119861120572120573

119899120574119902(119891 119909)

=

infin

sum

119896=1

119901119902

119899119896120574(119909) int

infin119860

0

119902minus119896

119887119902

119899119896120574(119905) 119891(

[119899]119902119905 + 120572

[119899]119902+ 120573

)119889119902119905

+ 119901119902

1198990120574(119909) 119891(

120572

[119899]119902+ 120573

)

(53)

where

119901119902

119899119896120574(119909) = 119902

11989622

Γ119902(119899120574 + 119896)

Γ119902(119896 + 1) Γ

119902(119899120574)

sdot(119902120574119909)

119896

(1 + 119902120574119909)(119899120574)+119896

119902

119887119902

119899119896120574(119909) = 120574119902

11989622

Γ119902(119899120574 + 119896 + 1)

Γ119902(119896) Γ119902(119899120574 + 1)

sdot(120574119905)119896minus1

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

119902

int

infin119860

0

119891 (119909) 119889119902119909 = (1 minus 119902)

infin

sum

119899=minusinfin

119891(119902119899

119860)

119902119899

119860 119860 gt 0

(54)

Notations used in (53) can be found in [18] For the operators(53) one can study their approximation properties based on119902-integers

Conflict of Interests

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

Acknowledgments

The authors would like to express their deep gratitude to theanonymous learned referee(s) and the editor for their valu-able suggestions and constructive comments which resultedin the subsequent improvement of this research article

References

[1] V Gupta D K Verma and P N Agrawal ldquoSimultaneousapproximation by certain Baskakov-Durrmeyer-Stancu opera-torsrdquo Journal of the Egyptian Mathematical Society vol 20 no3 pp 183ndash187 2012

[2] D K Verma VGupta and PN Agrawal ldquoSome approximationproperties of Baskakov-Durrmeyer-Stancu operatorsrdquo AppliedMathematics and Computation vol 218 no 11 pp 6549ndash65562012

[3] V N Mishra K Khatri L N Mishra and Deepmala ldquoInverseresult in simultaneous approximation by Baskakov-Durrmeyer-Stancu operatorsrdquo Journal of Inequalities and Applications vol2013 article 586 11 pages 2013

[4] V Gupta ldquoApproximation for modified Baskakov Durrmeyertype operatorsrdquoRockyMountain Journal ofMathematics vol 39no 3 pp 825ndash841 2009

[5] 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 no 4 article 125 2006

[6] V N Mishra and P Patel ldquoApproximation properties ofq-Baskakov-Durrmeyer-Stancu operatorsrdquo Mathematical Sci-ences vol 7 no 1 article 38 12 pages 2013

10 International Journal of Analysis

[7] P Patel and V N Mishra ldquoApproximation properties of certainsummation integral type operatorsrdquo Demonstratio Mathemat-ica vol 48 no 1 pp 77ndash90 2015

[8] 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

[9] 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

[10] W Li ldquoThe voronovskaja type expansion fomula of themodifiedBaskakov-Beta operatorsrdquo Journal of Baoji College of Arts andScience (Natural Science Edition) vol 2 article 004 2005

[11] G Freud andV Popov ldquoOn approximation by spline functionsrdquoin Proceedings of the Conference on Constructive Theory Func-tion Budapest Hungary 1969

[12] S Goldberg and A Meir ldquoMinimummoduli of ordinary differ-ential operatorsrdquo Proceedings of the London Mathematical Soci-ety vol 23 no 3 pp 1ndash15 1971

[13] V Gupta ldquoSome approxmation properties of q-durrmeyer ope-ratorsrdquo Applied Mathematics and Computation vol 197 no 1pp 172ndash178 2008

[14] A Aral and V Gupta ldquoOn the Durrmeyer type modificationof the q-Baskakov type operatorsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 3-4 pp 1171ndash1180 2010

[15] GM Phillips ldquoBernstein polynomials based on the q-integersrdquoAnnals of Numerical Mathematics vol 4 no 1ndash4 pp 511ndash5181997

[16] S Ostrovska ldquoOn the Lupas q-analogue of the Bernstein ope-ratorrdquoThe Rocky Mountain Journal of Mathematics vol 36 no5 pp 1615ndash1629 2006

[17] V N Mishra and P Patel ldquoOn generalized integral Bernsteinoperators based on q-integersrdquo Applied Mathematics and Com-putation vol 242 pp 931ndash944 2014

[18] V Kac and P Cheung Quantum Calculus UniversitextSpringer New York NY USA 2002

Submit your manuscripts athttpwwwhindawicom

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 International Journal of Analysis

The present a paper that deals with the study of simultaneousapproximation for the operators 119861120572120573

119899120574

2 Moments and Recurrence Relations

Lemma 1 If one defines the central moments for every119898 isin Nas

120583119899119898120574

(119909) = 119861120572120573

119899120574((119905 minus 119909)

119898

119909)

=

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905

+ 1199011198990120574

(119909) (120572

119899 + 120573minus 119909)

119898

(3)

then 1205831198990120574

(119909) = 1 1205831198991120574

(119909) = (120572minus120573119909)(119899+120573) and for 119899 gt 120574119898one has the following recurrence relation

(119899 minus 120574119898) (119899 + 120573) 120583119899119898+1120574

(119909) = 119899119909 (1 + 120574119909)

sdot 120583(1)

119899119898120574(119909) + 119898120583

119899119898minus1120574(119909)

+ 119898119899 + 1198992

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

sdot 120583119899119898120574

(119909)

+ 119898120574 (119899 + 120573) (120572

119899 + 120573minus 119909)

2

minus 119898119899(120572

119899 + 120573minus 119909)

sdot 120583119899119898minus1120574

(119909)

(4)

From the recurrence relation it can be easily verified that forall 119909 isin [0infin) one has 120583

119899119898120574(119909) = 119874(119899

minus[(119898+1)2]

) where [120572]

denotes the integral part of 120572

Proof Taking derivative of the above

120583(1)

119899119898120574(119909) = minus119898

infin

sum

119896=1

119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898minus1

119889119905 minus 1198981199011198990120574

(119909)

sdot (120572

119899 + 120573minus 119909)

119898minus1

+

infin

sum

119896=1

119901(1)

119899119896120574(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905 + 119901(1)

1198990120574(119909)

sdot (120572

119899 + 120573minus 119909)

119898

= minus119898120583119899119898minus1120574

(119909) +

infin

sum

119896=1

119901(1)

119899119896120574(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905 + 119901(1)

1198990120574(119909)

sdot (120572

119899 + 120573minus 119909)

119898

119909 (1 + 120574119909) 120583(1)

119899119898120574(119909) + 119898120583

119899119898minus1120574(119909)

=

infin

sum

119896=1

119909 (1 + 120574119909) 119901(1)

119899119896120574(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905 + 119909 (1 + 120574119909)

sdot 119901(1)

1198990120574(119909) (

120572

119899 + 120573minus 119909)

119898

(5)

Using 119909(1 + 120574119909)119901(1)

119899119896120574(119909) = (119896 minus 119899119909)119901

119899119896120574(119909) we get

119909 (1 + 120574119909) 120583(1)

119899119898120574(119909) + 119898120583

119899119898minus1120574(119909)

=

infin

sum

119896=1

(119896 minus 119899119909) 119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905 + (minus119899119909) 1199011198990120574

(119909)

sdot (120572

119899 + 120573minus 119909)

119898

=

infin

sum

119896=1

119896119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905 minus 119899119909120583119899119898120574

(119909) = 119868

minus 119899119909120583119899119898120574

(119909)

(6)

We can write 119868 as

119868 =

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

(119896 minus 1) minus (119899 + 2120574) 119905 119887119899119896120574

(119905)

sdot (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905 +

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905)

sdot (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905 + (119899 + 2120574)

infin

sum

119896=1

119901119899119896120574

(119909)

sdot int

infin

0

119905119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905 = 1198681

+ 1198682

(say)

(7)

To estimate 1198682using 119905 = ((119899 + 120573)119899)((119899119905 + 120572)(119899 + 120573) minus 119909) minus

(120572(119899 + 120573) minus 119909) we have

1198682=

(119899 + 2120574) (119899 + 120573)

119899

infin

sum

119896=1

119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898+1

119889119905 minus (120572

119899 + 120573

minus 119909)

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905

International Journal of Analysis 3

=(119899 + 2120574) (119899 + 120573)

119899

infin

sum

119896=1

119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898+1

119889119905 + 1199011198990120574

(119909)

sdot (120572

119899 + 120573minus 119909)

119898+1

minus (120572

119899 + 120573minus 119909)

sdot

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905

+ 1199011198990120574

(119909) (120572

119899 + 120573minus 119909)

119898

=(119899 + 2120574) (119899 + 120573)

119899120583119899119898+1120574

(119909) minus (120572

119899 + 120573minus 119909)

sdot 120583119899119898120574

(119909)

(8)

Next to estimate 1198681using the equality (119896 minus 1) minus (119899 +

2120574)119905119887119899119896120574

(119905) = 119905(1 + 120574119905)119887(1)

119899119896120574(119905) we have

1198681=

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119905119887(1)

119899119896120574(119905) (

119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905

+

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905

+ 120574

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

1199052

119887(1)

119899119896120574(119905) (

119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905

= 1198691+ 1198692

(say)

(9)

Again putting 119905 = ((119899 + 120573)119899)((119899119905 + 120572)(119899 + 120573) minus 119909) minus (120572(119899 +

120573) minus 119909) we get

1198691=

119899 + 120573

119899

infin

sum

119896=1

119901119899119896120574

(119909)

sdot int

infin

0

119887(1)

119899119896120574(119905) (

119899119905 + 120572

119899 + 120573minus 119909)

119898+1

119889119905 + (120572

119899 + 120573minus 119909)

sdot

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887(1)

119899119896120574(119905) (

119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905

+

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905

(10)

Now integrating by parts we get

1198691= minus (119898 + 1)

infin

sum

119896=1

119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905 + 119898(120572

119899 + 120573minus 119909)

sdot

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898minus1

119889119905

+

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905

= minus (119898 + 1)

sdot

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905

+ 1199011198990120574

(119909) (120572

119899 + 120573minus 119909)

119898

+ 119898(120572

119899 + 120573minus 119909)

sdot

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898minus1

119889119905

+ 1199011198990120574

(119909) (120572

119899 + 120573minus 119909)

119898minus1

+

infin

sum

119896=1

119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905 + 1199011198990120574

(119909) (120572

119899 + 120573

minus 119909)

119898

1198691= minus119898120583

119899119898120574(119909) + 119898(

120572

119899 + 120573minus 119909)120583

119899119898minus1120574(119909)

(11)

Proceeding in the similar manner we obtain the estimate 1198692

as

1198692= minus

120574 (119899 + 120573) (119898 + 2)

119899120583119899119898+1120574

(119909)

+ 2120574(119899 + 120573) (119898 + 1)

119899(

120572

119899 + 120573minus 119909)120583

119899119898120574(119909)

minus119898120574 (119899 + 120573)

119899(

120572

119899 + 120573minus 119909)

2

120583119899119898minus1120574

(119909)

(12)

Combining (6)ndash(12) we get

119909 (1 + 120574119909) 120583(1)

119899119898120574(119909) + 119898120583

119899119898minus1120574(119909) = minus119898120583

119899119898120574(119909)

+ 119898(120572

119899 + 120573minus 119909)120583

119899119898minus1120574(119909) minus

120574 (119899 + 120573) (119898 + 2)

119899

sdot 120583119899119898+1120574

(119909) + 2120574(119899 + 120573) (119898 + 1)

119899(

120572

119899 + 120573minus 119909)

4 International Journal of Analysis

sdot 120583119899119898120574

(119909) minus119898120574 (119899 + 120573)

119899(

120572

119899 + 120573minus 119909)

2

sdot 120583119899119898minus1120574

(119909) minus 119899119909120583119899119898120574

(119909)

+(119899 + 2120574) (119899 + 120573)

119899120583119899119898+1120574

(119909)

minus (120572

119899 + 120573minus 119909)120583

119899119898120574(119909)

(13)

Hence

(119899 minus 120574119898) (119899 + 120573) 120583119899119898+1120574

(119909) = 119899119909 (1 + 120574119909) 120583(1)

119899119898120574(119909)

+ 119898120583119899119898minus1120574

(119909) + 119898119899 + 1198992

119909

minus (2120574119898 minus 119899) (120572 minus (119899 + 120573) 119909) 120583119899119898120574

(119909)

+ 119898120574 (119899 + 120573) (120572

119899 + 120573minus 119909)

2

minus 119898119899(120572

119899 + 120573minus 119909)120583

119899119898minus1120574(119909)

(14)

This completes the proof of Lemma 1

Remark 2 (see [10]) For119898 isin Ncup0 if the119898th ordermomentis defined as

119880119899119898120574

(119909) =

infin

sum

119896=0

119901119899119896120574

(119909) (119896

119899minus 119909)

119898

(15)

then 1198801198990120574

(119909) = 1 1198801198991120574

(119909) = 0 and 119899119880119899119898+1120574

(119909) = 119909(1 +

120574119909)(119880(1)

119899119898120574(119909) + 119898119880

119899119898minus1120574(119909))

Consequently for all 119909 isin [0infin) we have 119880119899119898120574

(119909) =

119874(119899minus[(119898+1)2]

)

Remark 3 It is easily verified from Lemma 1 that for each 119909 isin

[0infin)

119861120572120573

119899120574(119905119898

119909) =119899119898

Γ (119899120574 + 119898) Γ (119899120574 minus 119898 + 1)

(119899 + 120573)119898

Γ (119899120574 + 1) Γ (119899120574)119909119898

+119898119899119898minus1

Γ (119899120574 + 119898 minus 1) Γ (119899120574 minus 119898 + 1)

(119899 + 120573)119898

Γ (119899120574 + 1) Γ (119899120574)119899 (119898 minus 1)

+ 120572(119899

120574minus 119898 + 1)119909

119898minus1

+120572119898 (119898 minus 1) 119899

119898minus2

Γ (119899120574 + 119898 minus 2) Γ (119899120574 minus 119898 + 2)

(119899 + 120573)119898

Γ (119899120574 + 1) Γ (119899120574)119899 (119898

minus 2) +120572 (119899120574 minus 119898 + 2)

2119909119898minus2

+ 119874 (119899minus2

)

(16)

Lemma 4 (see [10]) The polynomials119876119894119895119903120574

(119909) exist indepen-dent of 119899 and 119896 such that

119909 (1 + 120574119909)119903

119863119903

[119901119899119896120574

(119909)]

= sum

2119894+119895le119903

119894119895ge0

119899119894

(119896 minus 119899119909)119895

119876119894119895119903120574

(119909) 119901119899119896120574

(119909)

where 119863 equiv119889

119889119909

(17)

Lemma 5 If 119891 is 119903 times differentiable on [0infin) such that119891(119903minus1)

= 119874(119905120592

) 120592 gt 0 as 119905 rarr infin then for 119903 = 1 2 3 and119899 gt 120592 + 120574119903 one has

(119861120572120573

119899120574)(119903)

(119891 119909) =119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)

sdot

infin

sum

119896=0

119901119899+120574119903119896120574

(119909)

sdot int

infin

0

119887119899minus120574119903119896+119903120574

(119905) 119891(119903)

(119899119905 + 120572

119899 + 120573)119889119905

(18)

Proof First

(119861120572120573

119899120574)(1)

(119891 119909)

=

infin

sum

119896=1

119901(1)

119899119896120574(119909) int

infin

0

119887119899119896120574

(119905) 119891(119899119905 + 120572

119899 + 120573)119889119905

minus 119899 (1 + 120574119909)minus119899120574minus1

119891(120572

119899 + 120573)

(19)

Now using the identities

119901(1)

119899119896120574(119909) = 119899 119901

119899+120574119896minus1120574(119909) minus 119901

119899+120574119896120574(119909)

119887(1)

119899119896120574(119909) = (119899 + 120574) 119887

119899+120574119896minus1120574(119909) minus 119887

119899+120574119896120574(119909)

(20)

for 119896 ge 1 we have

(119861120572120573

119899120574)(1)

(119891 119909) =

infin

sum

119896=1

119899 119901119899+120574119896minus1120574

(119909) minus 119901119899+120574119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) 119891(119899119905 + 120572

119899 + 120573)119889119905 minus 119899 (1 + 120574119909)

minus119899120574minus1

sdot 119891 (120572

119899 + 120573) = 119899119901

119899+1205740120574(119909)

sdot int

infin

0

119887119899+1205741120574

(119905) 119891(119899119905 + 120572

119899 + 120573)119889119905 minus 119899 (1 + 120574119909)

minus119899120574minus1

sdot 119891 (120572

119899 + 120573) + 119899

infin

sum

119896=1

119901119899+120574119896120574

(119909)

sdot int

infin

0

119887119899119896+1120574

(119905) minus 119887119899119896120574

(119905) 119891(119899119905 + 120572

119899 + 120573)119889119905

International Journal of Analysis 5

(119861120572120573

119899120574)(1)

(119891 119909) = 119899 (1 + 120574119909)minus119899120574minus1

sdot int

infin

0

(119899 + 120574) (1 + 120574119905)minus119899120574minus2

119891(119899119905 + 120572

119899 + 120573)119889119905

+ 119899

infin

sum

119896=1

119901119899+120574119896120574

(119909)

sdot int

infin

0

(minus1

119899119887(1)

119899minus120574119896+1120574(119905)) 119891(

119899119905 + 120572

119899 + 120573)119889119905

minus 119899 (1 + 120574119909)minus119899120574minus1

119891(120572

119899 + 120573)

(21)

Integrating by parts we get

(119861120572120573

119899120574)(1)

(119891 119909) = 119899 (1 + 120574119909)minus119899120574minus1

119891(120572

119899 + 120573)

+1198992

119899 + 120573(1 + 120574119909)

minus119899120574minus1

sdot int

infin

0

(1 + 120574119905)minus119899120574minus1

119891(1)

(119899119905 + 120572

119899 + 120573)119889119905 +

119899

119899 + 120573

sdot

infin

sum

119896=1

119901119899+120574119896120574

(119909) int

infin

0

119887119899minus120574119896+1120574

(119905) 119891(1)

(119899119905 + 120572

119899 + 120573)119889119905

minus 119899 (1 + 120574119909)minus119899120574minus1

119891(120572

119899 + 120573)

(119861120572120573

119899120574)(1)

(119891 119909) =119899

119899 + 120573

infin

sum

119896=0

119901119899+120574119896120574

(119909)

sdot int

infin

0

119887119899minus120574119896+1120574

(119905) 119891(1)

(119899119905 + 120572

119899 + 120573)119889119905

(22)

Thus the result is true for 119903 = 1 We prove the result byinduction method Suppose that the result is true for 119903 = 119894then

(119861120572120573

119899120574)(119894)

(119891 119909) =119899119894

Γ (119899120574 + 119894) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

sdot

infin

sum

119896=0

119901119899+120574119894119896120574

(119909) int

infin

0

119887119899minus120574119894119896+119894120574

(119905) 119891(119894)

(119899119905 + 120572

119899 + 120573)119889119905

(23)

Thus using the identities (20) we have

(119861120572120573

119899120574)(119894+1)

(119891 119909)

=119899119894

Γ (119899120574 + 119894) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

infin

sum

119896=1

(119899

120574+ 119894)

sdot 119901119899+120574(119894+1)119896minus1120574

(119909) minus 119901119899+120574(119894+1)119896120574

(119909) int

infin

0

119887119899minus120574119894119896+119894120574

(119905)

sdot 119891(119894)

(119899119905 + 120572

119899 + 120573)119889119905 minus (

119899

120574+ 119894) (1 + 120574119909)

minus119899120574minus119894minus1

sdot int

infin

0

119887119899minus120574119894119894120574

(119905) 119891(119894)

(119899119905 + 120572

119899 + 120573)

=119899119894

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

119901119899+120574(119894+1)0120574

(119909)

sdot int

infin

0

119887119899minus1205741198941+119894120574

(119905) 119891(119894)

(119899119905 + 120572

119899 + 120573)119889119905

minus119899119894

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

119901119899+120574(119894+1)0120574

(119909)

sdot int

infin

0

119887119899minus120574119894119894120574

(119905) 119891(119894)

(119899119905 + 120572

119899 + 120573)119889119905

+119899119894

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

infin

sum

119896=1

119901119899+120574(119894+1)119896120574

(119909)

sdot int

infin

0

119887119899minus120574119894119896+119894+1120574

(119905) minus 119887119899minus120574119894119896+119894120574

(119905)

sdot 119891(119894)

(119899119905 + 120572

119899 + 120573)119889119905

=119899119894

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

119901119899+120574(119894+1)0120574

(119909)

sdot int

infin

0

(minus1

119899120574 minus 119894119887(1)

119899minus120574(119894minus1)1+119894120574(119905))119891

(119894)

(119899119905 + 120572

119899 + 120573)119889119905

+119899119894

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

infin

sum

119896=1

119901119899+120574(119894+1)119896120574

(119909)

sdot int

infin

0

(minus1

119899120574 minus 119894119887(1)

119899minus120574(119894minus1)119896+119894+1120574(119905))119891

(119894)

(119899119905 + 120572

119899 + 120573)119889119905

(24)

Integrating by parts we obtain

(119861120572120573

119899120574)(119894+1)

(119891 119909) =119899119894+1

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894+1

Γ (119899120574 + 1) Γ (119899120574)

sdot

infin

sum

119896=0

119901119899+120574(119894+1)119896120574

(119909)

sdot int

infin

0

119887119899minus120574(119894minus1)119896+119894+1120574

(119905) 119891(119894+1)

(119899119905 + 120572

119899 + 120573)119889119905

(25)

This completes the proof of Lemma 5

3 Direct Theorems

This section deals with the direct results we establish herepointwise approximation asymptotic formula and errorestimation in simultaneous approximation

6 International Journal of Analysis

We denote 119862120583[0infin) = 119891 isin 119862[0infin) |119891(119905)| le

119872119905120583 for some 119872 gt 0 120583 gt 0 and the norm sdot

120583on the

class 119862120583[0infin) is defined as 119891

120583= sup

0le119905ltinfin|119891(119905)|119905

minus120583

It canbe easily verified that the operators 119861120572120573

119899120574(119891 119909) are well defined

for 119891 isin 119862120583[0infin)

Theorem 6 Let 119891 isin 119862120583[0infin) and let 119891(119903) exist at a point

119909 isin (0infin) Then one has

lim119899rarrinfin

(119861120572120573

119899120574)(119903)

(119891 119909) = 119891(119903)

(119909) (26)

Proof By Taylorrsquos expansion of 119891 we have

119891 (119905) =

119903

sum

119894=0

119891(119894)

(119909)

119894(119905 minus 119909)

119894

+ 120598 (119905 119909) (119905 minus 119909)119903

(27)

where 120598(119905 119909) rarr 0 as 119905 rarr 119909 Hence

(119861120572120573

119899120574)(119903)

(119891 119909) =

119903

sum

119894=0

119891(119894)

(119909)

119894(119861120572120573

119899120574)(119903)

((119905 minus 119909)119894

119909)

+ (119861120572120573

119899120574)(119903)

(120598 (119905 119909) (119905 minus 119909)119903

119909)

= 1198771+ 1198772

(28)

First to estimate 1198771 using binomial expansion of ((119899119905 +

120572)(119899 + 120573) minus 119909)119894 and Remark 3 we have

1198771=

119903

sum

119894=0

119891(119894)

(119909)

119894

119894

sum

119895=0

(119894

119895) (minus119909)

119894minus119895

(119861120572120573

119899120574)(119903)

(119905119895

119909)

=119891(119903)

(119909)

119903119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)119903

= 119891(119903)

(119909) 119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)

997888rarr 119891(119903)

(119909) as 119899 997888rarr infin

(29)

Next applying Lemma 4 we obtain

1198772= int

infin

0

119882(119903)

119899120574(119905 119909) 120598 (119905 119909) (

119899119905 + 120572

119899 + 120573minus 119909)

119903

119889119905

100381610038161003816100381611987721003816100381610038161003816 le sum

2119894+119895le119903

119894119895ge0

119899119894

10038161003816100381610038161003816119876119894119895119903120574

(119909)10038161003816100381610038161003816

119909 (1 + 120574119909)119903

infin

sum

119896=1

|119896 minus 119899119909|119895

119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) |120598 (119905 119909)|

10038161003816100381610038161003816100381610038161003816

119899119905 + 120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

119903

119889119905

+Γ (119899120574 + 119903 + 2)

Γ (119899120574)(1 + 120574119909)

minus119899120574minus119903

|120598 (0 119909)|

sdot

10038161003816100381610038161003816100381610038161003816

120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

119903

(30)

The second term in the above expression tends to zero as 119899 rarr

infin Since 120598(119905 119909) rarr 0 as 119905 rarr 119909 for given 120576 gt 0 there existsa 120575 isin (0 1) such that |120598(119905 119909)| lt 120576 whenever 0 lt |119905 minus 119909| lt 120575If 120591 gt max120583 119903 where 120591 is any integer then we can find aconstant 119872

3gt 0 such that |120598(119905 119909)((119899119905 + 120572)(119899 + 120573) minus 119909)

119903

| le

1198723|(119899119905 + 120572)(119899 + 120573) minus 119909|

120591 for |119905 minus 119909| ge 120575 Therefore

100381610038161003816100381611987721003816100381610038161003816 le 119872

3sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=0

|119896 minus 119899119909|119895

119901119899119896120574

(119909)

sdot 120576 int|119905minus119909|lt120575

119887119899119896120574

(119909)

10038161003816100381610038161003816100381610038161003816

119899119905 + 120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

119903

119889119905

+ int|119905minus119909|ge120575

119887119899119896120574

(119905)

10038161003816100381610038161003816100381610038161003816

119899119905 + 120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

120591

119889119905 = 1198773+ 1198774

(31)

Applying the Cauchy-Schwarz inequality for integration andsummation respectively we obtain

100381610038161003816100381611987731003816100381610038161003816 le 120576119872

3sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=1

(119896 minus 119899119909)2119895

119901119899119896120574

(119909)

12

sdot

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

2119903

119889119905

12

(32)

Using Remark 2 and Lemma 1 we get 1198773le 120576119874(119899

1199032

)119874(119899minus1199032

)

= 120576 sdot 119874(1)

Again using the Cauchy-Schwarz inequality and Lemma1 we get

100381610038161003816100381611987741003816100381610038161003816 le 119872

4sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=1

|119896 minus 119899119909|119895

119901119899119896120574

(119909)

sdot int|119905minus119909|ge120575

119887119899119896120574

(119905)

10038161003816100381610038161003816100381610038161003816

119899119905 + 120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

120591

119889119905 le 1198724

sum

2119894+119895le119903

119894119895ge0

119899119894

sdot

infin

sum

119896=1

|119896 minus 119899119909|119895

119901119899119896120574

(119909) int|119905minus119909|ge120575

119887119899119896120574

(119905) 119889119905

12

sdot int|119905minus119909|ge120575

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

2120591

119889119905

12

le 1198724

sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=1

(119896 minus 119899119909)2119895

119901119899119896120574

(119909)

12

sdot

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

2120591

119889119905

12

= sum

2119894+119895le119903

119894119895ge0

119899119894

119874(1198991198952

)119874 (119899minus1205912

) = 119874 (119899(119903minus120591)2

) = 119900 (1)

(33)

Collecting the estimation of1198771ndash1198774 we get the required result

International Journal of Analysis 7

Theorem 7 Let 119891 isin 119862120583[0infin) If 119891(119903+2) exists at a point 119909 isin

(0infin) then

lim119899rarrinfin

119899 (119861120572120573

119899120574)(119903)

(119891 119909) minus 119891(119903)

(119909)

= 119903 (120574 (119903 minus 1) minus 120573) 119891(119903)

(119909)

+ 119903120574 (1 + 2119909) + 120572 minus 120573119909119891(119903+1)

(119909)

+ 119909 (1 + 120574119909) 119891(119903+2)

(119909)

(34)

Proof Using Taylorrsquos expansion of 119891 we have

119891 (119905) =

119903+2

sum

119894=0

119891(119894)

(119909)

119894(119905 minus 119909)

119894

+ 120598 (119905 119909) (119905 minus 119909)119903+2

(35)

where 120598(119905 119909) rarr 0 as 119905 rarr 119909 and 120598(119905 119909) = 119874((119905 minus 119909)120583

) 119905 rarr

infin for 120583 gt 0Applying Lemma 1 we have

119899 (119861120572120573

119899120574)(119903)

(119891 119909) minus 119891(119903)

(119909)

= 119899

119903+2

sum

119894=0

119891(119894)

(119909)

119894(119861120572120573

119899120574)(119903)

((119905 minus 119909)119894

119909) minus 119891(119903)

(119909)

+ 119899 (119861120572120573

119899120574)(119903)

(120598 (119905 119909) (119905 minus 119909)119903+2

119909)

= 1198641+ 1198642

(36)

First we have

1198641= 119899

119903+2

sum

119894=0

119891(119894)

(119909)

119894

119894

sum

119895=0

(119894

119895) (minus119909)

119894minus119895

(119861120572120573

119899120574)(119903)

(119905119895

119909)

minus 119899119891(119903)

(119909) =119891(119903)

(119909)

119903119899 (119861

120572120573

119899120574)(119903)

(119905119903

119909) minus 119903

+119891(119903+1)

(119909)

(119903 + 1)119899 (119903 + 1) (minus119909) (119861

120572120573

119899120574)(119903)

(119905119903

119909)

+ (119861120572120573

119899120574)(119903)

(119905119903+1

119909) +119891(119903+2)

(119909)

(119903 + 2)

sdot 119899 (119903 + 2) (119903 + 1)

21199092

(119861120572120573

119899120574)(119903)

(119905119903

119909) + (119903 + 2)

sdot (minus119909) (119861120572120573

119899120574)(119903)

(119905119903+1

119909) + (119861120572120573

119899120574)(119903)

(119905119903+2

119909)

= 119891(119903)

(119909) 119899 119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)minus 1

+119891(119903+1)

(119909)

(119903 + 1)119899(119903 + 1) (minus119909)

sdot119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)119903

+119899119903+1

Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903)

(119899 + 120573)119903+1

Γ (119899120574 + 1) Γ (119899120574)

(119903 + 1)119909

+(119903 + 1) 119899

119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903)

(119899 + 120573)119903+1

Γ (119899120574 + 1) Γ (119899120574)

119899119903 + 120572(119899

120574

minus 119903) 119903 +119891(119903+2)

(119909)

(119903 + 2)119899(

(119903 + 1) (119903 + 2)

21199092

sdot119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)119903 minus 119909 (119903 + 2)

sdot 119899119903+1

Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903)

(119899 + 120573)119903+1

Γ (119899120574 + 1) Γ (119899120574)

(119903 + 1)119909

+(119903 + 1) 119899

119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903)

(119899 + 120573)119903+1

Γ (119899120574 + 1) Γ (119899120574)

119899119903

+ 120572(119899

120574minus 119903) 119903

+119899119903+2

Γ (119899120574 + 119903 + 2) Γ (119899120574 minus 119903 minus 1)

(119899 + 120573)119903+2

Γ (119899120574 + 1) Γ (119899120574)

(119903 + 2)

21199092

+(119903 + 2) 119899

119903+1

Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903 minus 1)

(119899 + 120573)119903+2

Γ (119899120574 + 1) Γ (119899120574)

119899 (119903

+ 1) + 120572(119899

120574minus 119903 minus 1) (119903 + 1)119909

+120572 (119903 + 1) (119903 + 2) 119899

119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)119899119903

+120572 (119899120574 minus 119903)

2 119903)

(37)

Now the coefficients of119891(119903)(119909)119891(119903+1)(119909) and119891(119903+2)

(119909) in theabove expression tend to 119903(120574(119903 minus 1) minus 120573) 119903120574(1 + 2119909) + 120572 minus 120573119909and 119909(1 + 120574119909) respectively which follows by using inductionhypothesis on 119903 and taking the limit as 119899 rarr infin Hence inorder to prove (34) it is sufficient to show that 119864

2rarr 0

as 119899 rarr infin which follows along the lines of the proof ofTheorem 6 and by using Remark 2 and Lemmas 1 and 4

Remark 8 Particular case 120572 = 120573 = 0 was discussed inTheorem 41 in [4] which says that the coefficient of119891(119903+1)(119909)converges to 119903(1 + 2120574119909) but it converges to 119903120574(1 + 2119909) and weget this by putting 120572 = 120573 = 0 in the above theorem

Definition 9 The 119898th order modulus of continuity 120596119898(119891 120575

[119886 119887]) for a function continuous on [119886 119887] is defined by

120596119898(119891 120575 [119886 119887])

= sup 1003816100381610038161003816Δ119898

ℎ119891 (119909)

1003816100381610038161003816 |ℎ| le 120575 119909 119909 + ℎ isin [119886 119887]

(38)

For119898 = 1 120596119898(119891 120575) is usual modulus of continuity

8 International Journal of Analysis

Theorem 10 Let 119891 isin 119862120583[0infin) for some 120583 gt 0 and 0 lt 119886 lt

1198861lt 1198871lt 119887 lt infin Then for 119899 sufficiently large one has

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(119891 sdot) minus 119891(119903)

1003817100381710038171003817100381710038171003817119862[11988611198871]

le 11987211205962(119891(119903)

119899minus12

[1198861 1198871]) + 119872

2119899minus1 1003817100381710038171003817119891

1003817100381710038171003817120583

(39)

where1198721= 1198721(119903) and119872

2= 1198722(119903 119891)

Proof Let us assume that 0 lt 119886 lt 1198861

lt 1198871

lt 119887 lt infinFor sufficiently small 120578 gt 0 we define the function 119891

1205782

corresponding to 119891 isin 119862120583[119886 119887] and 119905 isin [119886

1 1198871] as follows

1198911205782

(119905) = 120578minus2

∬

1205782

minus1205782

(119891 (119905) minus Δ2

ℎ119891 (119905)) 119889119905

11198891199052 (40)

where ℎ = (1199051+ 1199052)2 and Δ

2

ℎis the second order forward

difference operator with step length ℎ For 119891 isin 119862[119886 119887] thefunctions 119891

1205782are known as the Steklov mean of order 2

which satisfy the following properties [11]

(a) 1198911205782

has continuous derivatives up to order 2 over[1198861 1198871]

(b) 119891(119903)1205782

119862[11988611198871]le 1198721120578minus119903

1205962(119891 120578 [119886 119887]) 119903 = 1 2

(c) 119891 minus 1198911205782

119862[11988611198871]le 11987221205962(119891 120578 [119886 119887])

(d) 1198911205782

119862[11988611198871]le 1198723119891120583

where119872119894 119894 = 1 2 3 are certain constants which are different

in each occurrence and are independent of 119891 and 120578We can write by linearity properties of 119861120572120573

119899120574

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(119891 sdot) minus 119891(119903)

1003817100381710038171003817100381710038171003817119862[11988611198871]

le

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(119891 minus 1198911205782

sdot)

1003817100381710038171003817100381710038171003817119862[11988611198871]

+

1003817100381710038171003817100381710038171003817((119861120572120573

119899120574)(119903)

1198911205782

sdot) minus 119891(119903)

1205782

1003817100381710038171003817100381710038171003817119862[11988611198871]

+10038171003817100381710038171003817119891(119903)

minus 119891(119903)

1205782

10038171003817100381710038171003817119862[11988611198871]

= 1198751+ 1198752+ 1198753

(41)

Since 119891(119903)

1205782= (119891(119903)

)1205782

(119905) by property (c) of the function 1198911205782

we get

1198753le 11987241205962(119891(119903)

120578 [119886 119887]) (42)

Next on an application of Theorem 7 it follows that

1198752le 1198725119899minus1

119903+2

sum

119894=119903

10038171003817100381710038171003817119891(119894)

1205782

10038171003817100381710038171003817119862[119886119887] (43)

Using the interpolation property due to Goldberg and Meir[12] for each 119895 = 119903 119903 + 1 119903 + 2 it follows that

10038171003817100381710038171003817119891(119894)

1205782

10038171003817100381710038171003817119862[119886119887]le 1198726100381710038171003817100381710038171198911205782

10038171003817100381710038171003817119862[119886119887]+10038171003817100381710038171003817119891(119903+2)

1205782

10038171003817100381710038171003817119862[119886119887] (44)

Therefore by applying properties (c) and (d) of the function1198911205782 we obtain

1198752le 1198727sdot 119899minus1

1003817100381710038171003817119891

1003817100381710038171003817120583 + 120575minus2

1205962(119891(119903)

120583 [119886 119887]) (45)

Finally we will estimate 1198751 choosing 119886

lowast 119887lowast satisfying theconditions 0 lt 119886 lt 119886

lowast

lt 1198861lt 1198871lt 119887lowast

lt 119887 lt infin Suppose ℏ(119905)denotes the characteristic function of the interval [119886lowast 119887lowast]Then

1198751le

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(ℏ (119905) (119891 (119905) minus 1198911205782

(119905)) sdot)

1003817100381710038171003817100381710038171003817119862[11988611198871]

+

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782

(119905)) sdot)

1003817100381710038171003817100381710038171003817119862[11988611198871]

= 1198754+ 1198755

(46)

By Lemma 5 we have

(119861120572120573

119899120574)(119903)

(ℏ (119905) (119891 (119905) minus 1198911205782

(119905)) 119909)

=119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)

infin

sum

119896=0

119901119899+120574119903119896120574

(119909)

sdot int

infin

0

119887119899minus120574119903119896+119903120574

(119905) ℏ (119905)

sdot (119891(119903)

(119899119905 + 120572

119899 + 120573) minus 119891

(119903)

1205782(119899119905 + 120572

119899 + 120573))119889119905

(47)

Hence100381710038171003817100381710038171003817(119861120572120573

119899120574)119903

(ℏ (119905) (119891 (119905) minus 1198911205782

(119905)) sdot)100381710038171003817100381710038171003817119862[11988611198871]

le 1198728

10038171003817100381710038171003817119891(119903)

minus 119891(119903)

1205782

10038171003817100381710038171003817119862[119886lowast 119887lowast]

(48)

Now for 119909 isin [1198861 1198871] and 119905 isin [0infin) [119886

lowast

119887lowast

] we choose a120575 gt 0 satisfying |(119899119905 + 120572)(119899 + 120573) minus 119909| ge 120575

Therefore by Lemma 4 and the Cauchy-Schwarz inequal-ity we have

119868 equiv (119861120572120573

119899120574)(119903)

((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782

(119905)) 119909)

|119868| le sum

2119894+119895le119903

119894119895ge0

119899119894

10038161003816100381610038161003816119876119894119895119903120574

(119909)10038161003816100381610038161003816

119909 (1 + 120574119909)119903

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|119895

sdot int

infin

0

119887119899119896120574

(119905) (1 minus ℏ (119905))

sdot

10038161003816100381610038161003816100381610038161003816119891 (

119899119905 + 120572

119899 + 120573) minus 1198911205782

(119899119905 + 120572

119899 + 120573)

10038161003816100381610038161003816100381610038161003816119889119905

+Γ (119899120574 + 119903)

Γ (119899120574)(1 + 120574119909)

minus119899120574minus119903

(1 minus ℏ (0))

sdot

10038161003816100381610038161003816100381610038161003816119891 (

120572

119899 + 120573) minus 1198911205782

(120572

119899 + 120573)

10038161003816100381610038161003816100381610038161003816

(49)

International Journal of Analysis 9

For sufficiently large 119899 the second term tends to zero Thus

|119868| le 1198729

10038171003817100381710038171198911003817100381710038171003817120583 sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|119895

sdot int|119905minus119909|ge120575

119887119899119896120574

(119905) 119889119905 le 1198729

10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898

sum

2119894+119895le119903

119894119895ge0

119899119894

sdot

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|119895

(int

infin

0

119887119899119896120574

(119905) 119889119905)

12

sdot (int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

4119898

119889119905)

12

le 1198729

10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898

sum

2119894+119895le119903

119894119895ge0

119899119894

sdot (

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|2119895

)

12

(

infin

sum

119896=1

119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

4119898

119889119905)

12

(50)

Hence by using Remark 2 and Lemma 1 we have

|119868| le 11987210

10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898

119874(119899(119894+(1198952)minus119898)

) le 11987211119899minus119902 1003817100381710038171003817119891

1003817100381710038171003817120583 (51)

where 119902 = 119898 minus (1199032) Now choosing 119898 gt 0 satisfying 119902 ge 1we obtain 119868 le 119872

11119899minus1

119891120583 Therefore by property (c) of the

function 1198911205782

(119905) we get

1198751le 1198728

10038171003817100381710038171003817119891(119903)

minus 119891(119903)

1205782

10038171003817100381710038171003817119862[119886lowast 119887lowast]+ 11987211119899minus1 1003817100381710038171003817119891

1003817100381710038171003817120583

le 119872121205962(119891(119903)

120578 [119886 119887]) + 11987211119899minus1 1003817100381710038171003817119891

1003817100381710038171003817120583

(52)

Choosing 120578 = 119899minus12 the theorem follows

Remark 11 In the last decade the applications of 119902-calculus inapproximation theory are one of themain areas of research In2008 Gupta [13] introduced 119902-Durrmeyer operators whoseapproximation properties were studied in [14] More work inthis direction can be seen in [15ndash17]

A Durrmeyer type 119902-analogue of the 119861120572120573

119899120574(119891 119909) is intro-

duced as follows

119861120572120573

119899120574119902(119891 119909)

=

infin

sum

119896=1

119901119902

119899119896120574(119909) int

infin119860

0

119902minus119896

119887119902

119899119896120574(119905) 119891(

[119899]119902119905 + 120572

[119899]119902+ 120573

)119889119902119905

+ 119901119902

1198990120574(119909) 119891(

120572

[119899]119902+ 120573

)

(53)

where

119901119902

119899119896120574(119909) = 119902

11989622

Γ119902(119899120574 + 119896)

Γ119902(119896 + 1) Γ

119902(119899120574)

sdot(119902120574119909)

119896

(1 + 119902120574119909)(119899120574)+119896

119902

119887119902

119899119896120574(119909) = 120574119902

11989622

Γ119902(119899120574 + 119896 + 1)

Γ119902(119896) Γ119902(119899120574 + 1)

sdot(120574119905)119896minus1

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

119902

int

infin119860

0

119891 (119909) 119889119902119909 = (1 minus 119902)

infin

sum

119899=minusinfin

119891(119902119899

119860)

119902119899

119860 119860 gt 0

(54)

Notations used in (53) can be found in [18] For the operators(53) one can study their approximation properties based on119902-integers

Conflict of Interests

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

Acknowledgments

The authors would like to express their deep gratitude to theanonymous learned referee(s) and the editor for their valu-able suggestions and constructive comments which resultedin the subsequent improvement of this research article

References

[1] V Gupta D K Verma and P N Agrawal ldquoSimultaneousapproximation by certain Baskakov-Durrmeyer-Stancu opera-torsrdquo Journal of the Egyptian Mathematical Society vol 20 no3 pp 183ndash187 2012

[2] D K Verma VGupta and PN Agrawal ldquoSome approximationproperties of Baskakov-Durrmeyer-Stancu operatorsrdquo AppliedMathematics and Computation vol 218 no 11 pp 6549ndash65562012

[3] V N Mishra K Khatri L N Mishra and Deepmala ldquoInverseresult in simultaneous approximation by Baskakov-Durrmeyer-Stancu operatorsrdquo Journal of Inequalities and Applications vol2013 article 586 11 pages 2013

[4] V Gupta ldquoApproximation for modified Baskakov Durrmeyertype operatorsrdquoRockyMountain Journal ofMathematics vol 39no 3 pp 825ndash841 2009

[5] 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 no 4 article 125 2006

[6] V N Mishra and P Patel ldquoApproximation properties ofq-Baskakov-Durrmeyer-Stancu operatorsrdquo Mathematical Sci-ences vol 7 no 1 article 38 12 pages 2013

10 International Journal of Analysis

[7] P Patel and V N Mishra ldquoApproximation properties of certainsummation integral type operatorsrdquo Demonstratio Mathemat-ica vol 48 no 1 pp 77ndash90 2015

[8] 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

[9] 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

[10] W Li ldquoThe voronovskaja type expansion fomula of themodifiedBaskakov-Beta operatorsrdquo Journal of Baoji College of Arts andScience (Natural Science Edition) vol 2 article 004 2005

[11] G Freud andV Popov ldquoOn approximation by spline functionsrdquoin Proceedings of the Conference on Constructive Theory Func-tion Budapest Hungary 1969

[12] S Goldberg and A Meir ldquoMinimummoduli of ordinary differ-ential operatorsrdquo Proceedings of the London Mathematical Soci-ety vol 23 no 3 pp 1ndash15 1971

[13] V Gupta ldquoSome approxmation properties of q-durrmeyer ope-ratorsrdquo Applied Mathematics and Computation vol 197 no 1pp 172ndash178 2008

[14] A Aral and V Gupta ldquoOn the Durrmeyer type modificationof the q-Baskakov type operatorsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 3-4 pp 1171ndash1180 2010

[15] GM Phillips ldquoBernstein polynomials based on the q-integersrdquoAnnals of Numerical Mathematics vol 4 no 1ndash4 pp 511ndash5181997

[16] S Ostrovska ldquoOn the Lupas q-analogue of the Bernstein ope-ratorrdquoThe Rocky Mountain Journal of Mathematics vol 36 no5 pp 1615ndash1629 2006

[17] V N Mishra and P Patel ldquoOn generalized integral Bernsteinoperators based on q-integersrdquo Applied Mathematics and Com-putation vol 242 pp 931ndash944 2014

[18] V Kac and P Cheung Quantum Calculus UniversitextSpringer New York NY USA 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Analysis 3

=(119899 + 2120574) (119899 + 120573)

119899

infin

sum

119896=1

119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898+1

119889119905 + 1199011198990120574

(119909)

sdot (120572

119899 + 120573minus 119909)

119898+1

minus (120572

119899 + 120573minus 119909)

sdot

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905

+ 1199011198990120574

(119909) (120572

119899 + 120573minus 119909)

119898

=(119899 + 2120574) (119899 + 120573)

119899120583119899119898+1120574

(119909) minus (120572

119899 + 120573minus 119909)

sdot 120583119899119898120574

(119909)

(8)

Next to estimate 1198681using the equality (119896 minus 1) minus (119899 +

2120574)119905119887119899119896120574

(119905) = 119905(1 + 120574119905)119887(1)

119899119896120574(119905) we have

1198681=

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119905119887(1)

119899119896120574(119905) (

119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905

+

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905

+ 120574

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

1199052

119887(1)

119899119896120574(119905) (

119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905

= 1198691+ 1198692

(say)

(9)

Again putting 119905 = ((119899 + 120573)119899)((119899119905 + 120572)(119899 + 120573) minus 119909) minus (120572(119899 +

120573) minus 119909) we get

1198691=

119899 + 120573

119899

infin

sum

119896=1

119901119899119896120574

(119909)

sdot int

infin

0

119887(1)

119899119896120574(119905) (

119899119905 + 120572

119899 + 120573minus 119909)

119898+1

119889119905 + (120572

119899 + 120573minus 119909)

sdot

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887(1)

119899119896120574(119905) (

119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905

+

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905

(10)

Now integrating by parts we get

1198691= minus (119898 + 1)

infin

sum

119896=1

119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905 + 119898(120572

119899 + 120573minus 119909)

sdot

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898minus1

119889119905

+

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905

= minus (119898 + 1)

sdot

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905

+ 1199011198990120574

(119909) (120572

119899 + 120573minus 119909)

119898

+ 119898(120572

119899 + 120573minus 119909)

sdot

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898minus1

119889119905

+ 1199011198990120574

(119909) (120572

119899 + 120573minus 119909)

119898minus1

+

infin

sum

119896=1

119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

119898

119889119905 + 1199011198990120574

(119909) (120572

119899 + 120573

minus 119909)

119898

1198691= minus119898120583

119899119898120574(119909) + 119898(

120572

119899 + 120573minus 119909)120583

119899119898minus1120574(119909)

(11)

Proceeding in the similar manner we obtain the estimate 1198692

as

1198692= minus

120574 (119899 + 120573) (119898 + 2)

119899120583119899119898+1120574

(119909)

+ 2120574(119899 + 120573) (119898 + 1)

119899(

120572

119899 + 120573minus 119909)120583

119899119898120574(119909)

minus119898120574 (119899 + 120573)

119899(

120572

119899 + 120573minus 119909)

2

120583119899119898minus1120574

(119909)

(12)

Combining (6)ndash(12) we get

119909 (1 + 120574119909) 120583(1)

119899119898120574(119909) + 119898120583

119899119898minus1120574(119909) = minus119898120583

119899119898120574(119909)

+ 119898(120572

119899 + 120573minus 119909)120583

119899119898minus1120574(119909) minus

120574 (119899 + 120573) (119898 + 2)

119899

sdot 120583119899119898+1120574

(119909) + 2120574(119899 + 120573) (119898 + 1)

119899(

120572

119899 + 120573minus 119909)

4 International Journal of Analysis

sdot 120583119899119898120574

(119909) minus119898120574 (119899 + 120573)

119899(

120572

119899 + 120573minus 119909)

2

sdot 120583119899119898minus1120574

(119909) minus 119899119909120583119899119898120574

(119909)

+(119899 + 2120574) (119899 + 120573)

119899120583119899119898+1120574

(119909)

minus (120572

119899 + 120573minus 119909)120583

119899119898120574(119909)

(13)

Hence

(119899 minus 120574119898) (119899 + 120573) 120583119899119898+1120574

(119909) = 119899119909 (1 + 120574119909) 120583(1)

119899119898120574(119909)

+ 119898120583119899119898minus1120574

(119909) + 119898119899 + 1198992

119909

minus (2120574119898 minus 119899) (120572 minus (119899 + 120573) 119909) 120583119899119898120574

(119909)

+ 119898120574 (119899 + 120573) (120572

119899 + 120573minus 119909)

2

minus 119898119899(120572

119899 + 120573minus 119909)120583

119899119898minus1120574(119909)

(14)

This completes the proof of Lemma 1

Remark 2 (see [10]) For119898 isin Ncup0 if the119898th ordermomentis defined as

119880119899119898120574

(119909) =

infin

sum

119896=0

119901119899119896120574

(119909) (119896

119899minus 119909)

119898

(15)

then 1198801198990120574

(119909) = 1 1198801198991120574

(119909) = 0 and 119899119880119899119898+1120574

(119909) = 119909(1 +

120574119909)(119880(1)

119899119898120574(119909) + 119898119880

119899119898minus1120574(119909))

Consequently for all 119909 isin [0infin) we have 119880119899119898120574

(119909) =

119874(119899minus[(119898+1)2]

)

Remark 3 It is easily verified from Lemma 1 that for each 119909 isin

[0infin)

119861120572120573

119899120574(119905119898

119909) =119899119898

Γ (119899120574 + 119898) Γ (119899120574 minus 119898 + 1)

(119899 + 120573)119898

Γ (119899120574 + 1) Γ (119899120574)119909119898

+119898119899119898minus1

Γ (119899120574 + 119898 minus 1) Γ (119899120574 minus 119898 + 1)

(119899 + 120573)119898

Γ (119899120574 + 1) Γ (119899120574)119899 (119898 minus 1)

+ 120572(119899

120574minus 119898 + 1)119909

119898minus1

+120572119898 (119898 minus 1) 119899

119898minus2

Γ (119899120574 + 119898 minus 2) Γ (119899120574 minus 119898 + 2)

(119899 + 120573)119898

Γ (119899120574 + 1) Γ (119899120574)119899 (119898

minus 2) +120572 (119899120574 minus 119898 + 2)

2119909119898minus2

+ 119874 (119899minus2

)

(16)

Lemma 4 (see [10]) The polynomials119876119894119895119903120574

(119909) exist indepen-dent of 119899 and 119896 such that

119909 (1 + 120574119909)119903

119863119903

[119901119899119896120574

(119909)]

= sum

2119894+119895le119903

119894119895ge0

119899119894

(119896 minus 119899119909)119895

119876119894119895119903120574

(119909) 119901119899119896120574

(119909)

where 119863 equiv119889

119889119909

(17)

Lemma 5 If 119891 is 119903 times differentiable on [0infin) such that119891(119903minus1)

= 119874(119905120592

) 120592 gt 0 as 119905 rarr infin then for 119903 = 1 2 3 and119899 gt 120592 + 120574119903 one has

(119861120572120573

119899120574)(119903)

(119891 119909) =119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)

sdot

infin

sum

119896=0

119901119899+120574119903119896120574

(119909)

sdot int

infin

0

119887119899minus120574119903119896+119903120574

(119905) 119891(119903)

(119899119905 + 120572

119899 + 120573)119889119905

(18)

Proof First

(119861120572120573

119899120574)(1)

(119891 119909)

=

infin

sum

119896=1

119901(1)

119899119896120574(119909) int

infin

0

119887119899119896120574

(119905) 119891(119899119905 + 120572

119899 + 120573)119889119905

minus 119899 (1 + 120574119909)minus119899120574minus1

119891(120572

119899 + 120573)

(19)

Now using the identities

119901(1)

119899119896120574(119909) = 119899 119901

119899+120574119896minus1120574(119909) minus 119901

119899+120574119896120574(119909)

119887(1)

119899119896120574(119909) = (119899 + 120574) 119887

119899+120574119896minus1120574(119909) minus 119887

119899+120574119896120574(119909)

(20)

for 119896 ge 1 we have

(119861120572120573

119899120574)(1)

(119891 119909) =

infin

sum

119896=1

119899 119901119899+120574119896minus1120574

(119909) minus 119901119899+120574119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) 119891(119899119905 + 120572

119899 + 120573)119889119905 minus 119899 (1 + 120574119909)

minus119899120574minus1

sdot 119891 (120572

119899 + 120573) = 119899119901

119899+1205740120574(119909)

sdot int

infin

0

119887119899+1205741120574

(119905) 119891(119899119905 + 120572

119899 + 120573)119889119905 minus 119899 (1 + 120574119909)

minus119899120574minus1

sdot 119891 (120572

119899 + 120573) + 119899

infin

sum

119896=1

119901119899+120574119896120574

(119909)

sdot int

infin

0

119887119899119896+1120574

(119905) minus 119887119899119896120574

(119905) 119891(119899119905 + 120572

119899 + 120573)119889119905

International Journal of Analysis 5

(119861120572120573

119899120574)(1)

(119891 119909) = 119899 (1 + 120574119909)minus119899120574minus1

sdot int

infin

0

(119899 + 120574) (1 + 120574119905)minus119899120574minus2

119891(119899119905 + 120572

119899 + 120573)119889119905

+ 119899

infin

sum

119896=1

119901119899+120574119896120574

(119909)

sdot int

infin

0

(minus1

119899119887(1)

119899minus120574119896+1120574(119905)) 119891(

119899119905 + 120572

119899 + 120573)119889119905

minus 119899 (1 + 120574119909)minus119899120574minus1

119891(120572

119899 + 120573)

(21)

Integrating by parts we get

(119861120572120573

119899120574)(1)

(119891 119909) = 119899 (1 + 120574119909)minus119899120574minus1

119891(120572

119899 + 120573)

+1198992

119899 + 120573(1 + 120574119909)

minus119899120574minus1

sdot int

infin

0

(1 + 120574119905)minus119899120574minus1

119891(1)

(119899119905 + 120572

119899 + 120573)119889119905 +

119899

119899 + 120573

sdot

infin

sum

119896=1

119901119899+120574119896120574

(119909) int

infin

0

119887119899minus120574119896+1120574

(119905) 119891(1)

(119899119905 + 120572

119899 + 120573)119889119905

minus 119899 (1 + 120574119909)minus119899120574minus1

119891(120572

119899 + 120573)

(119861120572120573

119899120574)(1)

(119891 119909) =119899

119899 + 120573

infin

sum

119896=0

119901119899+120574119896120574

(119909)

sdot int

infin

0

119887119899minus120574119896+1120574

(119905) 119891(1)

(119899119905 + 120572

119899 + 120573)119889119905

(22)

Thus the result is true for 119903 = 1 We prove the result byinduction method Suppose that the result is true for 119903 = 119894then

(119861120572120573

119899120574)(119894)

(119891 119909) =119899119894

Γ (119899120574 + 119894) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

sdot

infin

sum

119896=0

119901119899+120574119894119896120574

(119909) int

infin

0

119887119899minus120574119894119896+119894120574

(119905) 119891(119894)

(119899119905 + 120572

119899 + 120573)119889119905

(23)

Thus using the identities (20) we have

(119861120572120573

119899120574)(119894+1)

(119891 119909)

=119899119894

Γ (119899120574 + 119894) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

infin

sum

119896=1

(119899

120574+ 119894)

sdot 119901119899+120574(119894+1)119896minus1120574

(119909) minus 119901119899+120574(119894+1)119896120574

(119909) int

infin

0

119887119899minus120574119894119896+119894120574

(119905)

sdot 119891(119894)

(119899119905 + 120572

119899 + 120573)119889119905 minus (

119899

120574+ 119894) (1 + 120574119909)

minus119899120574minus119894minus1

sdot int

infin

0

119887119899minus120574119894119894120574

(119905) 119891(119894)

(119899119905 + 120572

119899 + 120573)

=119899119894

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

119901119899+120574(119894+1)0120574

(119909)

sdot int

infin

0

119887119899minus1205741198941+119894120574

(119905) 119891(119894)

(119899119905 + 120572

119899 + 120573)119889119905

minus119899119894

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

119901119899+120574(119894+1)0120574

(119909)

sdot int

infin

0

119887119899minus120574119894119894120574

(119905) 119891(119894)

(119899119905 + 120572

119899 + 120573)119889119905

+119899119894

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

infin

sum

119896=1

119901119899+120574(119894+1)119896120574

(119909)

sdot int

infin

0

119887119899minus120574119894119896+119894+1120574

(119905) minus 119887119899minus120574119894119896+119894120574

(119905)

sdot 119891(119894)

(119899119905 + 120572

119899 + 120573)119889119905

=119899119894

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

119901119899+120574(119894+1)0120574

(119909)

sdot int

infin

0

(minus1

119899120574 minus 119894119887(1)

119899minus120574(119894minus1)1+119894120574(119905))119891

(119894)

(119899119905 + 120572

119899 + 120573)119889119905

+119899119894

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

infin

sum

119896=1

119901119899+120574(119894+1)119896120574

(119909)

sdot int

infin

0

(minus1

119899120574 minus 119894119887(1)

119899minus120574(119894minus1)119896+119894+1120574(119905))119891

(119894)

(119899119905 + 120572

119899 + 120573)119889119905

(24)

Integrating by parts we obtain

(119861120572120573

119899120574)(119894+1)

(119891 119909) =119899119894+1

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894+1

Γ (119899120574 + 1) Γ (119899120574)

sdot

infin

sum

119896=0

119901119899+120574(119894+1)119896120574

(119909)

sdot int

infin

0

119887119899minus120574(119894minus1)119896+119894+1120574

(119905) 119891(119894+1)

(119899119905 + 120572

119899 + 120573)119889119905

(25)

This completes the proof of Lemma 5

3 Direct Theorems

This section deals with the direct results we establish herepointwise approximation asymptotic formula and errorestimation in simultaneous approximation

6 International Journal of Analysis

We denote 119862120583[0infin) = 119891 isin 119862[0infin) |119891(119905)| le

119872119905120583 for some 119872 gt 0 120583 gt 0 and the norm sdot

120583on the

class 119862120583[0infin) is defined as 119891

120583= sup

0le119905ltinfin|119891(119905)|119905

minus120583

It canbe easily verified that the operators 119861120572120573

119899120574(119891 119909) are well defined

for 119891 isin 119862120583[0infin)

Theorem 6 Let 119891 isin 119862120583[0infin) and let 119891(119903) exist at a point

119909 isin (0infin) Then one has

lim119899rarrinfin

(119861120572120573

119899120574)(119903)

(119891 119909) = 119891(119903)

(119909) (26)

Proof By Taylorrsquos expansion of 119891 we have

119891 (119905) =

119903

sum

119894=0

119891(119894)

(119909)

119894(119905 minus 119909)

119894

+ 120598 (119905 119909) (119905 minus 119909)119903

(27)

where 120598(119905 119909) rarr 0 as 119905 rarr 119909 Hence

(119861120572120573

119899120574)(119903)

(119891 119909) =

119903

sum

119894=0

119891(119894)

(119909)

119894(119861120572120573

119899120574)(119903)

((119905 minus 119909)119894

119909)

+ (119861120572120573

119899120574)(119903)

(120598 (119905 119909) (119905 minus 119909)119903

119909)

= 1198771+ 1198772

(28)

First to estimate 1198771 using binomial expansion of ((119899119905 +

120572)(119899 + 120573) minus 119909)119894 and Remark 3 we have

1198771=

119903

sum

119894=0

119891(119894)

(119909)

119894

119894

sum

119895=0

(119894

119895) (minus119909)

119894minus119895

(119861120572120573

119899120574)(119903)

(119905119895

119909)

=119891(119903)

(119909)

119903119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)119903

= 119891(119903)

(119909) 119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)

997888rarr 119891(119903)

(119909) as 119899 997888rarr infin

(29)

Next applying Lemma 4 we obtain

1198772= int

infin

0

119882(119903)

119899120574(119905 119909) 120598 (119905 119909) (

119899119905 + 120572

119899 + 120573minus 119909)

119903

119889119905

100381610038161003816100381611987721003816100381610038161003816 le sum

2119894+119895le119903

119894119895ge0

119899119894

10038161003816100381610038161003816119876119894119895119903120574

(119909)10038161003816100381610038161003816

119909 (1 + 120574119909)119903

infin

sum

119896=1

|119896 minus 119899119909|119895

119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) |120598 (119905 119909)|

10038161003816100381610038161003816100381610038161003816

119899119905 + 120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

119903

119889119905

+Γ (119899120574 + 119903 + 2)

Γ (119899120574)(1 + 120574119909)

minus119899120574minus119903

|120598 (0 119909)|

sdot

10038161003816100381610038161003816100381610038161003816

120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

119903

(30)

The second term in the above expression tends to zero as 119899 rarr

infin Since 120598(119905 119909) rarr 0 as 119905 rarr 119909 for given 120576 gt 0 there existsa 120575 isin (0 1) such that |120598(119905 119909)| lt 120576 whenever 0 lt |119905 minus 119909| lt 120575If 120591 gt max120583 119903 where 120591 is any integer then we can find aconstant 119872

3gt 0 such that |120598(119905 119909)((119899119905 + 120572)(119899 + 120573) minus 119909)

119903

| le

1198723|(119899119905 + 120572)(119899 + 120573) minus 119909|

120591 for |119905 minus 119909| ge 120575 Therefore

100381610038161003816100381611987721003816100381610038161003816 le 119872

3sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=0

|119896 minus 119899119909|119895

119901119899119896120574

(119909)

sdot 120576 int|119905minus119909|lt120575

119887119899119896120574

(119909)

10038161003816100381610038161003816100381610038161003816

119899119905 + 120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

119903

119889119905

+ int|119905minus119909|ge120575

119887119899119896120574

(119905)

10038161003816100381610038161003816100381610038161003816

119899119905 + 120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

120591

119889119905 = 1198773+ 1198774

(31)

Applying the Cauchy-Schwarz inequality for integration andsummation respectively we obtain

100381610038161003816100381611987731003816100381610038161003816 le 120576119872

3sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=1

(119896 minus 119899119909)2119895

119901119899119896120574

(119909)

12

sdot

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

2119903

119889119905

12

(32)

Using Remark 2 and Lemma 1 we get 1198773le 120576119874(119899

1199032

)119874(119899minus1199032

)

= 120576 sdot 119874(1)

Again using the Cauchy-Schwarz inequality and Lemma1 we get

100381610038161003816100381611987741003816100381610038161003816 le 119872

4sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=1

|119896 minus 119899119909|119895

119901119899119896120574

(119909)

sdot int|119905minus119909|ge120575

119887119899119896120574

(119905)

10038161003816100381610038161003816100381610038161003816

119899119905 + 120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

120591

119889119905 le 1198724

sum

2119894+119895le119903

119894119895ge0

119899119894

sdot

infin

sum

119896=1

|119896 minus 119899119909|119895

119901119899119896120574

(119909) int|119905minus119909|ge120575

119887119899119896120574

(119905) 119889119905

12

sdot int|119905minus119909|ge120575

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

2120591

119889119905

12

le 1198724

sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=1

(119896 minus 119899119909)2119895

119901119899119896120574

(119909)

12

sdot

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

2120591

119889119905

12

= sum

2119894+119895le119903

119894119895ge0

119899119894

119874(1198991198952

)119874 (119899minus1205912

) = 119874 (119899(119903minus120591)2

) = 119900 (1)

(33)

Collecting the estimation of1198771ndash1198774 we get the required result

International Journal of Analysis 7

Theorem 7 Let 119891 isin 119862120583[0infin) If 119891(119903+2) exists at a point 119909 isin

(0infin) then

lim119899rarrinfin

119899 (119861120572120573

119899120574)(119903)

(119891 119909) minus 119891(119903)

(119909)

= 119903 (120574 (119903 minus 1) minus 120573) 119891(119903)

(119909)

+ 119903120574 (1 + 2119909) + 120572 minus 120573119909119891(119903+1)

(119909)

+ 119909 (1 + 120574119909) 119891(119903+2)

(119909)

(34)

Proof Using Taylorrsquos expansion of 119891 we have

119891 (119905) =

119903+2

sum

119894=0

119891(119894)

(119909)

119894(119905 minus 119909)

119894

+ 120598 (119905 119909) (119905 minus 119909)119903+2

(35)

where 120598(119905 119909) rarr 0 as 119905 rarr 119909 and 120598(119905 119909) = 119874((119905 minus 119909)120583

) 119905 rarr

infin for 120583 gt 0Applying Lemma 1 we have

119899 (119861120572120573

119899120574)(119903)

(119891 119909) minus 119891(119903)

(119909)

= 119899

119903+2

sum

119894=0

119891(119894)

(119909)

119894(119861120572120573

119899120574)(119903)

((119905 minus 119909)119894

119909) minus 119891(119903)

(119909)

+ 119899 (119861120572120573

119899120574)(119903)

(120598 (119905 119909) (119905 minus 119909)119903+2

119909)

= 1198641+ 1198642

(36)

First we have

1198641= 119899

119903+2

sum

119894=0

119891(119894)

(119909)

119894

119894

sum

119895=0

(119894

119895) (minus119909)

119894minus119895

(119861120572120573

119899120574)(119903)

(119905119895

119909)

minus 119899119891(119903)

(119909) =119891(119903)

(119909)

119903119899 (119861

120572120573

119899120574)(119903)

(119905119903

119909) minus 119903

+119891(119903+1)

(119909)

(119903 + 1)119899 (119903 + 1) (minus119909) (119861

120572120573

119899120574)(119903)

(119905119903

119909)

+ (119861120572120573

119899120574)(119903)

(119905119903+1

119909) +119891(119903+2)

(119909)

(119903 + 2)

sdot 119899 (119903 + 2) (119903 + 1)

21199092

(119861120572120573

119899120574)(119903)

(119905119903

119909) + (119903 + 2)

sdot (minus119909) (119861120572120573

119899120574)(119903)

(119905119903+1

119909) + (119861120572120573

119899120574)(119903)

(119905119903+2

119909)

= 119891(119903)

(119909) 119899 119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)minus 1

+119891(119903+1)

(119909)

(119903 + 1)119899(119903 + 1) (minus119909)

sdot119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)119903

+119899119903+1

Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903)

(119899 + 120573)119903+1

Γ (119899120574 + 1) Γ (119899120574)

(119903 + 1)119909

+(119903 + 1) 119899

119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903)

(119899 + 120573)119903+1

Γ (119899120574 + 1) Γ (119899120574)

119899119903 + 120572(119899

120574

minus 119903) 119903 +119891(119903+2)

(119909)

(119903 + 2)119899(

(119903 + 1) (119903 + 2)

21199092

sdot119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)119903 minus 119909 (119903 + 2)

sdot 119899119903+1

Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903)

(119899 + 120573)119903+1

Γ (119899120574 + 1) Γ (119899120574)

(119903 + 1)119909

+(119903 + 1) 119899

119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903)

(119899 + 120573)119903+1

Γ (119899120574 + 1) Γ (119899120574)

119899119903

+ 120572(119899

120574minus 119903) 119903

+119899119903+2

Γ (119899120574 + 119903 + 2) Γ (119899120574 minus 119903 minus 1)

(119899 + 120573)119903+2

Γ (119899120574 + 1) Γ (119899120574)

(119903 + 2)

21199092

+(119903 + 2) 119899

119903+1

Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903 minus 1)

(119899 + 120573)119903+2

Γ (119899120574 + 1) Γ (119899120574)

119899 (119903

+ 1) + 120572(119899

120574minus 119903 minus 1) (119903 + 1)119909

+120572 (119903 + 1) (119903 + 2) 119899

119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)119899119903

+120572 (119899120574 minus 119903)

2 119903)

(37)

Now the coefficients of119891(119903)(119909)119891(119903+1)(119909) and119891(119903+2)

(119909) in theabove expression tend to 119903(120574(119903 minus 1) minus 120573) 119903120574(1 + 2119909) + 120572 minus 120573119909and 119909(1 + 120574119909) respectively which follows by using inductionhypothesis on 119903 and taking the limit as 119899 rarr infin Hence inorder to prove (34) it is sufficient to show that 119864

2rarr 0

as 119899 rarr infin which follows along the lines of the proof ofTheorem 6 and by using Remark 2 and Lemmas 1 and 4

Remark 8 Particular case 120572 = 120573 = 0 was discussed inTheorem 41 in [4] which says that the coefficient of119891(119903+1)(119909)converges to 119903(1 + 2120574119909) but it converges to 119903120574(1 + 2119909) and weget this by putting 120572 = 120573 = 0 in the above theorem

Definition 9 The 119898th order modulus of continuity 120596119898(119891 120575

[119886 119887]) for a function continuous on [119886 119887] is defined by

120596119898(119891 120575 [119886 119887])

= sup 1003816100381610038161003816Δ119898

ℎ119891 (119909)

1003816100381610038161003816 |ℎ| le 120575 119909 119909 + ℎ isin [119886 119887]

(38)

For119898 = 1 120596119898(119891 120575) is usual modulus of continuity

8 International Journal of Analysis

Theorem 10 Let 119891 isin 119862120583[0infin) for some 120583 gt 0 and 0 lt 119886 lt

1198861lt 1198871lt 119887 lt infin Then for 119899 sufficiently large one has

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(119891 sdot) minus 119891(119903)

1003817100381710038171003817100381710038171003817119862[11988611198871]

le 11987211205962(119891(119903)

119899minus12

[1198861 1198871]) + 119872

2119899minus1 1003817100381710038171003817119891

1003817100381710038171003817120583

(39)

where1198721= 1198721(119903) and119872

2= 1198722(119903 119891)

Proof Let us assume that 0 lt 119886 lt 1198861

lt 1198871

lt 119887 lt infinFor sufficiently small 120578 gt 0 we define the function 119891

1205782

corresponding to 119891 isin 119862120583[119886 119887] and 119905 isin [119886

1 1198871] as follows

1198911205782

(119905) = 120578minus2

∬

1205782

minus1205782

(119891 (119905) minus Δ2

ℎ119891 (119905)) 119889119905

11198891199052 (40)

where ℎ = (1199051+ 1199052)2 and Δ

2

ℎis the second order forward

difference operator with step length ℎ For 119891 isin 119862[119886 119887] thefunctions 119891

1205782are known as the Steklov mean of order 2

which satisfy the following properties [11]

(a) 1198911205782

has continuous derivatives up to order 2 over[1198861 1198871]

(b) 119891(119903)1205782

119862[11988611198871]le 1198721120578minus119903

1205962(119891 120578 [119886 119887]) 119903 = 1 2

(c) 119891 minus 1198911205782

119862[11988611198871]le 11987221205962(119891 120578 [119886 119887])

(d) 1198911205782

119862[11988611198871]le 1198723119891120583

where119872119894 119894 = 1 2 3 are certain constants which are different

in each occurrence and are independent of 119891 and 120578We can write by linearity properties of 119861120572120573

119899120574

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(119891 sdot) minus 119891(119903)

1003817100381710038171003817100381710038171003817119862[11988611198871]

le

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(119891 minus 1198911205782

sdot)

1003817100381710038171003817100381710038171003817119862[11988611198871]

+

1003817100381710038171003817100381710038171003817((119861120572120573

119899120574)(119903)

1198911205782

sdot) minus 119891(119903)

1205782

1003817100381710038171003817100381710038171003817119862[11988611198871]

+10038171003817100381710038171003817119891(119903)

minus 119891(119903)

1205782

10038171003817100381710038171003817119862[11988611198871]

= 1198751+ 1198752+ 1198753

(41)

Since 119891(119903)

1205782= (119891(119903)

)1205782

(119905) by property (c) of the function 1198911205782

we get

1198753le 11987241205962(119891(119903)

120578 [119886 119887]) (42)

Next on an application of Theorem 7 it follows that

1198752le 1198725119899minus1

119903+2

sum

119894=119903

10038171003817100381710038171003817119891(119894)

1205782

10038171003817100381710038171003817119862[119886119887] (43)

Using the interpolation property due to Goldberg and Meir[12] for each 119895 = 119903 119903 + 1 119903 + 2 it follows that

10038171003817100381710038171003817119891(119894)

1205782

10038171003817100381710038171003817119862[119886119887]le 1198726100381710038171003817100381710038171198911205782

10038171003817100381710038171003817119862[119886119887]+10038171003817100381710038171003817119891(119903+2)

1205782

10038171003817100381710038171003817119862[119886119887] (44)

Therefore by applying properties (c) and (d) of the function1198911205782 we obtain

1198752le 1198727sdot 119899minus1

1003817100381710038171003817119891

1003817100381710038171003817120583 + 120575minus2

1205962(119891(119903)

120583 [119886 119887]) (45)

Finally we will estimate 1198751 choosing 119886

lowast 119887lowast satisfying theconditions 0 lt 119886 lt 119886

lowast

lt 1198861lt 1198871lt 119887lowast

lt 119887 lt infin Suppose ℏ(119905)denotes the characteristic function of the interval [119886lowast 119887lowast]Then

1198751le

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(ℏ (119905) (119891 (119905) minus 1198911205782

(119905)) sdot)

1003817100381710038171003817100381710038171003817119862[11988611198871]

+

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782

(119905)) sdot)

1003817100381710038171003817100381710038171003817119862[11988611198871]

= 1198754+ 1198755

(46)

By Lemma 5 we have

(119861120572120573

119899120574)(119903)

(ℏ (119905) (119891 (119905) minus 1198911205782

(119905)) 119909)

=119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)

infin

sum

119896=0

119901119899+120574119903119896120574

(119909)

sdot int

infin

0

119887119899minus120574119903119896+119903120574

(119905) ℏ (119905)

sdot (119891(119903)

(119899119905 + 120572

119899 + 120573) minus 119891

(119903)

1205782(119899119905 + 120572

119899 + 120573))119889119905

(47)

Hence100381710038171003817100381710038171003817(119861120572120573

119899120574)119903

(ℏ (119905) (119891 (119905) minus 1198911205782

(119905)) sdot)100381710038171003817100381710038171003817119862[11988611198871]

le 1198728

10038171003817100381710038171003817119891(119903)

minus 119891(119903)

1205782

10038171003817100381710038171003817119862[119886lowast 119887lowast]

(48)

Now for 119909 isin [1198861 1198871] and 119905 isin [0infin) [119886

lowast

119887lowast

] we choose a120575 gt 0 satisfying |(119899119905 + 120572)(119899 + 120573) minus 119909| ge 120575

Therefore by Lemma 4 and the Cauchy-Schwarz inequal-ity we have

119868 equiv (119861120572120573

119899120574)(119903)

((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782

(119905)) 119909)

|119868| le sum

2119894+119895le119903

119894119895ge0

119899119894

10038161003816100381610038161003816119876119894119895119903120574

(119909)10038161003816100381610038161003816

119909 (1 + 120574119909)119903

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|119895

sdot int

infin

0

119887119899119896120574

(119905) (1 minus ℏ (119905))

sdot

10038161003816100381610038161003816100381610038161003816119891 (

119899119905 + 120572

119899 + 120573) minus 1198911205782

(119899119905 + 120572

119899 + 120573)

10038161003816100381610038161003816100381610038161003816119889119905

+Γ (119899120574 + 119903)

Γ (119899120574)(1 + 120574119909)

minus119899120574minus119903

(1 minus ℏ (0))

sdot

10038161003816100381610038161003816100381610038161003816119891 (

120572

119899 + 120573) minus 1198911205782

(120572

119899 + 120573)

10038161003816100381610038161003816100381610038161003816

(49)

International Journal of Analysis 9

For sufficiently large 119899 the second term tends to zero Thus

|119868| le 1198729

10038171003817100381710038171198911003817100381710038171003817120583 sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|119895

sdot int|119905minus119909|ge120575

119887119899119896120574

(119905) 119889119905 le 1198729

10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898

sum

2119894+119895le119903

119894119895ge0

119899119894

sdot

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|119895

(int

infin

0

119887119899119896120574

(119905) 119889119905)

12

sdot (int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

4119898

119889119905)

12

le 1198729

10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898

sum

2119894+119895le119903

119894119895ge0

119899119894

sdot (

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|2119895

)

12

(

infin

sum

119896=1

119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

4119898

119889119905)

12

(50)

Hence by using Remark 2 and Lemma 1 we have

|119868| le 11987210

10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898

119874(119899(119894+(1198952)minus119898)

) le 11987211119899minus119902 1003817100381710038171003817119891

1003817100381710038171003817120583 (51)

where 119902 = 119898 minus (1199032) Now choosing 119898 gt 0 satisfying 119902 ge 1we obtain 119868 le 119872

11119899minus1

119891120583 Therefore by property (c) of the

function 1198911205782

(119905) we get

1198751le 1198728

10038171003817100381710038171003817119891(119903)

minus 119891(119903)

1205782

10038171003817100381710038171003817119862[119886lowast 119887lowast]+ 11987211119899minus1 1003817100381710038171003817119891

1003817100381710038171003817120583

le 119872121205962(119891(119903)

120578 [119886 119887]) + 11987211119899minus1 1003817100381710038171003817119891

1003817100381710038171003817120583

(52)

Choosing 120578 = 119899minus12 the theorem follows

Remark 11 In the last decade the applications of 119902-calculus inapproximation theory are one of themain areas of research In2008 Gupta [13] introduced 119902-Durrmeyer operators whoseapproximation properties were studied in [14] More work inthis direction can be seen in [15ndash17]

A Durrmeyer type 119902-analogue of the 119861120572120573

119899120574(119891 119909) is intro-

duced as follows

119861120572120573

119899120574119902(119891 119909)

=

infin

sum

119896=1

119901119902

119899119896120574(119909) int

infin119860

0

119902minus119896

119887119902

119899119896120574(119905) 119891(

[119899]119902119905 + 120572

[119899]119902+ 120573

)119889119902119905

+ 119901119902

1198990120574(119909) 119891(

120572

[119899]119902+ 120573

)

(53)

where

119901119902

119899119896120574(119909) = 119902

11989622

Γ119902(119899120574 + 119896)

Γ119902(119896 + 1) Γ

119902(119899120574)

sdot(119902120574119909)

119896

(1 + 119902120574119909)(119899120574)+119896

119902

119887119902

119899119896120574(119909) = 120574119902

11989622

Γ119902(119899120574 + 119896 + 1)

Γ119902(119896) Γ119902(119899120574 + 1)

sdot(120574119905)119896minus1

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

119902

int

infin119860

0

119891 (119909) 119889119902119909 = (1 minus 119902)

infin

sum

119899=minusinfin

119891(119902119899

119860)

119902119899

119860 119860 gt 0

(54)

Notations used in (53) can be found in [18] For the operators(53) one can study their approximation properties based on119902-integers

Conflict of Interests

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

Acknowledgments

The authors would like to express their deep gratitude to theanonymous learned referee(s) and the editor for their valu-able suggestions and constructive comments which resultedin the subsequent improvement of this research article

References

[1] V Gupta D K Verma and P N Agrawal ldquoSimultaneousapproximation by certain Baskakov-Durrmeyer-Stancu opera-torsrdquo Journal of the Egyptian Mathematical Society vol 20 no3 pp 183ndash187 2012

[2] D K Verma VGupta and PN Agrawal ldquoSome approximationproperties of Baskakov-Durrmeyer-Stancu operatorsrdquo AppliedMathematics and Computation vol 218 no 11 pp 6549ndash65562012

[3] V N Mishra K Khatri L N Mishra and Deepmala ldquoInverseresult in simultaneous approximation by Baskakov-Durrmeyer-Stancu operatorsrdquo Journal of Inequalities and Applications vol2013 article 586 11 pages 2013

[4] V Gupta ldquoApproximation for modified Baskakov Durrmeyertype operatorsrdquoRockyMountain Journal ofMathematics vol 39no 3 pp 825ndash841 2009

[5] 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 no 4 article 125 2006

[6] V N Mishra and P Patel ldquoApproximation properties ofq-Baskakov-Durrmeyer-Stancu operatorsrdquo Mathematical Sci-ences vol 7 no 1 article 38 12 pages 2013

10 International Journal of Analysis

[7] P Patel and V N Mishra ldquoApproximation properties of certainsummation integral type operatorsrdquo Demonstratio Mathemat-ica vol 48 no 1 pp 77ndash90 2015

[8] 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

[9] 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

[10] W Li ldquoThe voronovskaja type expansion fomula of themodifiedBaskakov-Beta operatorsrdquo Journal of Baoji College of Arts andScience (Natural Science Edition) vol 2 article 004 2005

[11] G Freud andV Popov ldquoOn approximation by spline functionsrdquoin Proceedings of the Conference on Constructive Theory Func-tion Budapest Hungary 1969

[12] S Goldberg and A Meir ldquoMinimummoduli of ordinary differ-ential operatorsrdquo Proceedings of the London Mathematical Soci-ety vol 23 no 3 pp 1ndash15 1971

[13] V Gupta ldquoSome approxmation properties of q-durrmeyer ope-ratorsrdquo Applied Mathematics and Computation vol 197 no 1pp 172ndash178 2008

[14] A Aral and V Gupta ldquoOn the Durrmeyer type modificationof the q-Baskakov type operatorsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 3-4 pp 1171ndash1180 2010

[15] GM Phillips ldquoBernstein polynomials based on the q-integersrdquoAnnals of Numerical Mathematics vol 4 no 1ndash4 pp 511ndash5181997

[16] S Ostrovska ldquoOn the Lupas q-analogue of the Bernstein ope-ratorrdquoThe Rocky Mountain Journal of Mathematics vol 36 no5 pp 1615ndash1629 2006

[17] V N Mishra and P Patel ldquoOn generalized integral Bernsteinoperators based on q-integersrdquo Applied Mathematics and Com-putation vol 242 pp 931ndash944 2014

[18] V Kac and P Cheung Quantum Calculus UniversitextSpringer New York NY USA 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 International Journal of Analysis

sdot 120583119899119898120574

(119909) minus119898120574 (119899 + 120573)

119899(

120572

119899 + 120573minus 119909)

2

sdot 120583119899119898minus1120574

(119909) minus 119899119909120583119899119898120574

(119909)

+(119899 + 2120574) (119899 + 120573)

119899120583119899119898+1120574

(119909)

minus (120572

119899 + 120573minus 119909)120583

119899119898120574(119909)

(13)

Hence

(119899 minus 120574119898) (119899 + 120573) 120583119899119898+1120574

(119909) = 119899119909 (1 + 120574119909) 120583(1)

119899119898120574(119909)

+ 119898120583119899119898minus1120574

(119909) + 119898119899 + 1198992

119909

minus (2120574119898 minus 119899) (120572 minus (119899 + 120573) 119909) 120583119899119898120574

(119909)

+ 119898120574 (119899 + 120573) (120572

119899 + 120573minus 119909)

2

minus 119898119899(120572

119899 + 120573minus 119909)120583

119899119898minus1120574(119909)

(14)

This completes the proof of Lemma 1

Remark 2 (see [10]) For119898 isin Ncup0 if the119898th ordermomentis defined as

119880119899119898120574

(119909) =

infin

sum

119896=0

119901119899119896120574

(119909) (119896

119899minus 119909)

119898

(15)

then 1198801198990120574

(119909) = 1 1198801198991120574

(119909) = 0 and 119899119880119899119898+1120574

(119909) = 119909(1 +

120574119909)(119880(1)

119899119898120574(119909) + 119898119880

119899119898minus1120574(119909))

Consequently for all 119909 isin [0infin) we have 119880119899119898120574

(119909) =

119874(119899minus[(119898+1)2]

)

Remark 3 It is easily verified from Lemma 1 that for each 119909 isin

[0infin)

119861120572120573

119899120574(119905119898

119909) =119899119898

Γ (119899120574 + 119898) Γ (119899120574 minus 119898 + 1)

(119899 + 120573)119898

Γ (119899120574 + 1) Γ (119899120574)119909119898

+119898119899119898minus1

Γ (119899120574 + 119898 minus 1) Γ (119899120574 minus 119898 + 1)

(119899 + 120573)119898

Γ (119899120574 + 1) Γ (119899120574)119899 (119898 minus 1)

+ 120572(119899

120574minus 119898 + 1)119909

119898minus1

+120572119898 (119898 minus 1) 119899

119898minus2

Γ (119899120574 + 119898 minus 2) Γ (119899120574 minus 119898 + 2)

(119899 + 120573)119898

Γ (119899120574 + 1) Γ (119899120574)119899 (119898

minus 2) +120572 (119899120574 minus 119898 + 2)

2119909119898minus2

+ 119874 (119899minus2

)

(16)

Lemma 4 (see [10]) The polynomials119876119894119895119903120574

(119909) exist indepen-dent of 119899 and 119896 such that

119909 (1 + 120574119909)119903

119863119903

[119901119899119896120574

(119909)]

= sum

2119894+119895le119903

119894119895ge0

119899119894

(119896 minus 119899119909)119895

119876119894119895119903120574

(119909) 119901119899119896120574

(119909)

where 119863 equiv119889

119889119909

(17)

Lemma 5 If 119891 is 119903 times differentiable on [0infin) such that119891(119903minus1)

= 119874(119905120592

) 120592 gt 0 as 119905 rarr infin then for 119903 = 1 2 3 and119899 gt 120592 + 120574119903 one has

(119861120572120573

119899120574)(119903)

(119891 119909) =119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)

sdot

infin

sum

119896=0

119901119899+120574119903119896120574

(119909)

sdot int

infin

0

119887119899minus120574119903119896+119903120574

(119905) 119891(119903)

(119899119905 + 120572

119899 + 120573)119889119905

(18)

Proof First

(119861120572120573

119899120574)(1)

(119891 119909)

=

infin

sum

119896=1

119901(1)

119899119896120574(119909) int

infin

0

119887119899119896120574

(119905) 119891(119899119905 + 120572

119899 + 120573)119889119905

minus 119899 (1 + 120574119909)minus119899120574minus1

119891(120572

119899 + 120573)

(19)

Now using the identities

119901(1)

119899119896120574(119909) = 119899 119901

119899+120574119896minus1120574(119909) minus 119901

119899+120574119896120574(119909)

119887(1)

119899119896120574(119909) = (119899 + 120574) 119887

119899+120574119896minus1120574(119909) minus 119887

119899+120574119896120574(119909)

(20)

for 119896 ge 1 we have

(119861120572120573

119899120574)(1)

(119891 119909) =

infin

sum

119896=1

119899 119901119899+120574119896minus1120574

(119909) minus 119901119899+120574119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) 119891(119899119905 + 120572

119899 + 120573)119889119905 minus 119899 (1 + 120574119909)

minus119899120574minus1

sdot 119891 (120572

119899 + 120573) = 119899119901

119899+1205740120574(119909)

sdot int

infin

0

119887119899+1205741120574

(119905) 119891(119899119905 + 120572

119899 + 120573)119889119905 minus 119899 (1 + 120574119909)

minus119899120574minus1

sdot 119891 (120572

119899 + 120573) + 119899

infin

sum

119896=1

119901119899+120574119896120574

(119909)

sdot int

infin

0

119887119899119896+1120574

(119905) minus 119887119899119896120574

(119905) 119891(119899119905 + 120572

119899 + 120573)119889119905

International Journal of Analysis 5

(119861120572120573

119899120574)(1)

(119891 119909) = 119899 (1 + 120574119909)minus119899120574minus1

sdot int

infin

0

(119899 + 120574) (1 + 120574119905)minus119899120574minus2

119891(119899119905 + 120572

119899 + 120573)119889119905

+ 119899

infin

sum

119896=1

119901119899+120574119896120574

(119909)

sdot int

infin

0

(minus1

119899119887(1)

119899minus120574119896+1120574(119905)) 119891(

119899119905 + 120572

119899 + 120573)119889119905

minus 119899 (1 + 120574119909)minus119899120574minus1

119891(120572

119899 + 120573)

(21)

Integrating by parts we get

(119861120572120573

119899120574)(1)

(119891 119909) = 119899 (1 + 120574119909)minus119899120574minus1

119891(120572

119899 + 120573)

+1198992

119899 + 120573(1 + 120574119909)

minus119899120574minus1

sdot int

infin

0

(1 + 120574119905)minus119899120574minus1

119891(1)

(119899119905 + 120572

119899 + 120573)119889119905 +

119899

119899 + 120573

sdot

infin

sum

119896=1

119901119899+120574119896120574

(119909) int

infin

0

119887119899minus120574119896+1120574

(119905) 119891(1)

(119899119905 + 120572

119899 + 120573)119889119905

minus 119899 (1 + 120574119909)minus119899120574minus1

119891(120572

119899 + 120573)

(119861120572120573

119899120574)(1)

(119891 119909) =119899

119899 + 120573

infin

sum

119896=0

119901119899+120574119896120574

(119909)

sdot int

infin

0

119887119899minus120574119896+1120574

(119905) 119891(1)

(119899119905 + 120572

119899 + 120573)119889119905

(22)

Thus the result is true for 119903 = 1 We prove the result byinduction method Suppose that the result is true for 119903 = 119894then

(119861120572120573

119899120574)(119894)

(119891 119909) =119899119894

Γ (119899120574 + 119894) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

sdot

infin

sum

119896=0

119901119899+120574119894119896120574

(119909) int

infin

0

119887119899minus120574119894119896+119894120574

(119905) 119891(119894)

(119899119905 + 120572

119899 + 120573)119889119905

(23)

Thus using the identities (20) we have

(119861120572120573

119899120574)(119894+1)

(119891 119909)

=119899119894

Γ (119899120574 + 119894) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

infin

sum

119896=1

(119899

120574+ 119894)

sdot 119901119899+120574(119894+1)119896minus1120574

(119909) minus 119901119899+120574(119894+1)119896120574

(119909) int

infin

0

119887119899minus120574119894119896+119894120574

(119905)

sdot 119891(119894)

(119899119905 + 120572

119899 + 120573)119889119905 minus (

119899

120574+ 119894) (1 + 120574119909)

minus119899120574minus119894minus1

sdot int

infin

0

119887119899minus120574119894119894120574

(119905) 119891(119894)

(119899119905 + 120572

119899 + 120573)

=119899119894

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

119901119899+120574(119894+1)0120574

(119909)

sdot int

infin

0

119887119899minus1205741198941+119894120574

(119905) 119891(119894)

(119899119905 + 120572

119899 + 120573)119889119905

minus119899119894

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

119901119899+120574(119894+1)0120574

(119909)

sdot int

infin

0

119887119899minus120574119894119894120574

(119905) 119891(119894)

(119899119905 + 120572

119899 + 120573)119889119905

+119899119894

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

infin

sum

119896=1

119901119899+120574(119894+1)119896120574

(119909)

sdot int

infin

0

119887119899minus120574119894119896+119894+1120574

(119905) minus 119887119899minus120574119894119896+119894120574

(119905)

sdot 119891(119894)

(119899119905 + 120572

119899 + 120573)119889119905

=119899119894

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

119901119899+120574(119894+1)0120574

(119909)

sdot int

infin

0

(minus1

119899120574 minus 119894119887(1)

119899minus120574(119894minus1)1+119894120574(119905))119891

(119894)

(119899119905 + 120572

119899 + 120573)119889119905

+119899119894

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

infin

sum

119896=1

119901119899+120574(119894+1)119896120574

(119909)

sdot int

infin

0

(minus1

119899120574 minus 119894119887(1)

119899minus120574(119894minus1)119896+119894+1120574(119905))119891

(119894)

(119899119905 + 120572

119899 + 120573)119889119905

(24)

Integrating by parts we obtain

(119861120572120573

119899120574)(119894+1)

(119891 119909) =119899119894+1

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894+1

Γ (119899120574 + 1) Γ (119899120574)

sdot

infin

sum

119896=0

119901119899+120574(119894+1)119896120574

(119909)

sdot int

infin

0

119887119899minus120574(119894minus1)119896+119894+1120574

(119905) 119891(119894+1)

(119899119905 + 120572

119899 + 120573)119889119905

(25)

This completes the proof of Lemma 5

3 Direct Theorems

This section deals with the direct results we establish herepointwise approximation asymptotic formula and errorestimation in simultaneous approximation

6 International Journal of Analysis

We denote 119862120583[0infin) = 119891 isin 119862[0infin) |119891(119905)| le

119872119905120583 for some 119872 gt 0 120583 gt 0 and the norm sdot

120583on the

class 119862120583[0infin) is defined as 119891

120583= sup

0le119905ltinfin|119891(119905)|119905

minus120583

It canbe easily verified that the operators 119861120572120573

119899120574(119891 119909) are well defined

for 119891 isin 119862120583[0infin)

Theorem 6 Let 119891 isin 119862120583[0infin) and let 119891(119903) exist at a point

119909 isin (0infin) Then one has

lim119899rarrinfin

(119861120572120573

119899120574)(119903)

(119891 119909) = 119891(119903)

(119909) (26)

Proof By Taylorrsquos expansion of 119891 we have

119891 (119905) =

119903

sum

119894=0

119891(119894)

(119909)

119894(119905 minus 119909)

119894

+ 120598 (119905 119909) (119905 minus 119909)119903

(27)

where 120598(119905 119909) rarr 0 as 119905 rarr 119909 Hence

(119861120572120573

119899120574)(119903)

(119891 119909) =

119903

sum

119894=0

119891(119894)

(119909)

119894(119861120572120573

119899120574)(119903)

((119905 minus 119909)119894

119909)

+ (119861120572120573

119899120574)(119903)

(120598 (119905 119909) (119905 minus 119909)119903

119909)

= 1198771+ 1198772

(28)

First to estimate 1198771 using binomial expansion of ((119899119905 +

120572)(119899 + 120573) minus 119909)119894 and Remark 3 we have

1198771=

119903

sum

119894=0

119891(119894)

(119909)

119894

119894

sum

119895=0

(119894

119895) (minus119909)

119894minus119895

(119861120572120573

119899120574)(119903)

(119905119895

119909)

=119891(119903)

(119909)

119903119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)119903

= 119891(119903)

(119909) 119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)

997888rarr 119891(119903)

(119909) as 119899 997888rarr infin

(29)

Next applying Lemma 4 we obtain

1198772= int

infin

0

119882(119903)

119899120574(119905 119909) 120598 (119905 119909) (

119899119905 + 120572

119899 + 120573minus 119909)

119903

119889119905

100381610038161003816100381611987721003816100381610038161003816 le sum

2119894+119895le119903

119894119895ge0

119899119894

10038161003816100381610038161003816119876119894119895119903120574

(119909)10038161003816100381610038161003816

119909 (1 + 120574119909)119903

infin

sum

119896=1

|119896 minus 119899119909|119895

119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) |120598 (119905 119909)|

10038161003816100381610038161003816100381610038161003816

119899119905 + 120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

119903

119889119905

+Γ (119899120574 + 119903 + 2)

Γ (119899120574)(1 + 120574119909)

minus119899120574minus119903

|120598 (0 119909)|

sdot

10038161003816100381610038161003816100381610038161003816

120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

119903

(30)

The second term in the above expression tends to zero as 119899 rarr

infin Since 120598(119905 119909) rarr 0 as 119905 rarr 119909 for given 120576 gt 0 there existsa 120575 isin (0 1) such that |120598(119905 119909)| lt 120576 whenever 0 lt |119905 minus 119909| lt 120575If 120591 gt max120583 119903 where 120591 is any integer then we can find aconstant 119872

3gt 0 such that |120598(119905 119909)((119899119905 + 120572)(119899 + 120573) minus 119909)

119903

| le

1198723|(119899119905 + 120572)(119899 + 120573) minus 119909|

120591 for |119905 minus 119909| ge 120575 Therefore

100381610038161003816100381611987721003816100381610038161003816 le 119872

3sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=0

|119896 minus 119899119909|119895

119901119899119896120574

(119909)

sdot 120576 int|119905minus119909|lt120575

119887119899119896120574

(119909)

10038161003816100381610038161003816100381610038161003816

119899119905 + 120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

119903

119889119905

+ int|119905minus119909|ge120575

119887119899119896120574

(119905)

10038161003816100381610038161003816100381610038161003816

119899119905 + 120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

120591

119889119905 = 1198773+ 1198774

(31)

Applying the Cauchy-Schwarz inequality for integration andsummation respectively we obtain

100381610038161003816100381611987731003816100381610038161003816 le 120576119872

3sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=1

(119896 minus 119899119909)2119895

119901119899119896120574

(119909)

12

sdot

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

2119903

119889119905

12

(32)

Using Remark 2 and Lemma 1 we get 1198773le 120576119874(119899

1199032

)119874(119899minus1199032

)

= 120576 sdot 119874(1)

Again using the Cauchy-Schwarz inequality and Lemma1 we get

100381610038161003816100381611987741003816100381610038161003816 le 119872

4sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=1

|119896 minus 119899119909|119895

119901119899119896120574

(119909)

sdot int|119905minus119909|ge120575

119887119899119896120574

(119905)

10038161003816100381610038161003816100381610038161003816

119899119905 + 120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

120591

119889119905 le 1198724

sum

2119894+119895le119903

119894119895ge0

119899119894

sdot

infin

sum

119896=1

|119896 minus 119899119909|119895

119901119899119896120574

(119909) int|119905minus119909|ge120575

119887119899119896120574

(119905) 119889119905

12

sdot int|119905minus119909|ge120575

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

2120591

119889119905

12

le 1198724

sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=1

(119896 minus 119899119909)2119895

119901119899119896120574

(119909)

12

sdot

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

2120591

119889119905

12

= sum

2119894+119895le119903

119894119895ge0

119899119894

119874(1198991198952

)119874 (119899minus1205912

) = 119874 (119899(119903minus120591)2

) = 119900 (1)

(33)

Collecting the estimation of1198771ndash1198774 we get the required result

International Journal of Analysis 7

Theorem 7 Let 119891 isin 119862120583[0infin) If 119891(119903+2) exists at a point 119909 isin

(0infin) then

lim119899rarrinfin

119899 (119861120572120573

119899120574)(119903)

(119891 119909) minus 119891(119903)

(119909)

= 119903 (120574 (119903 minus 1) minus 120573) 119891(119903)

(119909)

+ 119903120574 (1 + 2119909) + 120572 minus 120573119909119891(119903+1)

(119909)

+ 119909 (1 + 120574119909) 119891(119903+2)

(119909)

(34)

Proof Using Taylorrsquos expansion of 119891 we have

119891 (119905) =

119903+2

sum

119894=0

119891(119894)

(119909)

119894(119905 minus 119909)

119894

+ 120598 (119905 119909) (119905 minus 119909)119903+2

(35)

where 120598(119905 119909) rarr 0 as 119905 rarr 119909 and 120598(119905 119909) = 119874((119905 minus 119909)120583

) 119905 rarr

infin for 120583 gt 0Applying Lemma 1 we have

119899 (119861120572120573

119899120574)(119903)

(119891 119909) minus 119891(119903)

(119909)

= 119899

119903+2

sum

119894=0

119891(119894)

(119909)

119894(119861120572120573

119899120574)(119903)

((119905 minus 119909)119894

119909) minus 119891(119903)

(119909)

+ 119899 (119861120572120573

119899120574)(119903)

(120598 (119905 119909) (119905 minus 119909)119903+2

119909)

= 1198641+ 1198642

(36)

First we have

1198641= 119899

119903+2

sum

119894=0

119891(119894)

(119909)

119894

119894

sum

119895=0

(119894

119895) (minus119909)

119894minus119895

(119861120572120573

119899120574)(119903)

(119905119895

119909)

minus 119899119891(119903)

(119909) =119891(119903)

(119909)

119903119899 (119861

120572120573

119899120574)(119903)

(119905119903

119909) minus 119903

+119891(119903+1)

(119909)

(119903 + 1)119899 (119903 + 1) (minus119909) (119861

120572120573

119899120574)(119903)

(119905119903

119909)

+ (119861120572120573

119899120574)(119903)

(119905119903+1

119909) +119891(119903+2)

(119909)

(119903 + 2)

sdot 119899 (119903 + 2) (119903 + 1)

21199092

(119861120572120573

119899120574)(119903)

(119905119903

119909) + (119903 + 2)

sdot (minus119909) (119861120572120573

119899120574)(119903)

(119905119903+1

119909) + (119861120572120573

119899120574)(119903)

(119905119903+2

119909)

= 119891(119903)

(119909) 119899 119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)minus 1

+119891(119903+1)

(119909)

(119903 + 1)119899(119903 + 1) (minus119909)

sdot119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)119903

+119899119903+1

Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903)

(119899 + 120573)119903+1

Γ (119899120574 + 1) Γ (119899120574)

(119903 + 1)119909

+(119903 + 1) 119899

119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903)

(119899 + 120573)119903+1

Γ (119899120574 + 1) Γ (119899120574)

119899119903 + 120572(119899

120574

minus 119903) 119903 +119891(119903+2)

(119909)

(119903 + 2)119899(

(119903 + 1) (119903 + 2)

21199092

sdot119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)119903 minus 119909 (119903 + 2)

sdot 119899119903+1

Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903)

(119899 + 120573)119903+1

Γ (119899120574 + 1) Γ (119899120574)

(119903 + 1)119909

+(119903 + 1) 119899

119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903)

(119899 + 120573)119903+1

Γ (119899120574 + 1) Γ (119899120574)

119899119903

+ 120572(119899

120574minus 119903) 119903

+119899119903+2

Γ (119899120574 + 119903 + 2) Γ (119899120574 minus 119903 minus 1)

(119899 + 120573)119903+2

Γ (119899120574 + 1) Γ (119899120574)

(119903 + 2)

21199092

+(119903 + 2) 119899

119903+1

Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903 minus 1)

(119899 + 120573)119903+2

Γ (119899120574 + 1) Γ (119899120574)

119899 (119903

+ 1) + 120572(119899

120574minus 119903 minus 1) (119903 + 1)119909

+120572 (119903 + 1) (119903 + 2) 119899

119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)119899119903

+120572 (119899120574 minus 119903)

2 119903)

(37)

Now the coefficients of119891(119903)(119909)119891(119903+1)(119909) and119891(119903+2)

(119909) in theabove expression tend to 119903(120574(119903 minus 1) minus 120573) 119903120574(1 + 2119909) + 120572 minus 120573119909and 119909(1 + 120574119909) respectively which follows by using inductionhypothesis on 119903 and taking the limit as 119899 rarr infin Hence inorder to prove (34) it is sufficient to show that 119864

2rarr 0

as 119899 rarr infin which follows along the lines of the proof ofTheorem 6 and by using Remark 2 and Lemmas 1 and 4

Remark 8 Particular case 120572 = 120573 = 0 was discussed inTheorem 41 in [4] which says that the coefficient of119891(119903+1)(119909)converges to 119903(1 + 2120574119909) but it converges to 119903120574(1 + 2119909) and weget this by putting 120572 = 120573 = 0 in the above theorem

Definition 9 The 119898th order modulus of continuity 120596119898(119891 120575

[119886 119887]) for a function continuous on [119886 119887] is defined by

120596119898(119891 120575 [119886 119887])

= sup 1003816100381610038161003816Δ119898

ℎ119891 (119909)

1003816100381610038161003816 |ℎ| le 120575 119909 119909 + ℎ isin [119886 119887]

(38)

For119898 = 1 120596119898(119891 120575) is usual modulus of continuity

8 International Journal of Analysis

Theorem 10 Let 119891 isin 119862120583[0infin) for some 120583 gt 0 and 0 lt 119886 lt

1198861lt 1198871lt 119887 lt infin Then for 119899 sufficiently large one has

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(119891 sdot) minus 119891(119903)

1003817100381710038171003817100381710038171003817119862[11988611198871]

le 11987211205962(119891(119903)

119899minus12

[1198861 1198871]) + 119872

2119899minus1 1003817100381710038171003817119891

1003817100381710038171003817120583

(39)

where1198721= 1198721(119903) and119872

2= 1198722(119903 119891)

Proof Let us assume that 0 lt 119886 lt 1198861

lt 1198871

lt 119887 lt infinFor sufficiently small 120578 gt 0 we define the function 119891

1205782

corresponding to 119891 isin 119862120583[119886 119887] and 119905 isin [119886

1 1198871] as follows

1198911205782

(119905) = 120578minus2

∬

1205782

minus1205782

(119891 (119905) minus Δ2

ℎ119891 (119905)) 119889119905

11198891199052 (40)

where ℎ = (1199051+ 1199052)2 and Δ

2

ℎis the second order forward

difference operator with step length ℎ For 119891 isin 119862[119886 119887] thefunctions 119891

1205782are known as the Steklov mean of order 2

which satisfy the following properties [11]

(a) 1198911205782

has continuous derivatives up to order 2 over[1198861 1198871]

(b) 119891(119903)1205782

119862[11988611198871]le 1198721120578minus119903

1205962(119891 120578 [119886 119887]) 119903 = 1 2

(c) 119891 minus 1198911205782

119862[11988611198871]le 11987221205962(119891 120578 [119886 119887])

(d) 1198911205782

119862[11988611198871]le 1198723119891120583

where119872119894 119894 = 1 2 3 are certain constants which are different

in each occurrence and are independent of 119891 and 120578We can write by linearity properties of 119861120572120573

119899120574

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(119891 sdot) minus 119891(119903)

1003817100381710038171003817100381710038171003817119862[11988611198871]

le

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(119891 minus 1198911205782

sdot)

1003817100381710038171003817100381710038171003817119862[11988611198871]

+

1003817100381710038171003817100381710038171003817((119861120572120573

119899120574)(119903)

1198911205782

sdot) minus 119891(119903)

1205782

1003817100381710038171003817100381710038171003817119862[11988611198871]

+10038171003817100381710038171003817119891(119903)

minus 119891(119903)

1205782

10038171003817100381710038171003817119862[11988611198871]

= 1198751+ 1198752+ 1198753

(41)

Since 119891(119903)

1205782= (119891(119903)

)1205782

(119905) by property (c) of the function 1198911205782

we get

1198753le 11987241205962(119891(119903)

120578 [119886 119887]) (42)

Next on an application of Theorem 7 it follows that

1198752le 1198725119899minus1

119903+2

sum

119894=119903

10038171003817100381710038171003817119891(119894)

1205782

10038171003817100381710038171003817119862[119886119887] (43)

Using the interpolation property due to Goldberg and Meir[12] for each 119895 = 119903 119903 + 1 119903 + 2 it follows that

10038171003817100381710038171003817119891(119894)

1205782

10038171003817100381710038171003817119862[119886119887]le 1198726100381710038171003817100381710038171198911205782

10038171003817100381710038171003817119862[119886119887]+10038171003817100381710038171003817119891(119903+2)

1205782

10038171003817100381710038171003817119862[119886119887] (44)

Therefore by applying properties (c) and (d) of the function1198911205782 we obtain

1198752le 1198727sdot 119899minus1

1003817100381710038171003817119891

1003817100381710038171003817120583 + 120575minus2

1205962(119891(119903)

120583 [119886 119887]) (45)

Finally we will estimate 1198751 choosing 119886

lowast 119887lowast satisfying theconditions 0 lt 119886 lt 119886

lowast

lt 1198861lt 1198871lt 119887lowast

lt 119887 lt infin Suppose ℏ(119905)denotes the characteristic function of the interval [119886lowast 119887lowast]Then

1198751le

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(ℏ (119905) (119891 (119905) minus 1198911205782

(119905)) sdot)

1003817100381710038171003817100381710038171003817119862[11988611198871]

+

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782

(119905)) sdot)

1003817100381710038171003817100381710038171003817119862[11988611198871]

= 1198754+ 1198755

(46)

By Lemma 5 we have

(119861120572120573

119899120574)(119903)

(ℏ (119905) (119891 (119905) minus 1198911205782

(119905)) 119909)

=119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)

infin

sum

119896=0

119901119899+120574119903119896120574

(119909)

sdot int

infin

0

119887119899minus120574119903119896+119903120574

(119905) ℏ (119905)

sdot (119891(119903)

(119899119905 + 120572

119899 + 120573) minus 119891

(119903)

1205782(119899119905 + 120572

119899 + 120573))119889119905

(47)

Hence100381710038171003817100381710038171003817(119861120572120573

119899120574)119903

(ℏ (119905) (119891 (119905) minus 1198911205782

(119905)) sdot)100381710038171003817100381710038171003817119862[11988611198871]

le 1198728

10038171003817100381710038171003817119891(119903)

minus 119891(119903)

1205782

10038171003817100381710038171003817119862[119886lowast 119887lowast]

(48)

Now for 119909 isin [1198861 1198871] and 119905 isin [0infin) [119886

lowast

119887lowast

] we choose a120575 gt 0 satisfying |(119899119905 + 120572)(119899 + 120573) minus 119909| ge 120575

Therefore by Lemma 4 and the Cauchy-Schwarz inequal-ity we have

119868 equiv (119861120572120573

119899120574)(119903)

((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782

(119905)) 119909)

|119868| le sum

2119894+119895le119903

119894119895ge0

119899119894

10038161003816100381610038161003816119876119894119895119903120574

(119909)10038161003816100381610038161003816

119909 (1 + 120574119909)119903

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|119895

sdot int

infin

0

119887119899119896120574

(119905) (1 minus ℏ (119905))

sdot

10038161003816100381610038161003816100381610038161003816119891 (

119899119905 + 120572

119899 + 120573) minus 1198911205782

(119899119905 + 120572

119899 + 120573)

10038161003816100381610038161003816100381610038161003816119889119905

+Γ (119899120574 + 119903)

Γ (119899120574)(1 + 120574119909)

minus119899120574minus119903

(1 minus ℏ (0))

sdot

10038161003816100381610038161003816100381610038161003816119891 (

120572

119899 + 120573) minus 1198911205782

(120572

119899 + 120573)

10038161003816100381610038161003816100381610038161003816

(49)

International Journal of Analysis 9

For sufficiently large 119899 the second term tends to zero Thus

|119868| le 1198729

10038171003817100381710038171198911003817100381710038171003817120583 sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|119895

sdot int|119905minus119909|ge120575

119887119899119896120574

(119905) 119889119905 le 1198729

10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898

sum

2119894+119895le119903

119894119895ge0

119899119894

sdot

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|119895

(int

infin

0

119887119899119896120574

(119905) 119889119905)

12

sdot (int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

4119898

119889119905)

12

le 1198729

10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898

sum

2119894+119895le119903

119894119895ge0

119899119894

sdot (

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|2119895

)

12

(

infin

sum

119896=1

119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

4119898

119889119905)

12

(50)

Hence by using Remark 2 and Lemma 1 we have

|119868| le 11987210

10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898

119874(119899(119894+(1198952)minus119898)

) le 11987211119899minus119902 1003817100381710038171003817119891

1003817100381710038171003817120583 (51)

where 119902 = 119898 minus (1199032) Now choosing 119898 gt 0 satisfying 119902 ge 1we obtain 119868 le 119872

11119899minus1

119891120583 Therefore by property (c) of the

function 1198911205782

(119905) we get

1198751le 1198728

10038171003817100381710038171003817119891(119903)

minus 119891(119903)

1205782

10038171003817100381710038171003817119862[119886lowast 119887lowast]+ 11987211119899minus1 1003817100381710038171003817119891

1003817100381710038171003817120583

le 119872121205962(119891(119903)

120578 [119886 119887]) + 11987211119899minus1 1003817100381710038171003817119891

1003817100381710038171003817120583

(52)

Choosing 120578 = 119899minus12 the theorem follows

Remark 11 In the last decade the applications of 119902-calculus inapproximation theory are one of themain areas of research In2008 Gupta [13] introduced 119902-Durrmeyer operators whoseapproximation properties were studied in [14] More work inthis direction can be seen in [15ndash17]

A Durrmeyer type 119902-analogue of the 119861120572120573

119899120574(119891 119909) is intro-

duced as follows

119861120572120573

119899120574119902(119891 119909)

=

infin

sum

119896=1

119901119902

119899119896120574(119909) int

infin119860

0

119902minus119896

119887119902

119899119896120574(119905) 119891(

[119899]119902119905 + 120572

[119899]119902+ 120573

)119889119902119905

+ 119901119902

1198990120574(119909) 119891(

120572

[119899]119902+ 120573

)

(53)

where

119901119902

119899119896120574(119909) = 119902

11989622

Γ119902(119899120574 + 119896)

Γ119902(119896 + 1) Γ

119902(119899120574)

sdot(119902120574119909)

119896

(1 + 119902120574119909)(119899120574)+119896

119902

119887119902

119899119896120574(119909) = 120574119902

11989622

Γ119902(119899120574 + 119896 + 1)

Γ119902(119896) Γ119902(119899120574 + 1)

sdot(120574119905)119896minus1

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

119902

int

infin119860

0

119891 (119909) 119889119902119909 = (1 minus 119902)

infin

sum

119899=minusinfin

119891(119902119899

119860)

119902119899

119860 119860 gt 0

(54)

Notations used in (53) can be found in [18] For the operators(53) one can study their approximation properties based on119902-integers

Conflict of Interests

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

Acknowledgments

The authors would like to express their deep gratitude to theanonymous learned referee(s) and the editor for their valu-able suggestions and constructive comments which resultedin the subsequent improvement of this research article

References

[1] V Gupta D K Verma and P N Agrawal ldquoSimultaneousapproximation by certain Baskakov-Durrmeyer-Stancu opera-torsrdquo Journal of the Egyptian Mathematical Society vol 20 no3 pp 183ndash187 2012

[2] D K Verma VGupta and PN Agrawal ldquoSome approximationproperties of Baskakov-Durrmeyer-Stancu operatorsrdquo AppliedMathematics and Computation vol 218 no 11 pp 6549ndash65562012

[3] V N Mishra K Khatri L N Mishra and Deepmala ldquoInverseresult in simultaneous approximation by Baskakov-Durrmeyer-Stancu operatorsrdquo Journal of Inequalities and Applications vol2013 article 586 11 pages 2013

[4] V Gupta ldquoApproximation for modified Baskakov Durrmeyertype operatorsrdquoRockyMountain Journal ofMathematics vol 39no 3 pp 825ndash841 2009

[5] 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 no 4 article 125 2006

[6] V N Mishra and P Patel ldquoApproximation properties ofq-Baskakov-Durrmeyer-Stancu operatorsrdquo Mathematical Sci-ences vol 7 no 1 article 38 12 pages 2013

10 International Journal of Analysis

[7] P Patel and V N Mishra ldquoApproximation properties of certainsummation integral type operatorsrdquo Demonstratio Mathemat-ica vol 48 no 1 pp 77ndash90 2015

[8] 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

[9] 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

[10] W Li ldquoThe voronovskaja type expansion fomula of themodifiedBaskakov-Beta operatorsrdquo Journal of Baoji College of Arts andScience (Natural Science Edition) vol 2 article 004 2005

[11] G Freud andV Popov ldquoOn approximation by spline functionsrdquoin Proceedings of the Conference on Constructive Theory Func-tion Budapest Hungary 1969

[12] S Goldberg and A Meir ldquoMinimummoduli of ordinary differ-ential operatorsrdquo Proceedings of the London Mathematical Soci-ety vol 23 no 3 pp 1ndash15 1971

[13] V Gupta ldquoSome approxmation properties of q-durrmeyer ope-ratorsrdquo Applied Mathematics and Computation vol 197 no 1pp 172ndash178 2008

[14] A Aral and V Gupta ldquoOn the Durrmeyer type modificationof the q-Baskakov type operatorsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 3-4 pp 1171ndash1180 2010

[15] GM Phillips ldquoBernstein polynomials based on the q-integersrdquoAnnals of Numerical Mathematics vol 4 no 1ndash4 pp 511ndash5181997

[16] S Ostrovska ldquoOn the Lupas q-analogue of the Bernstein ope-ratorrdquoThe Rocky Mountain Journal of Mathematics vol 36 no5 pp 1615ndash1629 2006

[17] V N Mishra and P Patel ldquoOn generalized integral Bernsteinoperators based on q-integersrdquo Applied Mathematics and Com-putation vol 242 pp 931ndash944 2014

[18] V Kac and P Cheung Quantum Calculus UniversitextSpringer New York NY USA 2002

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Function Spaces

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Analysis 5

(119861120572120573

119899120574)(1)

(119891 119909) = 119899 (1 + 120574119909)minus119899120574minus1

sdot int

infin

0

(119899 + 120574) (1 + 120574119905)minus119899120574minus2

119891(119899119905 + 120572

119899 + 120573)119889119905

+ 119899

infin

sum

119896=1

119901119899+120574119896120574

(119909)

sdot int

infin

0

(minus1

119899119887(1)

119899minus120574119896+1120574(119905)) 119891(

119899119905 + 120572

119899 + 120573)119889119905

minus 119899 (1 + 120574119909)minus119899120574minus1

119891(120572

119899 + 120573)

(21)

Integrating by parts we get

(119861120572120573

119899120574)(1)

(119891 119909) = 119899 (1 + 120574119909)minus119899120574minus1

119891(120572

119899 + 120573)

+1198992

119899 + 120573(1 + 120574119909)

minus119899120574minus1

sdot int

infin

0

(1 + 120574119905)minus119899120574minus1

119891(1)

(119899119905 + 120572

119899 + 120573)119889119905 +

119899

119899 + 120573

sdot

infin

sum

119896=1

119901119899+120574119896120574

(119909) int

infin

0

119887119899minus120574119896+1120574

(119905) 119891(1)

(119899119905 + 120572

119899 + 120573)119889119905

minus 119899 (1 + 120574119909)minus119899120574minus1

119891(120572

119899 + 120573)

(119861120572120573

119899120574)(1)

(119891 119909) =119899

119899 + 120573

infin

sum

119896=0

119901119899+120574119896120574

(119909)

sdot int

infin

0

119887119899minus120574119896+1120574

(119905) 119891(1)

(119899119905 + 120572

119899 + 120573)119889119905

(22)

Thus the result is true for 119903 = 1 We prove the result byinduction method Suppose that the result is true for 119903 = 119894then

(119861120572120573

119899120574)(119894)

(119891 119909) =119899119894

Γ (119899120574 + 119894) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

sdot

infin

sum

119896=0

119901119899+120574119894119896120574

(119909) int

infin

0

119887119899minus120574119894119896+119894120574

(119905) 119891(119894)

(119899119905 + 120572

119899 + 120573)119889119905

(23)

Thus using the identities (20) we have

(119861120572120573

119899120574)(119894+1)

(119891 119909)

=119899119894

Γ (119899120574 + 119894) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

infin

sum

119896=1

(119899

120574+ 119894)

sdot 119901119899+120574(119894+1)119896minus1120574

(119909) minus 119901119899+120574(119894+1)119896120574

(119909) int

infin

0

119887119899minus120574119894119896+119894120574

(119905)

sdot 119891(119894)

(119899119905 + 120572

119899 + 120573)119889119905 minus (

119899

120574+ 119894) (1 + 120574119909)

minus119899120574minus119894minus1

sdot int

infin

0

119887119899minus120574119894119894120574

(119905) 119891(119894)

(119899119905 + 120572

119899 + 120573)

=119899119894

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

119901119899+120574(119894+1)0120574

(119909)

sdot int

infin

0

119887119899minus1205741198941+119894120574

(119905) 119891(119894)

(119899119905 + 120572

119899 + 120573)119889119905

minus119899119894

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

119901119899+120574(119894+1)0120574

(119909)

sdot int

infin

0

119887119899minus120574119894119894120574

(119905) 119891(119894)

(119899119905 + 120572

119899 + 120573)119889119905

+119899119894

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

infin

sum

119896=1

119901119899+120574(119894+1)119896120574

(119909)

sdot int

infin

0

119887119899minus120574119894119896+119894+1120574

(119905) minus 119887119899minus120574119894119896+119894120574

(119905)

sdot 119891(119894)

(119899119905 + 120572

119899 + 120573)119889119905

=119899119894

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

119901119899+120574(119894+1)0120574

(119909)

sdot int

infin

0

(minus1

119899120574 minus 119894119887(1)

119899minus120574(119894minus1)1+119894120574(119905))119891

(119894)

(119899119905 + 120572

119899 + 120573)119889119905

+119899119894

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894

Γ (119899120574 + 1) Γ (119899120574)

infin

sum

119896=1

119901119899+120574(119894+1)119896120574

(119909)

sdot int

infin

0

(minus1

119899120574 minus 119894119887(1)

119899minus120574(119894minus1)119896+119894+1120574(119905))119891

(119894)

(119899119905 + 120572

119899 + 120573)119889119905

(24)

Integrating by parts we obtain

(119861120572120573

119899120574)(119894+1)

(119891 119909) =119899119894+1

Γ (119899120574 + 119894 + 1) Γ (119899120574 minus 119894 + 1)

(119899 + 120573)119894+1

Γ (119899120574 + 1) Γ (119899120574)

sdot

infin

sum

119896=0

119901119899+120574(119894+1)119896120574

(119909)

sdot int

infin

0

119887119899minus120574(119894minus1)119896+119894+1120574

(119905) 119891(119894+1)

(119899119905 + 120572

119899 + 120573)119889119905

(25)

This completes the proof of Lemma 5

3 Direct Theorems

This section deals with the direct results we establish herepointwise approximation asymptotic formula and errorestimation in simultaneous approximation

6 International Journal of Analysis

We denote 119862120583[0infin) = 119891 isin 119862[0infin) |119891(119905)| le

119872119905120583 for some 119872 gt 0 120583 gt 0 and the norm sdot

120583on the

class 119862120583[0infin) is defined as 119891

120583= sup

0le119905ltinfin|119891(119905)|119905

minus120583

It canbe easily verified that the operators 119861120572120573

119899120574(119891 119909) are well defined

for 119891 isin 119862120583[0infin)

Theorem 6 Let 119891 isin 119862120583[0infin) and let 119891(119903) exist at a point

119909 isin (0infin) Then one has

lim119899rarrinfin

(119861120572120573

119899120574)(119903)

(119891 119909) = 119891(119903)

(119909) (26)

Proof By Taylorrsquos expansion of 119891 we have

119891 (119905) =

119903

sum

119894=0

119891(119894)

(119909)

119894(119905 minus 119909)

119894

+ 120598 (119905 119909) (119905 minus 119909)119903

(27)

where 120598(119905 119909) rarr 0 as 119905 rarr 119909 Hence

(119861120572120573

119899120574)(119903)

(119891 119909) =

119903

sum

119894=0

119891(119894)

(119909)

119894(119861120572120573

119899120574)(119903)

((119905 minus 119909)119894

119909)

+ (119861120572120573

119899120574)(119903)

(120598 (119905 119909) (119905 minus 119909)119903

119909)

= 1198771+ 1198772

(28)

First to estimate 1198771 using binomial expansion of ((119899119905 +

120572)(119899 + 120573) minus 119909)119894 and Remark 3 we have

1198771=

119903

sum

119894=0

119891(119894)

(119909)

119894

119894

sum

119895=0

(119894

119895) (minus119909)

119894minus119895

(119861120572120573

119899120574)(119903)

(119905119895

119909)

=119891(119903)

(119909)

119903119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)119903

= 119891(119903)

(119909) 119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)

997888rarr 119891(119903)

(119909) as 119899 997888rarr infin

(29)

Next applying Lemma 4 we obtain

1198772= int

infin

0

119882(119903)

119899120574(119905 119909) 120598 (119905 119909) (

119899119905 + 120572

119899 + 120573minus 119909)

119903

119889119905

100381610038161003816100381611987721003816100381610038161003816 le sum

2119894+119895le119903

119894119895ge0

119899119894

10038161003816100381610038161003816119876119894119895119903120574

(119909)10038161003816100381610038161003816

119909 (1 + 120574119909)119903

infin

sum

119896=1

|119896 minus 119899119909|119895

119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) |120598 (119905 119909)|

10038161003816100381610038161003816100381610038161003816

119899119905 + 120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

119903

119889119905

+Γ (119899120574 + 119903 + 2)

Γ (119899120574)(1 + 120574119909)

minus119899120574minus119903

|120598 (0 119909)|

sdot

10038161003816100381610038161003816100381610038161003816

120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

119903

(30)

The second term in the above expression tends to zero as 119899 rarr

infin Since 120598(119905 119909) rarr 0 as 119905 rarr 119909 for given 120576 gt 0 there existsa 120575 isin (0 1) such that |120598(119905 119909)| lt 120576 whenever 0 lt |119905 minus 119909| lt 120575If 120591 gt max120583 119903 where 120591 is any integer then we can find aconstant 119872

3gt 0 such that |120598(119905 119909)((119899119905 + 120572)(119899 + 120573) minus 119909)

119903

| le

1198723|(119899119905 + 120572)(119899 + 120573) minus 119909|

120591 for |119905 minus 119909| ge 120575 Therefore

100381610038161003816100381611987721003816100381610038161003816 le 119872

3sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=0

|119896 minus 119899119909|119895

119901119899119896120574

(119909)

sdot 120576 int|119905minus119909|lt120575

119887119899119896120574

(119909)

10038161003816100381610038161003816100381610038161003816

119899119905 + 120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

119903

119889119905

+ int|119905minus119909|ge120575

119887119899119896120574

(119905)

10038161003816100381610038161003816100381610038161003816

119899119905 + 120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

120591

119889119905 = 1198773+ 1198774

(31)

Applying the Cauchy-Schwarz inequality for integration andsummation respectively we obtain

100381610038161003816100381611987731003816100381610038161003816 le 120576119872

3sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=1

(119896 minus 119899119909)2119895

119901119899119896120574

(119909)

12

sdot

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

2119903

119889119905

12

(32)

Using Remark 2 and Lemma 1 we get 1198773le 120576119874(119899

1199032

)119874(119899minus1199032

)

= 120576 sdot 119874(1)

Again using the Cauchy-Schwarz inequality and Lemma1 we get

100381610038161003816100381611987741003816100381610038161003816 le 119872

4sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=1

|119896 minus 119899119909|119895

119901119899119896120574

(119909)

sdot int|119905minus119909|ge120575

119887119899119896120574

(119905)

10038161003816100381610038161003816100381610038161003816

119899119905 + 120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

120591

119889119905 le 1198724

sum

2119894+119895le119903

119894119895ge0

119899119894

sdot

infin

sum

119896=1

|119896 minus 119899119909|119895

119901119899119896120574

(119909) int|119905minus119909|ge120575

119887119899119896120574

(119905) 119889119905

12

sdot int|119905minus119909|ge120575

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

2120591

119889119905

12

le 1198724

sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=1

(119896 minus 119899119909)2119895

119901119899119896120574

(119909)

12

sdot

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

2120591

119889119905

12

= sum

2119894+119895le119903

119894119895ge0

119899119894

119874(1198991198952

)119874 (119899minus1205912

) = 119874 (119899(119903minus120591)2

) = 119900 (1)

(33)

Collecting the estimation of1198771ndash1198774 we get the required result

International Journal of Analysis 7

Theorem 7 Let 119891 isin 119862120583[0infin) If 119891(119903+2) exists at a point 119909 isin

(0infin) then

lim119899rarrinfin

119899 (119861120572120573

119899120574)(119903)

(119891 119909) minus 119891(119903)

(119909)

= 119903 (120574 (119903 minus 1) minus 120573) 119891(119903)

(119909)

+ 119903120574 (1 + 2119909) + 120572 minus 120573119909119891(119903+1)

(119909)

+ 119909 (1 + 120574119909) 119891(119903+2)

(119909)

(34)

Proof Using Taylorrsquos expansion of 119891 we have

119891 (119905) =

119903+2

sum

119894=0

119891(119894)

(119909)

119894(119905 minus 119909)

119894

+ 120598 (119905 119909) (119905 minus 119909)119903+2

(35)

where 120598(119905 119909) rarr 0 as 119905 rarr 119909 and 120598(119905 119909) = 119874((119905 minus 119909)120583

) 119905 rarr

infin for 120583 gt 0Applying Lemma 1 we have

119899 (119861120572120573

119899120574)(119903)

(119891 119909) minus 119891(119903)

(119909)

= 119899

119903+2

sum

119894=0

119891(119894)

(119909)

119894(119861120572120573

119899120574)(119903)

((119905 minus 119909)119894

119909) minus 119891(119903)

(119909)

+ 119899 (119861120572120573

119899120574)(119903)

(120598 (119905 119909) (119905 minus 119909)119903+2

119909)

= 1198641+ 1198642

(36)

First we have

1198641= 119899

119903+2

sum

119894=0

119891(119894)

(119909)

119894

119894

sum

119895=0

(119894

119895) (minus119909)

119894minus119895

(119861120572120573

119899120574)(119903)

(119905119895

119909)

minus 119899119891(119903)

(119909) =119891(119903)

(119909)

119903119899 (119861

120572120573

119899120574)(119903)

(119905119903

119909) minus 119903

+119891(119903+1)

(119909)

(119903 + 1)119899 (119903 + 1) (minus119909) (119861

120572120573

119899120574)(119903)

(119905119903

119909)

+ (119861120572120573

119899120574)(119903)

(119905119903+1

119909) +119891(119903+2)

(119909)

(119903 + 2)

sdot 119899 (119903 + 2) (119903 + 1)

21199092

(119861120572120573

119899120574)(119903)

(119905119903

119909) + (119903 + 2)

sdot (minus119909) (119861120572120573

119899120574)(119903)

(119905119903+1

119909) + (119861120572120573

119899120574)(119903)

(119905119903+2

119909)

= 119891(119903)

(119909) 119899 119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)minus 1

+119891(119903+1)

(119909)

(119903 + 1)119899(119903 + 1) (minus119909)

sdot119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)119903

+119899119903+1

Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903)

(119899 + 120573)119903+1

Γ (119899120574 + 1) Γ (119899120574)

(119903 + 1)119909

+(119903 + 1) 119899

119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903)

(119899 + 120573)119903+1

Γ (119899120574 + 1) Γ (119899120574)

119899119903 + 120572(119899

120574

minus 119903) 119903 +119891(119903+2)

(119909)

(119903 + 2)119899(

(119903 + 1) (119903 + 2)

21199092

sdot119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)119903 minus 119909 (119903 + 2)

sdot 119899119903+1

Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903)

(119899 + 120573)119903+1

Γ (119899120574 + 1) Γ (119899120574)

(119903 + 1)119909

+(119903 + 1) 119899

119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903)

(119899 + 120573)119903+1

Γ (119899120574 + 1) Γ (119899120574)

119899119903

+ 120572(119899

120574minus 119903) 119903

+119899119903+2

Γ (119899120574 + 119903 + 2) Γ (119899120574 minus 119903 minus 1)

(119899 + 120573)119903+2

Γ (119899120574 + 1) Γ (119899120574)

(119903 + 2)

21199092

+(119903 + 2) 119899

119903+1

Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903 minus 1)

(119899 + 120573)119903+2

Γ (119899120574 + 1) Γ (119899120574)

119899 (119903

+ 1) + 120572(119899

120574minus 119903 minus 1) (119903 + 1)119909

+120572 (119903 + 1) (119903 + 2) 119899

119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)119899119903

+120572 (119899120574 minus 119903)

2 119903)

(37)

Now the coefficients of119891(119903)(119909)119891(119903+1)(119909) and119891(119903+2)

(119909) in theabove expression tend to 119903(120574(119903 minus 1) minus 120573) 119903120574(1 + 2119909) + 120572 minus 120573119909and 119909(1 + 120574119909) respectively which follows by using inductionhypothesis on 119903 and taking the limit as 119899 rarr infin Hence inorder to prove (34) it is sufficient to show that 119864

2rarr 0

as 119899 rarr infin which follows along the lines of the proof ofTheorem 6 and by using Remark 2 and Lemmas 1 and 4

Remark 8 Particular case 120572 = 120573 = 0 was discussed inTheorem 41 in [4] which says that the coefficient of119891(119903+1)(119909)converges to 119903(1 + 2120574119909) but it converges to 119903120574(1 + 2119909) and weget this by putting 120572 = 120573 = 0 in the above theorem

Definition 9 The 119898th order modulus of continuity 120596119898(119891 120575

[119886 119887]) for a function continuous on [119886 119887] is defined by

120596119898(119891 120575 [119886 119887])

= sup 1003816100381610038161003816Δ119898

ℎ119891 (119909)

1003816100381610038161003816 |ℎ| le 120575 119909 119909 + ℎ isin [119886 119887]

(38)

For119898 = 1 120596119898(119891 120575) is usual modulus of continuity

8 International Journal of Analysis

Theorem 10 Let 119891 isin 119862120583[0infin) for some 120583 gt 0 and 0 lt 119886 lt

1198861lt 1198871lt 119887 lt infin Then for 119899 sufficiently large one has

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(119891 sdot) minus 119891(119903)

1003817100381710038171003817100381710038171003817119862[11988611198871]

le 11987211205962(119891(119903)

119899minus12

[1198861 1198871]) + 119872

2119899minus1 1003817100381710038171003817119891

1003817100381710038171003817120583

(39)

where1198721= 1198721(119903) and119872

2= 1198722(119903 119891)

Proof Let us assume that 0 lt 119886 lt 1198861

lt 1198871

lt 119887 lt infinFor sufficiently small 120578 gt 0 we define the function 119891

1205782

corresponding to 119891 isin 119862120583[119886 119887] and 119905 isin [119886

1 1198871] as follows

1198911205782

(119905) = 120578minus2

∬

1205782

minus1205782

(119891 (119905) minus Δ2

ℎ119891 (119905)) 119889119905

11198891199052 (40)

where ℎ = (1199051+ 1199052)2 and Δ

2

ℎis the second order forward

difference operator with step length ℎ For 119891 isin 119862[119886 119887] thefunctions 119891

1205782are known as the Steklov mean of order 2

which satisfy the following properties [11]

(a) 1198911205782

has continuous derivatives up to order 2 over[1198861 1198871]

(b) 119891(119903)1205782

119862[11988611198871]le 1198721120578minus119903

1205962(119891 120578 [119886 119887]) 119903 = 1 2

(c) 119891 minus 1198911205782

119862[11988611198871]le 11987221205962(119891 120578 [119886 119887])

(d) 1198911205782

119862[11988611198871]le 1198723119891120583

where119872119894 119894 = 1 2 3 are certain constants which are different

in each occurrence and are independent of 119891 and 120578We can write by linearity properties of 119861120572120573

119899120574

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(119891 sdot) minus 119891(119903)

1003817100381710038171003817100381710038171003817119862[11988611198871]

le

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(119891 minus 1198911205782

sdot)

1003817100381710038171003817100381710038171003817119862[11988611198871]

+

1003817100381710038171003817100381710038171003817((119861120572120573

119899120574)(119903)

1198911205782

sdot) minus 119891(119903)

1205782

1003817100381710038171003817100381710038171003817119862[11988611198871]

+10038171003817100381710038171003817119891(119903)

minus 119891(119903)

1205782

10038171003817100381710038171003817119862[11988611198871]

= 1198751+ 1198752+ 1198753

(41)

Since 119891(119903)

1205782= (119891(119903)

)1205782

(119905) by property (c) of the function 1198911205782

we get

1198753le 11987241205962(119891(119903)

120578 [119886 119887]) (42)

Next on an application of Theorem 7 it follows that

1198752le 1198725119899minus1

119903+2

sum

119894=119903

10038171003817100381710038171003817119891(119894)

1205782

10038171003817100381710038171003817119862[119886119887] (43)

Using the interpolation property due to Goldberg and Meir[12] for each 119895 = 119903 119903 + 1 119903 + 2 it follows that

10038171003817100381710038171003817119891(119894)

1205782

10038171003817100381710038171003817119862[119886119887]le 1198726100381710038171003817100381710038171198911205782

10038171003817100381710038171003817119862[119886119887]+10038171003817100381710038171003817119891(119903+2)

1205782

10038171003817100381710038171003817119862[119886119887] (44)

Therefore by applying properties (c) and (d) of the function1198911205782 we obtain

1198752le 1198727sdot 119899minus1

1003817100381710038171003817119891

1003817100381710038171003817120583 + 120575minus2

1205962(119891(119903)

120583 [119886 119887]) (45)

Finally we will estimate 1198751 choosing 119886

lowast 119887lowast satisfying theconditions 0 lt 119886 lt 119886

lowast

lt 1198861lt 1198871lt 119887lowast

lt 119887 lt infin Suppose ℏ(119905)denotes the characteristic function of the interval [119886lowast 119887lowast]Then

1198751le

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(ℏ (119905) (119891 (119905) minus 1198911205782

(119905)) sdot)

1003817100381710038171003817100381710038171003817119862[11988611198871]

+

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782

(119905)) sdot)

1003817100381710038171003817100381710038171003817119862[11988611198871]

= 1198754+ 1198755

(46)

By Lemma 5 we have

(119861120572120573

119899120574)(119903)

(ℏ (119905) (119891 (119905) minus 1198911205782

(119905)) 119909)

=119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)

infin

sum

119896=0

119901119899+120574119903119896120574

(119909)

sdot int

infin

0

119887119899minus120574119903119896+119903120574

(119905) ℏ (119905)

sdot (119891(119903)

(119899119905 + 120572

119899 + 120573) minus 119891

(119903)

1205782(119899119905 + 120572

119899 + 120573))119889119905

(47)

Hence100381710038171003817100381710038171003817(119861120572120573

119899120574)119903

(ℏ (119905) (119891 (119905) minus 1198911205782

(119905)) sdot)100381710038171003817100381710038171003817119862[11988611198871]

le 1198728

10038171003817100381710038171003817119891(119903)

minus 119891(119903)

1205782

10038171003817100381710038171003817119862[119886lowast 119887lowast]

(48)

Now for 119909 isin [1198861 1198871] and 119905 isin [0infin) [119886

lowast

119887lowast

] we choose a120575 gt 0 satisfying |(119899119905 + 120572)(119899 + 120573) minus 119909| ge 120575

Therefore by Lemma 4 and the Cauchy-Schwarz inequal-ity we have

119868 equiv (119861120572120573

119899120574)(119903)

((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782

(119905)) 119909)

|119868| le sum

2119894+119895le119903

119894119895ge0

119899119894

10038161003816100381610038161003816119876119894119895119903120574

(119909)10038161003816100381610038161003816

119909 (1 + 120574119909)119903

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|119895

sdot int

infin

0

119887119899119896120574

(119905) (1 minus ℏ (119905))

sdot

10038161003816100381610038161003816100381610038161003816119891 (

119899119905 + 120572

119899 + 120573) minus 1198911205782

(119899119905 + 120572

119899 + 120573)

10038161003816100381610038161003816100381610038161003816119889119905

+Γ (119899120574 + 119903)

Γ (119899120574)(1 + 120574119909)

minus119899120574minus119903

(1 minus ℏ (0))

sdot

10038161003816100381610038161003816100381610038161003816119891 (

120572

119899 + 120573) minus 1198911205782

(120572

119899 + 120573)

10038161003816100381610038161003816100381610038161003816

(49)

International Journal of Analysis 9

For sufficiently large 119899 the second term tends to zero Thus

|119868| le 1198729

10038171003817100381710038171198911003817100381710038171003817120583 sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|119895

sdot int|119905minus119909|ge120575

119887119899119896120574

(119905) 119889119905 le 1198729

10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898

sum

2119894+119895le119903

119894119895ge0

119899119894

sdot

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|119895

(int

infin

0

119887119899119896120574

(119905) 119889119905)

12

sdot (int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

4119898

119889119905)

12

le 1198729

10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898

sum

2119894+119895le119903

119894119895ge0

119899119894

sdot (

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|2119895

)

12

(

infin

sum

119896=1

119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

4119898

119889119905)

12

(50)

Hence by using Remark 2 and Lemma 1 we have

|119868| le 11987210

10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898

119874(119899(119894+(1198952)minus119898)

) le 11987211119899minus119902 1003817100381710038171003817119891

1003817100381710038171003817120583 (51)

where 119902 = 119898 minus (1199032) Now choosing 119898 gt 0 satisfying 119902 ge 1we obtain 119868 le 119872

11119899minus1

119891120583 Therefore by property (c) of the

function 1198911205782

(119905) we get

1198751le 1198728

10038171003817100381710038171003817119891(119903)

minus 119891(119903)

1205782

10038171003817100381710038171003817119862[119886lowast 119887lowast]+ 11987211119899minus1 1003817100381710038171003817119891

1003817100381710038171003817120583

le 119872121205962(119891(119903)

120578 [119886 119887]) + 11987211119899minus1 1003817100381710038171003817119891

1003817100381710038171003817120583

(52)

Choosing 120578 = 119899minus12 the theorem follows

Remark 11 In the last decade the applications of 119902-calculus inapproximation theory are one of themain areas of research In2008 Gupta [13] introduced 119902-Durrmeyer operators whoseapproximation properties were studied in [14] More work inthis direction can be seen in [15ndash17]

A Durrmeyer type 119902-analogue of the 119861120572120573

119899120574(119891 119909) is intro-

duced as follows

119861120572120573

119899120574119902(119891 119909)

=

infin

sum

119896=1

119901119902

119899119896120574(119909) int

infin119860

0

119902minus119896

119887119902

119899119896120574(119905) 119891(

[119899]119902119905 + 120572

[119899]119902+ 120573

)119889119902119905

+ 119901119902

1198990120574(119909) 119891(

120572

[119899]119902+ 120573

)

(53)

where

119901119902

119899119896120574(119909) = 119902

11989622

Γ119902(119899120574 + 119896)

Γ119902(119896 + 1) Γ

119902(119899120574)

sdot(119902120574119909)

119896

(1 + 119902120574119909)(119899120574)+119896

119902

119887119902

119899119896120574(119909) = 120574119902

11989622

Γ119902(119899120574 + 119896 + 1)

Γ119902(119896) Γ119902(119899120574 + 1)

sdot(120574119905)119896minus1

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

119902

int

infin119860

0

119891 (119909) 119889119902119909 = (1 minus 119902)

infin

sum

119899=minusinfin

119891(119902119899

119860)

119902119899

119860 119860 gt 0

(54)

Notations used in (53) can be found in [18] For the operators(53) one can study their approximation properties based on119902-integers

Conflict of Interests

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

Acknowledgments

The authors would like to express their deep gratitude to theanonymous learned referee(s) and the editor for their valu-able suggestions and constructive comments which resultedin the subsequent improvement of this research article

References

[1] V Gupta D K Verma and P N Agrawal ldquoSimultaneousapproximation by certain Baskakov-Durrmeyer-Stancu opera-torsrdquo Journal of the Egyptian Mathematical Society vol 20 no3 pp 183ndash187 2012

[2] D K Verma VGupta and PN Agrawal ldquoSome approximationproperties of Baskakov-Durrmeyer-Stancu operatorsrdquo AppliedMathematics and Computation vol 218 no 11 pp 6549ndash65562012

[3] V N Mishra K Khatri L N Mishra and Deepmala ldquoInverseresult in simultaneous approximation by Baskakov-Durrmeyer-Stancu operatorsrdquo Journal of Inequalities and Applications vol2013 article 586 11 pages 2013

[4] V Gupta ldquoApproximation for modified Baskakov Durrmeyertype operatorsrdquoRockyMountain Journal ofMathematics vol 39no 3 pp 825ndash841 2009

[5] 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 no 4 article 125 2006

[6] V N Mishra and P Patel ldquoApproximation properties ofq-Baskakov-Durrmeyer-Stancu operatorsrdquo Mathematical Sci-ences vol 7 no 1 article 38 12 pages 2013

10 International Journal of Analysis

[7] P Patel and V N Mishra ldquoApproximation properties of certainsummation integral type operatorsrdquo Demonstratio Mathemat-ica vol 48 no 1 pp 77ndash90 2015

[8] 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

[9] 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

[10] W Li ldquoThe voronovskaja type expansion fomula of themodifiedBaskakov-Beta operatorsrdquo Journal of Baoji College of Arts andScience (Natural Science Edition) vol 2 article 004 2005

[11] G Freud andV Popov ldquoOn approximation by spline functionsrdquoin Proceedings of the Conference on Constructive Theory Func-tion Budapest Hungary 1969

[12] S Goldberg and A Meir ldquoMinimummoduli of ordinary differ-ential operatorsrdquo Proceedings of the London Mathematical Soci-ety vol 23 no 3 pp 1ndash15 1971

[13] V Gupta ldquoSome approxmation properties of q-durrmeyer ope-ratorsrdquo Applied Mathematics and Computation vol 197 no 1pp 172ndash178 2008

[14] A Aral and V Gupta ldquoOn the Durrmeyer type modificationof the q-Baskakov type operatorsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 3-4 pp 1171ndash1180 2010

[15] GM Phillips ldquoBernstein polynomials based on the q-integersrdquoAnnals of Numerical Mathematics vol 4 no 1ndash4 pp 511ndash5181997

[16] S Ostrovska ldquoOn the Lupas q-analogue of the Bernstein ope-ratorrdquoThe Rocky Mountain Journal of Mathematics vol 36 no5 pp 1615ndash1629 2006

[17] V N Mishra and P Patel ldquoOn generalized integral Bernsteinoperators based on q-integersrdquo Applied Mathematics and Com-putation vol 242 pp 931ndash944 2014

[18] V Kac and P Cheung Quantum Calculus UniversitextSpringer New York NY USA 2002

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 International Journal of Analysis

We denote 119862120583[0infin) = 119891 isin 119862[0infin) |119891(119905)| le

119872119905120583 for some 119872 gt 0 120583 gt 0 and the norm sdot

120583on the

class 119862120583[0infin) is defined as 119891

120583= sup

0le119905ltinfin|119891(119905)|119905

minus120583

It canbe easily verified that the operators 119861120572120573

119899120574(119891 119909) are well defined

for 119891 isin 119862120583[0infin)

Theorem 6 Let 119891 isin 119862120583[0infin) and let 119891(119903) exist at a point

119909 isin (0infin) Then one has

lim119899rarrinfin

(119861120572120573

119899120574)(119903)

(119891 119909) = 119891(119903)

(119909) (26)

Proof By Taylorrsquos expansion of 119891 we have

119891 (119905) =

119903

sum

119894=0

119891(119894)

(119909)

119894(119905 minus 119909)

119894

+ 120598 (119905 119909) (119905 minus 119909)119903

(27)

where 120598(119905 119909) rarr 0 as 119905 rarr 119909 Hence

(119861120572120573

119899120574)(119903)

(119891 119909) =

119903

sum

119894=0

119891(119894)

(119909)

119894(119861120572120573

119899120574)(119903)

((119905 minus 119909)119894

119909)

+ (119861120572120573

119899120574)(119903)

(120598 (119905 119909) (119905 minus 119909)119903

119909)

= 1198771+ 1198772

(28)

First to estimate 1198771 using binomial expansion of ((119899119905 +

120572)(119899 + 120573) minus 119909)119894 and Remark 3 we have

1198771=

119903

sum

119894=0

119891(119894)

(119909)

119894

119894

sum

119895=0

(119894

119895) (minus119909)

119894minus119895

(119861120572120573

119899120574)(119903)

(119905119895

119909)

=119891(119903)

(119909)

119903119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)119903

= 119891(119903)

(119909) 119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)

997888rarr 119891(119903)

(119909) as 119899 997888rarr infin

(29)

Next applying Lemma 4 we obtain

1198772= int

infin

0

119882(119903)

119899120574(119905 119909) 120598 (119905 119909) (

119899119905 + 120572

119899 + 120573minus 119909)

119903

119889119905

100381610038161003816100381611987721003816100381610038161003816 le sum

2119894+119895le119903

119894119895ge0

119899119894

10038161003816100381610038161003816119876119894119895119903120574

(119909)10038161003816100381610038161003816

119909 (1 + 120574119909)119903

infin

sum

119896=1

|119896 minus 119899119909|119895

119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) |120598 (119905 119909)|

10038161003816100381610038161003816100381610038161003816

119899119905 + 120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

119903

119889119905

+Γ (119899120574 + 119903 + 2)

Γ (119899120574)(1 + 120574119909)

minus119899120574minus119903

|120598 (0 119909)|

sdot

10038161003816100381610038161003816100381610038161003816

120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

119903

(30)

The second term in the above expression tends to zero as 119899 rarr

infin Since 120598(119905 119909) rarr 0 as 119905 rarr 119909 for given 120576 gt 0 there existsa 120575 isin (0 1) such that |120598(119905 119909)| lt 120576 whenever 0 lt |119905 minus 119909| lt 120575If 120591 gt max120583 119903 where 120591 is any integer then we can find aconstant 119872

3gt 0 such that |120598(119905 119909)((119899119905 + 120572)(119899 + 120573) minus 119909)

119903

| le

1198723|(119899119905 + 120572)(119899 + 120573) minus 119909|

120591 for |119905 minus 119909| ge 120575 Therefore

100381610038161003816100381611987721003816100381610038161003816 le 119872

3sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=0

|119896 minus 119899119909|119895

119901119899119896120574

(119909)

sdot 120576 int|119905minus119909|lt120575

119887119899119896120574

(119909)

10038161003816100381610038161003816100381610038161003816

119899119905 + 120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

119903

119889119905

+ int|119905minus119909|ge120575

119887119899119896120574

(119905)

10038161003816100381610038161003816100381610038161003816

119899119905 + 120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

120591

119889119905 = 1198773+ 1198774

(31)

Applying the Cauchy-Schwarz inequality for integration andsummation respectively we obtain

100381610038161003816100381611987731003816100381610038161003816 le 120576119872

3sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=1

(119896 minus 119899119909)2119895

119901119899119896120574

(119909)

12

sdot

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

2119903

119889119905

12

(32)

Using Remark 2 and Lemma 1 we get 1198773le 120576119874(119899

1199032

)119874(119899minus1199032

)

= 120576 sdot 119874(1)

Again using the Cauchy-Schwarz inequality and Lemma1 we get

100381610038161003816100381611987741003816100381610038161003816 le 119872

4sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=1

|119896 minus 119899119909|119895

119901119899119896120574

(119909)

sdot int|119905minus119909|ge120575

119887119899119896120574

(119905)

10038161003816100381610038161003816100381610038161003816

119899119905 + 120572

119899 + 120573minus 119909

10038161003816100381610038161003816100381610038161003816

120591

119889119905 le 1198724

sum

2119894+119895le119903

119894119895ge0

119899119894

sdot

infin

sum

119896=1

|119896 minus 119899119909|119895

119901119899119896120574

(119909) int|119905minus119909|ge120575

119887119899119896120574

(119905) 119889119905

12

sdot int|119905minus119909|ge120575

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

2120591

119889119905

12

le 1198724

sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=1

(119896 minus 119899119909)2119895

119901119899119896120574

(119909)

12

sdot

infin

sum

119896=1

119901119899119896120574

(119909) int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

2120591

119889119905

12

= sum

2119894+119895le119903

119894119895ge0

119899119894

119874(1198991198952

)119874 (119899minus1205912

) = 119874 (119899(119903minus120591)2

) = 119900 (1)

(33)

Collecting the estimation of1198771ndash1198774 we get the required result

International Journal of Analysis 7

Theorem 7 Let 119891 isin 119862120583[0infin) If 119891(119903+2) exists at a point 119909 isin

(0infin) then

lim119899rarrinfin

119899 (119861120572120573

119899120574)(119903)

(119891 119909) minus 119891(119903)

(119909)

= 119903 (120574 (119903 minus 1) minus 120573) 119891(119903)

(119909)

+ 119903120574 (1 + 2119909) + 120572 minus 120573119909119891(119903+1)

(119909)

+ 119909 (1 + 120574119909) 119891(119903+2)

(119909)

(34)

Proof Using Taylorrsquos expansion of 119891 we have

119891 (119905) =

119903+2

sum

119894=0

119891(119894)

(119909)

119894(119905 minus 119909)

119894

+ 120598 (119905 119909) (119905 minus 119909)119903+2

(35)

where 120598(119905 119909) rarr 0 as 119905 rarr 119909 and 120598(119905 119909) = 119874((119905 minus 119909)120583

) 119905 rarr

infin for 120583 gt 0Applying Lemma 1 we have

119899 (119861120572120573

119899120574)(119903)

(119891 119909) minus 119891(119903)

(119909)

= 119899

119903+2

sum

119894=0

119891(119894)

(119909)

119894(119861120572120573

119899120574)(119903)

((119905 minus 119909)119894

119909) minus 119891(119903)

(119909)

+ 119899 (119861120572120573

119899120574)(119903)

(120598 (119905 119909) (119905 minus 119909)119903+2

119909)

= 1198641+ 1198642

(36)

First we have

1198641= 119899

119903+2

sum

119894=0

119891(119894)

(119909)

119894

119894

sum

119895=0

(119894

119895) (minus119909)

119894minus119895

(119861120572120573

119899120574)(119903)

(119905119895

119909)

minus 119899119891(119903)

(119909) =119891(119903)

(119909)

119903119899 (119861

120572120573

119899120574)(119903)

(119905119903

119909) minus 119903

+119891(119903+1)

(119909)

(119903 + 1)119899 (119903 + 1) (minus119909) (119861

120572120573

119899120574)(119903)

(119905119903

119909)

+ (119861120572120573

119899120574)(119903)

(119905119903+1

119909) +119891(119903+2)

(119909)

(119903 + 2)

sdot 119899 (119903 + 2) (119903 + 1)

21199092

(119861120572120573

119899120574)(119903)

(119905119903

119909) + (119903 + 2)

sdot (minus119909) (119861120572120573

119899120574)(119903)

(119905119903+1

119909) + (119861120572120573

119899120574)(119903)

(119905119903+2

119909)

= 119891(119903)

(119909) 119899 119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)minus 1

+119891(119903+1)

(119909)

(119903 + 1)119899(119903 + 1) (minus119909)

sdot119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)119903

+119899119903+1

Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903)

(119899 + 120573)119903+1

Γ (119899120574 + 1) Γ (119899120574)

(119903 + 1)119909

+(119903 + 1) 119899

119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903)

(119899 + 120573)119903+1

Γ (119899120574 + 1) Γ (119899120574)

119899119903 + 120572(119899

120574

minus 119903) 119903 +119891(119903+2)

(119909)

(119903 + 2)119899(

(119903 + 1) (119903 + 2)

21199092

sdot119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)119903 minus 119909 (119903 + 2)

sdot 119899119903+1

Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903)

(119899 + 120573)119903+1

Γ (119899120574 + 1) Γ (119899120574)

(119903 + 1)119909

+(119903 + 1) 119899

119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903)

(119899 + 120573)119903+1

Γ (119899120574 + 1) Γ (119899120574)

119899119903

+ 120572(119899

120574minus 119903) 119903

+119899119903+2

Γ (119899120574 + 119903 + 2) Γ (119899120574 minus 119903 minus 1)

(119899 + 120573)119903+2

Γ (119899120574 + 1) Γ (119899120574)

(119903 + 2)

21199092

+(119903 + 2) 119899

119903+1

Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903 minus 1)

(119899 + 120573)119903+2

Γ (119899120574 + 1) Γ (119899120574)

119899 (119903

+ 1) + 120572(119899

120574minus 119903 minus 1) (119903 + 1)119909

+120572 (119903 + 1) (119903 + 2) 119899

119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)119899119903

+120572 (119899120574 minus 119903)

2 119903)

(37)

Now the coefficients of119891(119903)(119909)119891(119903+1)(119909) and119891(119903+2)

(119909) in theabove expression tend to 119903(120574(119903 minus 1) minus 120573) 119903120574(1 + 2119909) + 120572 minus 120573119909and 119909(1 + 120574119909) respectively which follows by using inductionhypothesis on 119903 and taking the limit as 119899 rarr infin Hence inorder to prove (34) it is sufficient to show that 119864

2rarr 0

as 119899 rarr infin which follows along the lines of the proof ofTheorem 6 and by using Remark 2 and Lemmas 1 and 4

Remark 8 Particular case 120572 = 120573 = 0 was discussed inTheorem 41 in [4] which says that the coefficient of119891(119903+1)(119909)converges to 119903(1 + 2120574119909) but it converges to 119903120574(1 + 2119909) and weget this by putting 120572 = 120573 = 0 in the above theorem

Definition 9 The 119898th order modulus of continuity 120596119898(119891 120575

[119886 119887]) for a function continuous on [119886 119887] is defined by

120596119898(119891 120575 [119886 119887])

= sup 1003816100381610038161003816Δ119898

ℎ119891 (119909)

1003816100381610038161003816 |ℎ| le 120575 119909 119909 + ℎ isin [119886 119887]

(38)

For119898 = 1 120596119898(119891 120575) is usual modulus of continuity

8 International Journal of Analysis

Theorem 10 Let 119891 isin 119862120583[0infin) for some 120583 gt 0 and 0 lt 119886 lt

1198861lt 1198871lt 119887 lt infin Then for 119899 sufficiently large one has

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(119891 sdot) minus 119891(119903)

1003817100381710038171003817100381710038171003817119862[11988611198871]

le 11987211205962(119891(119903)

119899minus12

[1198861 1198871]) + 119872

2119899minus1 1003817100381710038171003817119891

1003817100381710038171003817120583

(39)

where1198721= 1198721(119903) and119872

2= 1198722(119903 119891)

Proof Let us assume that 0 lt 119886 lt 1198861

lt 1198871

lt 119887 lt infinFor sufficiently small 120578 gt 0 we define the function 119891

1205782

corresponding to 119891 isin 119862120583[119886 119887] and 119905 isin [119886

1 1198871] as follows

1198911205782

(119905) = 120578minus2

∬

1205782

minus1205782

(119891 (119905) minus Δ2

ℎ119891 (119905)) 119889119905

11198891199052 (40)

where ℎ = (1199051+ 1199052)2 and Δ

2

ℎis the second order forward

difference operator with step length ℎ For 119891 isin 119862[119886 119887] thefunctions 119891

1205782are known as the Steklov mean of order 2

which satisfy the following properties [11]

(a) 1198911205782

has continuous derivatives up to order 2 over[1198861 1198871]

(b) 119891(119903)1205782

119862[11988611198871]le 1198721120578minus119903

1205962(119891 120578 [119886 119887]) 119903 = 1 2

(c) 119891 minus 1198911205782

119862[11988611198871]le 11987221205962(119891 120578 [119886 119887])

(d) 1198911205782

119862[11988611198871]le 1198723119891120583

where119872119894 119894 = 1 2 3 are certain constants which are different

in each occurrence and are independent of 119891 and 120578We can write by linearity properties of 119861120572120573

119899120574

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(119891 sdot) minus 119891(119903)

1003817100381710038171003817100381710038171003817119862[11988611198871]

le

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(119891 minus 1198911205782

sdot)

1003817100381710038171003817100381710038171003817119862[11988611198871]

+

1003817100381710038171003817100381710038171003817((119861120572120573

119899120574)(119903)

1198911205782

sdot) minus 119891(119903)

1205782

1003817100381710038171003817100381710038171003817119862[11988611198871]

+10038171003817100381710038171003817119891(119903)

minus 119891(119903)

1205782

10038171003817100381710038171003817119862[11988611198871]

= 1198751+ 1198752+ 1198753

(41)

Since 119891(119903)

1205782= (119891(119903)

)1205782

(119905) by property (c) of the function 1198911205782

we get

1198753le 11987241205962(119891(119903)

120578 [119886 119887]) (42)

Next on an application of Theorem 7 it follows that

1198752le 1198725119899minus1

119903+2

sum

119894=119903

10038171003817100381710038171003817119891(119894)

1205782

10038171003817100381710038171003817119862[119886119887] (43)

Using the interpolation property due to Goldberg and Meir[12] for each 119895 = 119903 119903 + 1 119903 + 2 it follows that

10038171003817100381710038171003817119891(119894)

1205782

10038171003817100381710038171003817119862[119886119887]le 1198726100381710038171003817100381710038171198911205782

10038171003817100381710038171003817119862[119886119887]+10038171003817100381710038171003817119891(119903+2)

1205782

10038171003817100381710038171003817119862[119886119887] (44)

Therefore by applying properties (c) and (d) of the function1198911205782 we obtain

1198752le 1198727sdot 119899minus1

1003817100381710038171003817119891

1003817100381710038171003817120583 + 120575minus2

1205962(119891(119903)

120583 [119886 119887]) (45)

Finally we will estimate 1198751 choosing 119886

lowast 119887lowast satisfying theconditions 0 lt 119886 lt 119886

lowast

lt 1198861lt 1198871lt 119887lowast

lt 119887 lt infin Suppose ℏ(119905)denotes the characteristic function of the interval [119886lowast 119887lowast]Then

1198751le

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(ℏ (119905) (119891 (119905) minus 1198911205782

(119905)) sdot)

1003817100381710038171003817100381710038171003817119862[11988611198871]

+

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782

(119905)) sdot)

1003817100381710038171003817100381710038171003817119862[11988611198871]

= 1198754+ 1198755

(46)

By Lemma 5 we have

(119861120572120573

119899120574)(119903)

(ℏ (119905) (119891 (119905) minus 1198911205782

(119905)) 119909)

=119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)

infin

sum

119896=0

119901119899+120574119903119896120574

(119909)

sdot int

infin

0

119887119899minus120574119903119896+119903120574

(119905) ℏ (119905)

sdot (119891(119903)

(119899119905 + 120572

119899 + 120573) minus 119891

(119903)

1205782(119899119905 + 120572

119899 + 120573))119889119905

(47)

Hence100381710038171003817100381710038171003817(119861120572120573

119899120574)119903

(ℏ (119905) (119891 (119905) minus 1198911205782

(119905)) sdot)100381710038171003817100381710038171003817119862[11988611198871]

le 1198728

10038171003817100381710038171003817119891(119903)

minus 119891(119903)

1205782

10038171003817100381710038171003817119862[119886lowast 119887lowast]

(48)

Now for 119909 isin [1198861 1198871] and 119905 isin [0infin) [119886

lowast

119887lowast

] we choose a120575 gt 0 satisfying |(119899119905 + 120572)(119899 + 120573) minus 119909| ge 120575

Therefore by Lemma 4 and the Cauchy-Schwarz inequal-ity we have

119868 equiv (119861120572120573

119899120574)(119903)

((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782

(119905)) 119909)

|119868| le sum

2119894+119895le119903

119894119895ge0

119899119894

10038161003816100381610038161003816119876119894119895119903120574

(119909)10038161003816100381610038161003816

119909 (1 + 120574119909)119903

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|119895

sdot int

infin

0

119887119899119896120574

(119905) (1 minus ℏ (119905))

sdot

10038161003816100381610038161003816100381610038161003816119891 (

119899119905 + 120572

119899 + 120573) minus 1198911205782

(119899119905 + 120572

119899 + 120573)

10038161003816100381610038161003816100381610038161003816119889119905

+Γ (119899120574 + 119903)

Γ (119899120574)(1 + 120574119909)

minus119899120574minus119903

(1 minus ℏ (0))

sdot

10038161003816100381610038161003816100381610038161003816119891 (

120572

119899 + 120573) minus 1198911205782

(120572

119899 + 120573)

10038161003816100381610038161003816100381610038161003816

(49)

International Journal of Analysis 9

For sufficiently large 119899 the second term tends to zero Thus

|119868| le 1198729

10038171003817100381710038171198911003817100381710038171003817120583 sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|119895

sdot int|119905minus119909|ge120575

119887119899119896120574

(119905) 119889119905 le 1198729

10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898

sum

2119894+119895le119903

119894119895ge0

119899119894

sdot

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|119895

(int

infin

0

119887119899119896120574

(119905) 119889119905)

12

sdot (int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

4119898

119889119905)

12

le 1198729

10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898

sum

2119894+119895le119903

119894119895ge0

119899119894

sdot (

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|2119895

)

12

(

infin

sum

119896=1

119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

4119898

119889119905)

12

(50)

Hence by using Remark 2 and Lemma 1 we have

|119868| le 11987210

10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898

119874(119899(119894+(1198952)minus119898)

) le 11987211119899minus119902 1003817100381710038171003817119891

1003817100381710038171003817120583 (51)

where 119902 = 119898 minus (1199032) Now choosing 119898 gt 0 satisfying 119902 ge 1we obtain 119868 le 119872

11119899minus1

119891120583 Therefore by property (c) of the

function 1198911205782

(119905) we get

1198751le 1198728

10038171003817100381710038171003817119891(119903)

minus 119891(119903)

1205782

10038171003817100381710038171003817119862[119886lowast 119887lowast]+ 11987211119899minus1 1003817100381710038171003817119891

1003817100381710038171003817120583

le 119872121205962(119891(119903)

120578 [119886 119887]) + 11987211119899minus1 1003817100381710038171003817119891

1003817100381710038171003817120583

(52)

Choosing 120578 = 119899minus12 the theorem follows

Remark 11 In the last decade the applications of 119902-calculus inapproximation theory are one of themain areas of research In2008 Gupta [13] introduced 119902-Durrmeyer operators whoseapproximation properties were studied in [14] More work inthis direction can be seen in [15ndash17]

A Durrmeyer type 119902-analogue of the 119861120572120573

119899120574(119891 119909) is intro-

duced as follows

119861120572120573

119899120574119902(119891 119909)

=

infin

sum

119896=1

119901119902

119899119896120574(119909) int

infin119860

0

119902minus119896

119887119902

119899119896120574(119905) 119891(

[119899]119902119905 + 120572

[119899]119902+ 120573

)119889119902119905

+ 119901119902

1198990120574(119909) 119891(

120572

[119899]119902+ 120573

)

(53)

where

119901119902

119899119896120574(119909) = 119902

11989622

Γ119902(119899120574 + 119896)

Γ119902(119896 + 1) Γ

119902(119899120574)

sdot(119902120574119909)

119896

(1 + 119902120574119909)(119899120574)+119896

119902

119887119902

119899119896120574(119909) = 120574119902

11989622

Γ119902(119899120574 + 119896 + 1)

Γ119902(119896) Γ119902(119899120574 + 1)

sdot(120574119905)119896minus1

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

119902

int

infin119860

0

119891 (119909) 119889119902119909 = (1 minus 119902)

infin

sum

119899=minusinfin

119891(119902119899

119860)

119902119899

119860 119860 gt 0

(54)

Notations used in (53) can be found in [18] For the operators(53) one can study their approximation properties based on119902-integers

Conflict of Interests

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

Acknowledgments

The authors would like to express their deep gratitude to theanonymous learned referee(s) and the editor for their valu-able suggestions and constructive comments which resultedin the subsequent improvement of this research article

References

[1] V Gupta D K Verma and P N Agrawal ldquoSimultaneousapproximation by certain Baskakov-Durrmeyer-Stancu opera-torsrdquo Journal of the Egyptian Mathematical Society vol 20 no3 pp 183ndash187 2012

[2] D K Verma VGupta and PN Agrawal ldquoSome approximationproperties of Baskakov-Durrmeyer-Stancu operatorsrdquo AppliedMathematics and Computation vol 218 no 11 pp 6549ndash65562012

[3] V N Mishra K Khatri L N Mishra and Deepmala ldquoInverseresult in simultaneous approximation by Baskakov-Durrmeyer-Stancu operatorsrdquo Journal of Inequalities and Applications vol2013 article 586 11 pages 2013

[4] V Gupta ldquoApproximation for modified Baskakov Durrmeyertype operatorsrdquoRockyMountain Journal ofMathematics vol 39no 3 pp 825ndash841 2009

[5] 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 no 4 article 125 2006

[6] V N Mishra and P Patel ldquoApproximation properties ofq-Baskakov-Durrmeyer-Stancu operatorsrdquo Mathematical Sci-ences vol 7 no 1 article 38 12 pages 2013

10 International Journal of Analysis

[7] P Patel and V N Mishra ldquoApproximation properties of certainsummation integral type operatorsrdquo Demonstratio Mathemat-ica vol 48 no 1 pp 77ndash90 2015

[8] 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

[9] 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

[10] W Li ldquoThe voronovskaja type expansion fomula of themodifiedBaskakov-Beta operatorsrdquo Journal of Baoji College of Arts andScience (Natural Science Edition) vol 2 article 004 2005

[11] G Freud andV Popov ldquoOn approximation by spline functionsrdquoin Proceedings of the Conference on Constructive Theory Func-tion Budapest Hungary 1969

[12] S Goldberg and A Meir ldquoMinimummoduli of ordinary differ-ential operatorsrdquo Proceedings of the London Mathematical Soci-ety vol 23 no 3 pp 1ndash15 1971

[13] V Gupta ldquoSome approxmation properties of q-durrmeyer ope-ratorsrdquo Applied Mathematics and Computation vol 197 no 1pp 172ndash178 2008

[14] A Aral and V Gupta ldquoOn the Durrmeyer type modificationof the q-Baskakov type operatorsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 3-4 pp 1171ndash1180 2010

[15] GM Phillips ldquoBernstein polynomials based on the q-integersrdquoAnnals of Numerical Mathematics vol 4 no 1ndash4 pp 511ndash5181997

[16] S Ostrovska ldquoOn the Lupas q-analogue of the Bernstein ope-ratorrdquoThe Rocky Mountain Journal of Mathematics vol 36 no5 pp 1615ndash1629 2006

[17] V N Mishra and P Patel ldquoOn generalized integral Bernsteinoperators based on q-integersrdquo Applied Mathematics and Com-putation vol 242 pp 931ndash944 2014

[18] V Kac and P Cheung Quantum Calculus UniversitextSpringer New York NY USA 2002

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Analysis 7

Theorem 7 Let 119891 isin 119862120583[0infin) If 119891(119903+2) exists at a point 119909 isin

(0infin) then

lim119899rarrinfin

119899 (119861120572120573

119899120574)(119903)

(119891 119909) minus 119891(119903)

(119909)

= 119903 (120574 (119903 minus 1) minus 120573) 119891(119903)

(119909)

+ 119903120574 (1 + 2119909) + 120572 minus 120573119909119891(119903+1)

(119909)

+ 119909 (1 + 120574119909) 119891(119903+2)

(119909)

(34)

Proof Using Taylorrsquos expansion of 119891 we have

119891 (119905) =

119903+2

sum

119894=0

119891(119894)

(119909)

119894(119905 minus 119909)

119894

+ 120598 (119905 119909) (119905 minus 119909)119903+2

(35)

where 120598(119905 119909) rarr 0 as 119905 rarr 119909 and 120598(119905 119909) = 119874((119905 minus 119909)120583

) 119905 rarr

infin for 120583 gt 0Applying Lemma 1 we have

119899 (119861120572120573

119899120574)(119903)

(119891 119909) minus 119891(119903)

(119909)

= 119899

119903+2

sum

119894=0

119891(119894)

(119909)

119894(119861120572120573

119899120574)(119903)

((119905 minus 119909)119894

119909) minus 119891(119903)

(119909)

+ 119899 (119861120572120573

119899120574)(119903)

(120598 (119905 119909) (119905 minus 119909)119903+2

119909)

= 1198641+ 1198642

(36)

First we have

1198641= 119899

119903+2

sum

119894=0

119891(119894)

(119909)

119894

119894

sum

119895=0

(119894

119895) (minus119909)

119894minus119895

(119861120572120573

119899120574)(119903)

(119905119895

119909)

minus 119899119891(119903)

(119909) =119891(119903)

(119909)

119903119899 (119861

120572120573

119899120574)(119903)

(119905119903

119909) minus 119903

+119891(119903+1)

(119909)

(119903 + 1)119899 (119903 + 1) (minus119909) (119861

120572120573

119899120574)(119903)

(119905119903

119909)

+ (119861120572120573

119899120574)(119903)

(119905119903+1

119909) +119891(119903+2)

(119909)

(119903 + 2)

sdot 119899 (119903 + 2) (119903 + 1)

21199092

(119861120572120573

119899120574)(119903)

(119905119903

119909) + (119903 + 2)

sdot (minus119909) (119861120572120573

119899120574)(119903)

(119905119903+1

119909) + (119861120572120573

119899120574)(119903)

(119905119903+2

119909)

= 119891(119903)

(119909) 119899 119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)minus 1

+119891(119903+1)

(119909)

(119903 + 1)119899(119903 + 1) (minus119909)

sdot119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)119903

+119899119903+1

Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903)

(119899 + 120573)119903+1

Γ (119899120574 + 1) Γ (119899120574)

(119903 + 1)119909

+(119903 + 1) 119899

119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903)

(119899 + 120573)119903+1

Γ (119899120574 + 1) Γ (119899120574)

119899119903 + 120572(119899

120574

minus 119903) 119903 +119891(119903+2)

(119909)

(119903 + 2)119899(

(119903 + 1) (119903 + 2)

21199092

sdot119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)119903 minus 119909 (119903 + 2)

sdot 119899119903+1

Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903)

(119899 + 120573)119903+1

Γ (119899120574 + 1) Γ (119899120574)

(119903 + 1)119909

+(119903 + 1) 119899

119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903)

(119899 + 120573)119903+1

Γ (119899120574 + 1) Γ (119899120574)

119899119903

+ 120572(119899

120574minus 119903) 119903

+119899119903+2

Γ (119899120574 + 119903 + 2) Γ (119899120574 minus 119903 minus 1)

(119899 + 120573)119903+2

Γ (119899120574 + 1) Γ (119899120574)

(119903 + 2)

21199092

+(119903 + 2) 119899

119903+1

Γ (119899120574 + 119903 + 1) Γ (119899120574 minus 119903 minus 1)

(119899 + 120573)119903+2

Γ (119899120574 + 1) Γ (119899120574)

119899 (119903

+ 1) + 120572(119899

120574minus 119903 minus 1) (119903 + 1)119909

+120572 (119903 + 1) (119903 + 2) 119899

119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)119899119903

+120572 (119899120574 minus 119903)

2 119903)

(37)

Now the coefficients of119891(119903)(119909)119891(119903+1)(119909) and119891(119903+2)

(119909) in theabove expression tend to 119903(120574(119903 minus 1) minus 120573) 119903120574(1 + 2119909) + 120572 minus 120573119909and 119909(1 + 120574119909) respectively which follows by using inductionhypothesis on 119903 and taking the limit as 119899 rarr infin Hence inorder to prove (34) it is sufficient to show that 119864

2rarr 0

as 119899 rarr infin which follows along the lines of the proof ofTheorem 6 and by using Remark 2 and Lemmas 1 and 4

Remark 8 Particular case 120572 = 120573 = 0 was discussed inTheorem 41 in [4] which says that the coefficient of119891(119903+1)(119909)converges to 119903(1 + 2120574119909) but it converges to 119903120574(1 + 2119909) and weget this by putting 120572 = 120573 = 0 in the above theorem

Definition 9 The 119898th order modulus of continuity 120596119898(119891 120575

[119886 119887]) for a function continuous on [119886 119887] is defined by

120596119898(119891 120575 [119886 119887])

= sup 1003816100381610038161003816Δ119898

ℎ119891 (119909)

1003816100381610038161003816 |ℎ| le 120575 119909 119909 + ℎ isin [119886 119887]

(38)

For119898 = 1 120596119898(119891 120575) is usual modulus of continuity

8 International Journal of Analysis

Theorem 10 Let 119891 isin 119862120583[0infin) for some 120583 gt 0 and 0 lt 119886 lt

1198861lt 1198871lt 119887 lt infin Then for 119899 sufficiently large one has

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(119891 sdot) minus 119891(119903)

1003817100381710038171003817100381710038171003817119862[11988611198871]

le 11987211205962(119891(119903)

119899minus12

[1198861 1198871]) + 119872

2119899minus1 1003817100381710038171003817119891

1003817100381710038171003817120583

(39)

where1198721= 1198721(119903) and119872

2= 1198722(119903 119891)

Proof Let us assume that 0 lt 119886 lt 1198861

lt 1198871

lt 119887 lt infinFor sufficiently small 120578 gt 0 we define the function 119891

1205782

corresponding to 119891 isin 119862120583[119886 119887] and 119905 isin [119886

1 1198871] as follows

1198911205782

(119905) = 120578minus2

∬

1205782

minus1205782

(119891 (119905) minus Δ2

ℎ119891 (119905)) 119889119905

11198891199052 (40)

where ℎ = (1199051+ 1199052)2 and Δ

2

ℎis the second order forward

difference operator with step length ℎ For 119891 isin 119862[119886 119887] thefunctions 119891

1205782are known as the Steklov mean of order 2

which satisfy the following properties [11]

(a) 1198911205782

has continuous derivatives up to order 2 over[1198861 1198871]

(b) 119891(119903)1205782

119862[11988611198871]le 1198721120578minus119903

1205962(119891 120578 [119886 119887]) 119903 = 1 2

(c) 119891 minus 1198911205782

119862[11988611198871]le 11987221205962(119891 120578 [119886 119887])

(d) 1198911205782

119862[11988611198871]le 1198723119891120583

where119872119894 119894 = 1 2 3 are certain constants which are different

in each occurrence and are independent of 119891 and 120578We can write by linearity properties of 119861120572120573

119899120574

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(119891 sdot) minus 119891(119903)

1003817100381710038171003817100381710038171003817119862[11988611198871]

le

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(119891 minus 1198911205782

sdot)

1003817100381710038171003817100381710038171003817119862[11988611198871]

+

1003817100381710038171003817100381710038171003817((119861120572120573

119899120574)(119903)

1198911205782

sdot) minus 119891(119903)

1205782

1003817100381710038171003817100381710038171003817119862[11988611198871]

+10038171003817100381710038171003817119891(119903)

minus 119891(119903)

1205782

10038171003817100381710038171003817119862[11988611198871]

= 1198751+ 1198752+ 1198753

(41)

Since 119891(119903)

1205782= (119891(119903)

)1205782

(119905) by property (c) of the function 1198911205782

we get

1198753le 11987241205962(119891(119903)

120578 [119886 119887]) (42)

Next on an application of Theorem 7 it follows that

1198752le 1198725119899minus1

119903+2

sum

119894=119903

10038171003817100381710038171003817119891(119894)

1205782

10038171003817100381710038171003817119862[119886119887] (43)

Using the interpolation property due to Goldberg and Meir[12] for each 119895 = 119903 119903 + 1 119903 + 2 it follows that

10038171003817100381710038171003817119891(119894)

1205782

10038171003817100381710038171003817119862[119886119887]le 1198726100381710038171003817100381710038171198911205782

10038171003817100381710038171003817119862[119886119887]+10038171003817100381710038171003817119891(119903+2)

1205782

10038171003817100381710038171003817119862[119886119887] (44)

Therefore by applying properties (c) and (d) of the function1198911205782 we obtain

1198752le 1198727sdot 119899minus1

1003817100381710038171003817119891

1003817100381710038171003817120583 + 120575minus2

1205962(119891(119903)

120583 [119886 119887]) (45)

Finally we will estimate 1198751 choosing 119886

lowast 119887lowast satisfying theconditions 0 lt 119886 lt 119886

lowast

lt 1198861lt 1198871lt 119887lowast

lt 119887 lt infin Suppose ℏ(119905)denotes the characteristic function of the interval [119886lowast 119887lowast]Then

1198751le

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(ℏ (119905) (119891 (119905) minus 1198911205782

(119905)) sdot)

1003817100381710038171003817100381710038171003817119862[11988611198871]

+

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782

(119905)) sdot)

1003817100381710038171003817100381710038171003817119862[11988611198871]

= 1198754+ 1198755

(46)

By Lemma 5 we have

(119861120572120573

119899120574)(119903)

(ℏ (119905) (119891 (119905) minus 1198911205782

(119905)) 119909)

=119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)

infin

sum

119896=0

119901119899+120574119903119896120574

(119909)

sdot int

infin

0

119887119899minus120574119903119896+119903120574

(119905) ℏ (119905)

sdot (119891(119903)

(119899119905 + 120572

119899 + 120573) minus 119891

(119903)

1205782(119899119905 + 120572

119899 + 120573))119889119905

(47)

Hence100381710038171003817100381710038171003817(119861120572120573

119899120574)119903

(ℏ (119905) (119891 (119905) minus 1198911205782

(119905)) sdot)100381710038171003817100381710038171003817119862[11988611198871]

le 1198728

10038171003817100381710038171003817119891(119903)

minus 119891(119903)

1205782

10038171003817100381710038171003817119862[119886lowast 119887lowast]

(48)

Now for 119909 isin [1198861 1198871] and 119905 isin [0infin) [119886

lowast

119887lowast

] we choose a120575 gt 0 satisfying |(119899119905 + 120572)(119899 + 120573) minus 119909| ge 120575

Therefore by Lemma 4 and the Cauchy-Schwarz inequal-ity we have

119868 equiv (119861120572120573

119899120574)(119903)

((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782

(119905)) 119909)

|119868| le sum

2119894+119895le119903

119894119895ge0

119899119894

10038161003816100381610038161003816119876119894119895119903120574

(119909)10038161003816100381610038161003816

119909 (1 + 120574119909)119903

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|119895

sdot int

infin

0

119887119899119896120574

(119905) (1 minus ℏ (119905))

sdot

10038161003816100381610038161003816100381610038161003816119891 (

119899119905 + 120572

119899 + 120573) minus 1198911205782

(119899119905 + 120572

119899 + 120573)

10038161003816100381610038161003816100381610038161003816119889119905

+Γ (119899120574 + 119903)

Γ (119899120574)(1 + 120574119909)

minus119899120574minus119903

(1 minus ℏ (0))

sdot

10038161003816100381610038161003816100381610038161003816119891 (

120572

119899 + 120573) minus 1198911205782

(120572

119899 + 120573)

10038161003816100381610038161003816100381610038161003816

(49)

International Journal of Analysis 9

For sufficiently large 119899 the second term tends to zero Thus

|119868| le 1198729

10038171003817100381710038171198911003817100381710038171003817120583 sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|119895

sdot int|119905minus119909|ge120575

119887119899119896120574

(119905) 119889119905 le 1198729

10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898

sum

2119894+119895le119903

119894119895ge0

119899119894

sdot

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|119895

(int

infin

0

119887119899119896120574

(119905) 119889119905)

12

sdot (int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

4119898

119889119905)

12

le 1198729

10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898

sum

2119894+119895le119903

119894119895ge0

119899119894

sdot (

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|2119895

)

12

(

infin

sum

119896=1

119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

4119898

119889119905)

12

(50)

Hence by using Remark 2 and Lemma 1 we have

|119868| le 11987210

10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898

119874(119899(119894+(1198952)minus119898)

) le 11987211119899minus119902 1003817100381710038171003817119891

1003817100381710038171003817120583 (51)

where 119902 = 119898 minus (1199032) Now choosing 119898 gt 0 satisfying 119902 ge 1we obtain 119868 le 119872

11119899minus1

119891120583 Therefore by property (c) of the

function 1198911205782

(119905) we get

1198751le 1198728

10038171003817100381710038171003817119891(119903)

minus 119891(119903)

1205782

10038171003817100381710038171003817119862[119886lowast 119887lowast]+ 11987211119899minus1 1003817100381710038171003817119891

1003817100381710038171003817120583

le 119872121205962(119891(119903)

120578 [119886 119887]) + 11987211119899minus1 1003817100381710038171003817119891

1003817100381710038171003817120583

(52)

Choosing 120578 = 119899minus12 the theorem follows

Remark 11 In the last decade the applications of 119902-calculus inapproximation theory are one of themain areas of research In2008 Gupta [13] introduced 119902-Durrmeyer operators whoseapproximation properties were studied in [14] More work inthis direction can be seen in [15ndash17]

A Durrmeyer type 119902-analogue of the 119861120572120573

119899120574(119891 119909) is intro-

duced as follows

119861120572120573

119899120574119902(119891 119909)

=

infin

sum

119896=1

119901119902

119899119896120574(119909) int

infin119860

0

119902minus119896

119887119902

119899119896120574(119905) 119891(

[119899]119902119905 + 120572

[119899]119902+ 120573

)119889119902119905

+ 119901119902

1198990120574(119909) 119891(

120572

[119899]119902+ 120573

)

(53)

where

119901119902

119899119896120574(119909) = 119902

11989622

Γ119902(119899120574 + 119896)

Γ119902(119896 + 1) Γ

119902(119899120574)

sdot(119902120574119909)

119896

(1 + 119902120574119909)(119899120574)+119896

119902

119887119902

119899119896120574(119909) = 120574119902

11989622

Γ119902(119899120574 + 119896 + 1)

Γ119902(119896) Γ119902(119899120574 + 1)

sdot(120574119905)119896minus1

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

119902

int

infin119860

0

119891 (119909) 119889119902119909 = (1 minus 119902)

infin

sum

119899=minusinfin

119891(119902119899

119860)

119902119899

119860 119860 gt 0

(54)

Notations used in (53) can be found in [18] For the operators(53) one can study their approximation properties based on119902-integers

Conflict of Interests

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

Acknowledgments

The authors would like to express their deep gratitude to theanonymous learned referee(s) and the editor for their valu-able suggestions and constructive comments which resultedin the subsequent improvement of this research article

References

[1] V Gupta D K Verma and P N Agrawal ldquoSimultaneousapproximation by certain Baskakov-Durrmeyer-Stancu opera-torsrdquo Journal of the Egyptian Mathematical Society vol 20 no3 pp 183ndash187 2012

[2] D K Verma VGupta and PN Agrawal ldquoSome approximationproperties of Baskakov-Durrmeyer-Stancu operatorsrdquo AppliedMathematics and Computation vol 218 no 11 pp 6549ndash65562012

[3] V N Mishra K Khatri L N Mishra and Deepmala ldquoInverseresult in simultaneous approximation by Baskakov-Durrmeyer-Stancu operatorsrdquo Journal of Inequalities and Applications vol2013 article 586 11 pages 2013

[4] V Gupta ldquoApproximation for modified Baskakov Durrmeyertype operatorsrdquoRockyMountain Journal ofMathematics vol 39no 3 pp 825ndash841 2009

[5] 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 no 4 article 125 2006

[6] V N Mishra and P Patel ldquoApproximation properties ofq-Baskakov-Durrmeyer-Stancu operatorsrdquo Mathematical Sci-ences vol 7 no 1 article 38 12 pages 2013

10 International Journal of Analysis

[7] P Patel and V N Mishra ldquoApproximation properties of certainsummation integral type operatorsrdquo Demonstratio Mathemat-ica vol 48 no 1 pp 77ndash90 2015

[8] 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

[9] 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

[10] W Li ldquoThe voronovskaja type expansion fomula of themodifiedBaskakov-Beta operatorsrdquo Journal of Baoji College of Arts andScience (Natural Science Edition) vol 2 article 004 2005

[11] G Freud andV Popov ldquoOn approximation by spline functionsrdquoin Proceedings of the Conference on Constructive Theory Func-tion Budapest Hungary 1969

[12] S Goldberg and A Meir ldquoMinimummoduli of ordinary differ-ential operatorsrdquo Proceedings of the London Mathematical Soci-ety vol 23 no 3 pp 1ndash15 1971

[13] V Gupta ldquoSome approxmation properties of q-durrmeyer ope-ratorsrdquo Applied Mathematics and Computation vol 197 no 1pp 172ndash178 2008

[14] A Aral and V Gupta ldquoOn the Durrmeyer type modificationof the q-Baskakov type operatorsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 3-4 pp 1171ndash1180 2010

[15] GM Phillips ldquoBernstein polynomials based on the q-integersrdquoAnnals of Numerical Mathematics vol 4 no 1ndash4 pp 511ndash5181997

[16] S Ostrovska ldquoOn the Lupas q-analogue of the Bernstein ope-ratorrdquoThe Rocky Mountain Journal of Mathematics vol 36 no5 pp 1615ndash1629 2006

[17] V N Mishra and P Patel ldquoOn generalized integral Bernsteinoperators based on q-integersrdquo Applied Mathematics and Com-putation vol 242 pp 931ndash944 2014

[18] V Kac and P Cheung Quantum Calculus UniversitextSpringer New York NY USA 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 International Journal of Analysis

Theorem 10 Let 119891 isin 119862120583[0infin) for some 120583 gt 0 and 0 lt 119886 lt

1198861lt 1198871lt 119887 lt infin Then for 119899 sufficiently large one has

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(119891 sdot) minus 119891(119903)

1003817100381710038171003817100381710038171003817119862[11988611198871]

le 11987211205962(119891(119903)

119899minus12

[1198861 1198871]) + 119872

2119899minus1 1003817100381710038171003817119891

1003817100381710038171003817120583

(39)

where1198721= 1198721(119903) and119872

2= 1198722(119903 119891)

Proof Let us assume that 0 lt 119886 lt 1198861

lt 1198871

lt 119887 lt infinFor sufficiently small 120578 gt 0 we define the function 119891

1205782

corresponding to 119891 isin 119862120583[119886 119887] and 119905 isin [119886

1 1198871] as follows

1198911205782

(119905) = 120578minus2

∬

1205782

minus1205782

(119891 (119905) minus Δ2

ℎ119891 (119905)) 119889119905

11198891199052 (40)

where ℎ = (1199051+ 1199052)2 and Δ

2

ℎis the second order forward

difference operator with step length ℎ For 119891 isin 119862[119886 119887] thefunctions 119891

1205782are known as the Steklov mean of order 2

which satisfy the following properties [11]

(a) 1198911205782

has continuous derivatives up to order 2 over[1198861 1198871]

(b) 119891(119903)1205782

119862[11988611198871]le 1198721120578minus119903

1205962(119891 120578 [119886 119887]) 119903 = 1 2

(c) 119891 minus 1198911205782

119862[11988611198871]le 11987221205962(119891 120578 [119886 119887])

(d) 1198911205782

119862[11988611198871]le 1198723119891120583

where119872119894 119894 = 1 2 3 are certain constants which are different

in each occurrence and are independent of 119891 and 120578We can write by linearity properties of 119861120572120573

119899120574

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(119891 sdot) minus 119891(119903)

1003817100381710038171003817100381710038171003817119862[11988611198871]

le

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(119891 minus 1198911205782

sdot)

1003817100381710038171003817100381710038171003817119862[11988611198871]

+

1003817100381710038171003817100381710038171003817((119861120572120573

119899120574)(119903)

1198911205782

sdot) minus 119891(119903)

1205782

1003817100381710038171003817100381710038171003817119862[11988611198871]

+10038171003817100381710038171003817119891(119903)

minus 119891(119903)

1205782

10038171003817100381710038171003817119862[11988611198871]

= 1198751+ 1198752+ 1198753

(41)

Since 119891(119903)

1205782= (119891(119903)

)1205782

(119905) by property (c) of the function 1198911205782

we get

1198753le 11987241205962(119891(119903)

120578 [119886 119887]) (42)

Next on an application of Theorem 7 it follows that

1198752le 1198725119899minus1

119903+2

sum

119894=119903

10038171003817100381710038171003817119891(119894)

1205782

10038171003817100381710038171003817119862[119886119887] (43)

Using the interpolation property due to Goldberg and Meir[12] for each 119895 = 119903 119903 + 1 119903 + 2 it follows that

10038171003817100381710038171003817119891(119894)

1205782

10038171003817100381710038171003817119862[119886119887]le 1198726100381710038171003817100381710038171198911205782

10038171003817100381710038171003817119862[119886119887]+10038171003817100381710038171003817119891(119903+2)

1205782

10038171003817100381710038171003817119862[119886119887] (44)

Therefore by applying properties (c) and (d) of the function1198911205782 we obtain

1198752le 1198727sdot 119899minus1

1003817100381710038171003817119891

1003817100381710038171003817120583 + 120575minus2

1205962(119891(119903)

120583 [119886 119887]) (45)

Finally we will estimate 1198751 choosing 119886

lowast 119887lowast satisfying theconditions 0 lt 119886 lt 119886

lowast

lt 1198861lt 1198871lt 119887lowast

lt 119887 lt infin Suppose ℏ(119905)denotes the characteristic function of the interval [119886lowast 119887lowast]Then

1198751le

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

(ℏ (119905) (119891 (119905) minus 1198911205782

(119905)) sdot)

1003817100381710038171003817100381710038171003817119862[11988611198871]

+

1003817100381710038171003817100381710038171003817(119861120572120573

119899120574)(119903)

((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782

(119905)) sdot)

1003817100381710038171003817100381710038171003817119862[11988611198871]

= 1198754+ 1198755

(46)

By Lemma 5 we have

(119861120572120573

119899120574)(119903)

(ℏ (119905) (119891 (119905) minus 1198911205782

(119905)) 119909)

=119899119903

Γ (119899120574 + 119903) Γ (119899120574 minus 119903 + 1)

(119899 + 120573)119903

Γ (119899120574 + 1) Γ (119899120574)

infin

sum

119896=0

119901119899+120574119903119896120574

(119909)

sdot int

infin

0

119887119899minus120574119903119896+119903120574

(119905) ℏ (119905)

sdot (119891(119903)

(119899119905 + 120572

119899 + 120573) minus 119891

(119903)

1205782(119899119905 + 120572

119899 + 120573))119889119905

(47)

Hence100381710038171003817100381710038171003817(119861120572120573

119899120574)119903

(ℏ (119905) (119891 (119905) minus 1198911205782

(119905)) sdot)100381710038171003817100381710038171003817119862[11988611198871]

le 1198728

10038171003817100381710038171003817119891(119903)

minus 119891(119903)

1205782

10038171003817100381710038171003817119862[119886lowast 119887lowast]

(48)

Now for 119909 isin [1198861 1198871] and 119905 isin [0infin) [119886

lowast

119887lowast

] we choose a120575 gt 0 satisfying |(119899119905 + 120572)(119899 + 120573) minus 119909| ge 120575

Therefore by Lemma 4 and the Cauchy-Schwarz inequal-ity we have

119868 equiv (119861120572120573

119899120574)(119903)

((1 minus ℏ (119905)) (119891 (119905) minus 1198911205782

(119905)) 119909)

|119868| le sum

2119894+119895le119903

119894119895ge0

119899119894

10038161003816100381610038161003816119876119894119895119903120574

(119909)10038161003816100381610038161003816

119909 (1 + 120574119909)119903

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|119895

sdot int

infin

0

119887119899119896120574

(119905) (1 minus ℏ (119905))

sdot

10038161003816100381610038161003816100381610038161003816119891 (

119899119905 + 120572

119899 + 120573) minus 1198911205782

(119899119905 + 120572

119899 + 120573)

10038161003816100381610038161003816100381610038161003816119889119905

+Γ (119899120574 + 119903)

Γ (119899120574)(1 + 120574119909)

minus119899120574minus119903

(1 minus ℏ (0))

sdot

10038161003816100381610038161003816100381610038161003816119891 (

120572

119899 + 120573) minus 1198911205782

(120572

119899 + 120573)

10038161003816100381610038161003816100381610038161003816

(49)

International Journal of Analysis 9

For sufficiently large 119899 the second term tends to zero Thus

|119868| le 1198729

10038171003817100381710038171198911003817100381710038171003817120583 sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|119895

sdot int|119905minus119909|ge120575

119887119899119896120574

(119905) 119889119905 le 1198729

10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898

sum

2119894+119895le119903

119894119895ge0

119899119894

sdot

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|119895

(int

infin

0

119887119899119896120574

(119905) 119889119905)

12

sdot (int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

4119898

119889119905)

12

le 1198729

10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898

sum

2119894+119895le119903

119894119895ge0

119899119894

sdot (

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|2119895

)

12

(

infin

sum

119896=1

119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

4119898

119889119905)

12

(50)

Hence by using Remark 2 and Lemma 1 we have

|119868| le 11987210

10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898

119874(119899(119894+(1198952)minus119898)

) le 11987211119899minus119902 1003817100381710038171003817119891

1003817100381710038171003817120583 (51)

where 119902 = 119898 minus (1199032) Now choosing 119898 gt 0 satisfying 119902 ge 1we obtain 119868 le 119872

11119899minus1

119891120583 Therefore by property (c) of the

function 1198911205782

(119905) we get

1198751le 1198728

10038171003817100381710038171003817119891(119903)

minus 119891(119903)

1205782

10038171003817100381710038171003817119862[119886lowast 119887lowast]+ 11987211119899minus1 1003817100381710038171003817119891

1003817100381710038171003817120583

le 119872121205962(119891(119903)

120578 [119886 119887]) + 11987211119899minus1 1003817100381710038171003817119891

1003817100381710038171003817120583

(52)

Choosing 120578 = 119899minus12 the theorem follows

Remark 11 In the last decade the applications of 119902-calculus inapproximation theory are one of themain areas of research In2008 Gupta [13] introduced 119902-Durrmeyer operators whoseapproximation properties were studied in [14] More work inthis direction can be seen in [15ndash17]

A Durrmeyer type 119902-analogue of the 119861120572120573

119899120574(119891 119909) is intro-

duced as follows

119861120572120573

119899120574119902(119891 119909)

=

infin

sum

119896=1

119901119902

119899119896120574(119909) int

infin119860

0

119902minus119896

119887119902

119899119896120574(119905) 119891(

[119899]119902119905 + 120572

[119899]119902+ 120573

)119889119902119905

+ 119901119902

1198990120574(119909) 119891(

120572

[119899]119902+ 120573

)

(53)

where

119901119902

119899119896120574(119909) = 119902

11989622

Γ119902(119899120574 + 119896)

Γ119902(119896 + 1) Γ

119902(119899120574)

sdot(119902120574119909)

119896

(1 + 119902120574119909)(119899120574)+119896

119902

119887119902

119899119896120574(119909) = 120574119902

11989622

Γ119902(119899120574 + 119896 + 1)

Γ119902(119896) Γ119902(119899120574 + 1)

sdot(120574119905)119896minus1

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

119902

int

infin119860

0

119891 (119909) 119889119902119909 = (1 minus 119902)

infin

sum

119899=minusinfin

119891(119902119899

119860)

119902119899

119860 119860 gt 0

(54)

Notations used in (53) can be found in [18] For the operators(53) one can study their approximation properties based on119902-integers

Conflict of Interests

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

Acknowledgments

The authors would like to express their deep gratitude to theanonymous learned referee(s) and the editor for their valu-able suggestions and constructive comments which resultedin the subsequent improvement of this research article

References

[1] V Gupta D K Verma and P N Agrawal ldquoSimultaneousapproximation by certain Baskakov-Durrmeyer-Stancu opera-torsrdquo Journal of the Egyptian Mathematical Society vol 20 no3 pp 183ndash187 2012

[2] D K Verma VGupta and PN Agrawal ldquoSome approximationproperties of Baskakov-Durrmeyer-Stancu operatorsrdquo AppliedMathematics and Computation vol 218 no 11 pp 6549ndash65562012

[3] V N Mishra K Khatri L N Mishra and Deepmala ldquoInverseresult in simultaneous approximation by Baskakov-Durrmeyer-Stancu operatorsrdquo Journal of Inequalities and Applications vol2013 article 586 11 pages 2013

[4] V Gupta ldquoApproximation for modified Baskakov Durrmeyertype operatorsrdquoRockyMountain Journal ofMathematics vol 39no 3 pp 825ndash841 2009

[5] 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 no 4 article 125 2006

[6] V N Mishra and P Patel ldquoApproximation properties ofq-Baskakov-Durrmeyer-Stancu operatorsrdquo Mathematical Sci-ences vol 7 no 1 article 38 12 pages 2013

10 International Journal of Analysis

[7] P Patel and V N Mishra ldquoApproximation properties of certainsummation integral type operatorsrdquo Demonstratio Mathemat-ica vol 48 no 1 pp 77ndash90 2015

[8] 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

[9] 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

[10] W Li ldquoThe voronovskaja type expansion fomula of themodifiedBaskakov-Beta operatorsrdquo Journal of Baoji College of Arts andScience (Natural Science Edition) vol 2 article 004 2005

[11] G Freud andV Popov ldquoOn approximation by spline functionsrdquoin Proceedings of the Conference on Constructive Theory Func-tion Budapest Hungary 1969

[12] S Goldberg and A Meir ldquoMinimummoduli of ordinary differ-ential operatorsrdquo Proceedings of the London Mathematical Soci-ety vol 23 no 3 pp 1ndash15 1971

[13] V Gupta ldquoSome approxmation properties of q-durrmeyer ope-ratorsrdquo Applied Mathematics and Computation vol 197 no 1pp 172ndash178 2008

[14] A Aral and V Gupta ldquoOn the Durrmeyer type modificationof the q-Baskakov type operatorsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 3-4 pp 1171ndash1180 2010

[15] GM Phillips ldquoBernstein polynomials based on the q-integersrdquoAnnals of Numerical Mathematics vol 4 no 1ndash4 pp 511ndash5181997

[16] S Ostrovska ldquoOn the Lupas q-analogue of the Bernstein ope-ratorrdquoThe Rocky Mountain Journal of Mathematics vol 36 no5 pp 1615ndash1629 2006

[17] V N Mishra and P Patel ldquoOn generalized integral Bernsteinoperators based on q-integersrdquo Applied Mathematics and Com-putation vol 242 pp 931ndash944 2014

[18] V Kac and P Cheung Quantum Calculus UniversitextSpringer New York NY USA 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Analysis 9

For sufficiently large 119899 the second term tends to zero Thus

|119868| le 1198729

10038171003817100381710038171198911003817100381710038171003817120583 sum

2119894+119895le119903

119894119895ge0

119899119894

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|119895

sdot int|119905minus119909|ge120575

119887119899119896120574

(119905) 119889119905 le 1198729

10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898

sum

2119894+119895le119903

119894119895ge0

119899119894

sdot

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|119895

(int

infin

0

119887119899119896120574

(119905) 119889119905)

12

sdot (int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

4119898

119889119905)

12

le 1198729

10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898

sum

2119894+119895le119903

119894119895ge0

119899119894

sdot (

infin

sum

119896=1

119901119899119896120574

(119909) |119896 minus 119899119909|2119895

)

12

(

infin

sum

119896=1

119901119899119896120574

(119909)

sdot int

infin

0

119887119899119896120574

(119905) (119899119905 + 120572

119899 + 120573minus 119909)

4119898

119889119905)

12

(50)

Hence by using Remark 2 and Lemma 1 we have

|119868| le 11987210

10038171003817100381710038171198911003817100381710038171003817120583 120575minus2119898

119874(119899(119894+(1198952)minus119898)

) le 11987211119899minus119902 1003817100381710038171003817119891

1003817100381710038171003817120583 (51)

where 119902 = 119898 minus (1199032) Now choosing 119898 gt 0 satisfying 119902 ge 1we obtain 119868 le 119872

11119899minus1

119891120583 Therefore by property (c) of the

function 1198911205782

(119905) we get

1198751le 1198728

10038171003817100381710038171003817119891(119903)

minus 119891(119903)

1205782

10038171003817100381710038171003817119862[119886lowast 119887lowast]+ 11987211119899minus1 1003817100381710038171003817119891

1003817100381710038171003817120583

le 119872121205962(119891(119903)

120578 [119886 119887]) + 11987211119899minus1 1003817100381710038171003817119891

1003817100381710038171003817120583

(52)

Choosing 120578 = 119899minus12 the theorem follows

Remark 11 In the last decade the applications of 119902-calculus inapproximation theory are one of themain areas of research In2008 Gupta [13] introduced 119902-Durrmeyer operators whoseapproximation properties were studied in [14] More work inthis direction can be seen in [15ndash17]

A Durrmeyer type 119902-analogue of the 119861120572120573

119899120574(119891 119909) is intro-

duced as follows

119861120572120573

119899120574119902(119891 119909)

=

infin

sum

119896=1

119901119902

119899119896120574(119909) int

infin119860

0

119902minus119896

119887119902

119899119896120574(119905) 119891(

[119899]119902119905 + 120572

[119899]119902+ 120573

)119889119902119905

+ 119901119902

1198990120574(119909) 119891(

120572

[119899]119902+ 120573

)

(53)

where

119901119902

119899119896120574(119909) = 119902

11989622

Γ119902(119899120574 + 119896)

Γ119902(119896 + 1) Γ

119902(119899120574)

sdot(119902120574119909)

119896

(1 + 119902120574119909)(119899120574)+119896

119902

119887119902

119899119896120574(119909) = 120574119902

11989622

Γ119902(119899120574 + 119896 + 1)

Γ119902(119896) Γ119902(119899120574 + 1)

sdot(120574119905)119896minus1

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

119902

int

infin119860

0

119891 (119909) 119889119902119909 = (1 minus 119902)

infin

sum

119899=minusinfin

119891(119902119899

119860)

119902119899

119860 119860 gt 0

(54)

Notations used in (53) can be found in [18] For the operators(53) one can study their approximation properties based on119902-integers

Conflict of Interests

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

Acknowledgments

The authors would like to express their deep gratitude to theanonymous learned referee(s) and the editor for their valu-able suggestions and constructive comments which resultedin the subsequent improvement of this research article

References

[1] V Gupta D K Verma and P N Agrawal ldquoSimultaneousapproximation by certain Baskakov-Durrmeyer-Stancu opera-torsrdquo Journal of the Egyptian Mathematical Society vol 20 no3 pp 183ndash187 2012

[2] D K Verma VGupta and PN Agrawal ldquoSome approximationproperties of Baskakov-Durrmeyer-Stancu operatorsrdquo AppliedMathematics and Computation vol 218 no 11 pp 6549ndash65562012

[3] V N Mishra K Khatri L N Mishra and Deepmala ldquoInverseresult in simultaneous approximation by Baskakov-Durrmeyer-Stancu operatorsrdquo Journal of Inequalities and Applications vol2013 article 586 11 pages 2013

[4] V Gupta ldquoApproximation for modified Baskakov Durrmeyertype operatorsrdquoRockyMountain Journal ofMathematics vol 39no 3 pp 825ndash841 2009

[5] 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 no 4 article 125 2006

[6] V N Mishra and P Patel ldquoApproximation properties ofq-Baskakov-Durrmeyer-Stancu operatorsrdquo Mathematical Sci-ences vol 7 no 1 article 38 12 pages 2013

10 International Journal of Analysis

[7] P Patel and V N Mishra ldquoApproximation properties of certainsummation integral type operatorsrdquo Demonstratio Mathemat-ica vol 48 no 1 pp 77ndash90 2015

[8] 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

[9] 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

[10] W Li ldquoThe voronovskaja type expansion fomula of themodifiedBaskakov-Beta operatorsrdquo Journal of Baoji College of Arts andScience (Natural Science Edition) vol 2 article 004 2005

[11] G Freud andV Popov ldquoOn approximation by spline functionsrdquoin Proceedings of the Conference on Constructive Theory Func-tion Budapest Hungary 1969

[12] S Goldberg and A Meir ldquoMinimummoduli of ordinary differ-ential operatorsrdquo Proceedings of the London Mathematical Soci-ety vol 23 no 3 pp 1ndash15 1971

[13] V Gupta ldquoSome approxmation properties of q-durrmeyer ope-ratorsrdquo Applied Mathematics and Computation vol 197 no 1pp 172ndash178 2008

[14] A Aral and V Gupta ldquoOn the Durrmeyer type modificationof the q-Baskakov type operatorsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 3-4 pp 1171ndash1180 2010

[15] GM Phillips ldquoBernstein polynomials based on the q-integersrdquoAnnals of Numerical Mathematics vol 4 no 1ndash4 pp 511ndash5181997

[16] S Ostrovska ldquoOn the Lupas q-analogue of the Bernstein ope-ratorrdquoThe Rocky Mountain Journal of Mathematics vol 36 no5 pp 1615ndash1629 2006

[17] V N Mishra and P Patel ldquoOn generalized integral Bernsteinoperators based on q-integersrdquo Applied Mathematics and Com-putation vol 242 pp 931ndash944 2014

[18] V Kac and P Cheung Quantum Calculus UniversitextSpringer New York NY USA 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 International Journal of Analysis

[7] P Patel and V N Mishra ldquoApproximation properties of certainsummation integral type operatorsrdquo Demonstratio Mathemat-ica vol 48 no 1 pp 77ndash90 2015

[8] 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

[9] 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

[10] W Li ldquoThe voronovskaja type expansion fomula of themodifiedBaskakov-Beta operatorsrdquo Journal of Baoji College of Arts andScience (Natural Science Edition) vol 2 article 004 2005

[11] G Freud andV Popov ldquoOn approximation by spline functionsrdquoin Proceedings of the Conference on Constructive Theory Func-tion Budapest Hungary 1969

[12] S Goldberg and A Meir ldquoMinimummoduli of ordinary differ-ential operatorsrdquo Proceedings of the London Mathematical Soci-ety vol 23 no 3 pp 1ndash15 1971

[13] V Gupta ldquoSome approxmation properties of q-durrmeyer ope-ratorsrdquo Applied Mathematics and Computation vol 197 no 1pp 172ndash178 2008

[14] A Aral and V Gupta ldquoOn the Durrmeyer type modificationof the q-Baskakov type operatorsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 3-4 pp 1171ndash1180 2010

[15] GM Phillips ldquoBernstein polynomials based on the q-integersrdquoAnnals of Numerical Mathematics vol 4 no 1ndash4 pp 511ndash5181997

[16] S Ostrovska ldquoOn the Lupas q-analogue of the Bernstein ope-ratorrdquoThe Rocky Mountain Journal of Mathematics vol 36 no5 pp 1615ndash1629 2006

[17] V N Mishra and P Patel ldquoOn generalized integral Bernsteinoperators based on q-integersrdquo Applied Mathematics and Com-putation vol 242 pp 931ndash944 2014

[18] V Kac and P Cheung Quantum Calculus UniversitextSpringer New York NY USA 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of