research article rate of convergence of modified baskakov...
Post on 18-Sep-2020
1 Views
Preview:
TRANSCRIPT
Research ArticleRate of Convergence of Modified Baskakov-Durrmeyer TypeOperators for Functions of Bounded Variation
Prashantkumar Patel12 and Vishnu Narayan Mishra13
1 Department of Applied Mathematics amp Humanities Sardar Vallabhbhai National Institute of TechnologySurat Gujarat 395 007 India
2Department of Mathematics St Xavierrsquos College Ahmedabad Gujarat 380 009 India3 L 1627 Awadh Puri Colony Beniganj Phase-III Opposite-Industrial Training Institute (ITI) Ayodhya Main RoadFaizabad Uttar Pradesh 224 001 India
Correspondence should be addressed to Prashantkumar Patel prashant225gmailcom
Received 19 April 2014 Accepted 11 June 2014 Published 2 July 2014
Academic Editor Abdelalim A Elsadany
Copyright copy 2014 P Patel and V N Mishra This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We study a certain integral modification of well-known Baskakov operators with weight function of beta basis function Weestablish rate of convergence for these operators for functions having derivative of bounded variation Also we discuss Stancutype generalization of these operators
1 Introduction
The integral modification of Baskakov operators havingweight function of some beta basis function are defined asthe following for 119909 isin [0infin) 120574 gt 0
119861119899120574(119891 119909) =
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) 119891 (119905) 119889119905
+ (1 + 120574119909)minus119899
119891 (0)
= int
infin
0
119882119899120574(119909 119905) 119891 (119905) 119889119905
(1)
where
119901119899119896120574
(119909) =
Γ (119899120574 + 119896)
Γ (119896 + 1) Γ (119899120574)
sdot
(120574119909)119896
(1 + 120574119909)(119899120574)+119896
119887119899119896120574
(119905) =
120574Γ (119899120574 + 119896 + 1)
Γ (119896) Γ (119899120574 + 1)
sdot
(120574119905)119896minus1
(1 + 120574119905)(119899120574)+119896+1
119882119899120574(119909 119905) =
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) 119889119905 + (1 + 120574119909)minus119899120574
120575 (119905)
(2)
120575(119905) being the Dirac delta functionThe operators defined by (1) were introduced by Gupta
[1] these operators are different from the usual Baskakov-Durrmeyer operators Actually these operators satisfy condi-tion 119861
119899120574(119886119905 + 119887 119909) = 119886119909 + 119887 where 119886 and 119887 are constants In
[1] the author estimated some direct results in simultaneousapproximation for these operators (1) In particular case 120574 =1 the operators (1) reduce to the operators studied in [2 3]
In recent years a lot of work has been done on suchoperators We refer to some of the important papers on therecent development on similar type of operators [4ndash9] Therate of convergence for certainDurrmeyer type operators andthe generalizations is one of the important areas of researchin recent years In present article we extend the studies andhere we estimate the rate of convergence for functions havingderivative of bounded variation
Hindawi Publishing CorporationJournal of Difference EquationsVolume 2014 Article ID 235480 6 pageshttpdxdoiorg1011552014235480
2 Journal of Difference Equations
We denote 120601119899120574(119909 119905) = int
119905
0
119882119899120574(119909 119904)119889119904 then in particular
we have
120601119899120574(119909infin) = int
infin
0
119882119899120574(119909 119904) 119889119904 = 1 (3)
By 119863119861119903(0infin) 119903 ge 0 we denote the class of absolutely
continuous functions 119891 defined on the interval (0infin) suchthat
(i) 119891(119905) = 119874(119905119903) 119905 rarr infin(ii) having a derivative 1198911015840 on the interval (0infin) coincid-
ing ae with a function which is of bounded variationon every finite subinterval of (0infin)
It can be observed that all function 119891 isin 119861119863119903(0infin) possess
for each 119888 gt 0 a representation
119891 (119909) = 119891 (119888) + int
119909
119888
120595 (119905) 119889119905 119909 ge 119888 (4)
2 Rate of Convergence for 119861119899120574
Lemma 1 (see [1]) Let the function 119879119899119898120574
(119909)119898 isin N cup 0 bedefined as
119879119899119898120574
(119909) = 119861119899120574((119905 minus 119909)
119898
119909)
=
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119905 minus 119909)119898
119889119905
+ (1 + 120574119909)minus119899120574
(minus119909)119898
(5)
Then it is easily verified that for each 119909 isin (0infin) 1198791198990120574
(119909) = 11198791198991120574
(119909) = 0 and 1198791198992120574
(119909) = 2119909(1 + 120574119909)(119899 minus 120574) and also thefollowing recurrence relation holds
(119899 minus 120574119898)119879119899119898+1120574
(119909) = 119909 (1 + 120574119909)
times [119879(1)
119899119898120574(119909) + 2119898119879
119899119898minus1120574(119909)]
+ 119898 (1 + 2120574119909) 119879119899119898120574
(119909)
(6)
From the recurrence relation it can be easily be verified that forall 119909 isin [0infin) we have 119879
119899119898120574(119909) = 119874(119899
minus[(119898+1)2]
)
Remark 2 FromLemma 1 usingCauchy-Schwarz inequalityit follows that
119861119899120574(|119905 minus 119909| 119909) le [119861
119899120574((119905 minus 119909)
2
119909)]
12
le radic
2119909 (1 + 120574119909)
119899 minus 120574
(7)
Lemma 3 Let 119909 isin (0infin) and119882119899120574(119909 119905) be the kernel defined
in (1) Then for 119899 being sufficiently large one has
(a) 120601119899120574(119909 119910) = int
119910
0
119882119899120574(119909 119905)119889119905 le 2119909(1 + 120574119909)(119899 minus
120574)(119909 minus 119910)2
0 le 119910 lt 119909
(b) 1 minus 120601119899120574(119909 119911) = int
infin
119911
119882119899120574(119909 119905)119889119905 le 2119909(1 + 120574119909)(119899 minus
120574)(119911 minus 119909)2
119909 lt 119911 lt infin
Proof First we prove (a) by using Lemma 1 we have
int
119910
0
119882119899120574(119909 119905) 119889119905 le int
119910
0
(119909 minus 119905)2
(119909 minus 119910)2119882119899120574(119909 119905) 119889119905
le (119909 minus 119910)minus2
1198791198992120574
(119909)
le
2119909 (1 + 120574119909)
(119899 minus 120574) (119909 minus 119910)2
(8)
The proof of (b) is similar we omit the details
Theorem 4 Let 119891 isin 119863119861119903(0infin) 119903 isin N and 119909 isin (0infin) Then
for 119899 being sufficiently large we have10038161003816100381610038161003816119861119899120574(119891 119909) minus 119891 (119909)
10038161003816100381610038161003816
le
2 (1 + 120574119909)
119899 minus 120574
[radic119899]
sum
119896=1
119909+119909119896
⋁
119909minus119909119896
((1198911015840
)119909
)
+
119909
radic119899
119909+119909radic119899
⋁
119909minus119909radic119899
((1198911015840
)119909
) +
2 (1 + 120574119909)
(119899 minus 120574) 119909
times [
10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891
1015840
(119909+
)
10038161003816100381610038161003816+1003816100381610038161003816119891 (119909)
1003816100381610038161003816]
+ radic
2119909 (1 + 120574119909)
119899 minus 120574
[1198722119903
119874(119899minus1199032
) +
100381610038161003816100381610038161198911015840
(119909+
)
10038161003816100381610038161003816]
+ radic
119909 (1 + 120574119909)
2 (119899 minus 120574)
100381610038161003816100381610038161198911015840
(119909+
) minus 1198911015840
(119909minus
)
10038161003816100381610038161003816
(9)
where the auxiliary function 119891119909is given by
119891119909(119905) =
119891 (119905) minus 119891 (119909minus
) 0 le 119905 lt 119909
0 119905 = 119909
119891 (119905) minus 119891 (119909+
) 119909 lt 119905 lt infin
(10)
⋁119887
119886119891(119909) denotes the total variation of 119891
119909on [119886 119887]
Proof By the application of mean value theorem we have
119861119899120574(119891 119909) minus 119891 (119909) = int
infin
0
119882119899120574(119909 119905) (119891 (119905) minus 119891 (119909)) 119889119905
= int
infin
0
int
119905
119909
119882119899120574(119909 119905) (119891
1015840
(119906) 119889119906) 119889119905
(11)
Also using the identity
1198911015840
(119906) =
1198911015840
(119909+
) + 1198911015840
(119909minus
)
2
+ (1198911015840
)119909
(119906)
+
1198911015840
(119909+
) minus 1198911015840
(119909minus
)
2
sgn (119906 minus 119909)
+ [1198911015840
(119909) minus
1198911015840
(119909+
) + 1198911015840
(119909minus
)
2
] 120594119909(119906)
(12)
Journal of Difference Equations 3
where
120594119909(119906) =
1 119906 = 119909
0 119906 = 119909
(13)
we can see that
int
infin
0
(int
119905
119909
(1198911015840
(119909) minus
1198911015840
(119909+
) + 1198911015840
(119909minus
)
2
)120594119909(119906) 119889119906)
times119882119899120574(119905 119909) 119889119905 = 0
(14)
Also
int
infin
0
(int
119905
119909
(
1198911015840
(119909+
) minus 1198911015840
(119909minus
)
2
) sgn (119906 minus 119909) 119889119906)119882119899120574(119905 119909) 119889119905
=
1198911015840
(119909+
) minus 1198911015840
(119909minus
)
2
119861119899120574(|119905 minus 119909| 119909)
int
infin
0
(int
119905
119909
(
1198911015840
(119909+
) + 1198911015840
(119909minus
)
2
) 119889119906)119882119899120574(119905 119909) 119889119905
=
1198911015840
(119909+
) + 1198911015840
(119909minus
)
2
119861119899120574(119905 minus 119909 119909)
(15)
Substitute value of 1198911015840(119906) from (12) in (11) and using (14) and(15) we get
10038161003816100381610038161003816119861119899120574(119891 119909) minus 119891 (119909)
10038161003816100381610038161003816le
10038161003816100381610038161003816100381610038161003816
int
infin
119909
(int
119905
119909
(1198911015840
)119909
(119906) 119889119906)119882119899120574(119909 119905) 119889119905
10038161003816100381610038161003816100381610038161003816
+
10038161003816100381610038161003816100381610038161003816
int
119909
0
(int
119905
119909
(1198911015840
)119909
(119906) 119889119906)119882119899120574(119909 119905) 119889119905
10038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161198911015840
(119909+
) minus 1198911015840
(119909minus
)
10038161003816100381610038161003816
2
119861119899120574(|119905 minus 119909| 119909)
+
100381610038161003816100381610038161198911015840
(119909+
) + 1198911015840
(119909minus
)
10038161003816100381610038161003816
2
119861119899120574(119905 minus 119909 119909)
(16)
Using Lemma 1 and Remark 2 we obtain
10038161003816100381610038161003816119861119899120574(119891 119909) minus 119891 (119909)
10038161003816100381610038161003816le
10038161003816100381610038161003816100381610038161003816
int
infin
119909
(int
119905
119909
(1198911015840
)119909
(119906) 119889119906)119882119899120574(119909 119905) 119889119905
10038161003816100381610038161003816100381610038161003816
+
10038161003816100381610038161003816100381610038161003816
int
119909
0
(int
119905
119909
(1198911015840
)119909
(119906) 119889119906)119882119899120574(119909 119905) 119889119905
10038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161198911015840
(119909+
) minus 1198911015840
(119909minus
)
10038161003816100381610038161003816
2
radic
2119909 (1 + 120574119909)
119899 minus 120574
=
10038161003816100381610038161003816119875119899120574(119891 119909)
10038161003816100381610038161003816+
10038161003816100381610038161003816119876119899120574(119891 119909)
10038161003816100381610038161003816
+
100381610038161003816100381610038161198911015840
(119909+
) minus 1198911015840
(119909minus
)
10038161003816100381610038161003816
2
radic
2119909 (1 + 120574119909)
119899 minus 120574
(17)
On applying Lemma 3 with 119910 = 119909 minus 119909radic119899 and integrating byparts we have
10038161003816100381610038161003816119876119899120574(119891 119909)
10038161003816100381610038161003816=
10038161003816100381610038161003816100381610038161003816
int
119909
0
int
119905
119909
(1198911015840
)119909
(119906) 119889119906 119889119905 (120601119899120574(119909 119910))
10038161003816100381610038161003816100381610038161003816
=
10038161003816100381610038161003816100381610038161003816
int
119909
0
120601119899120574(119909 119910) (119891
1015840
)119909
(119905) 119889119905
10038161003816100381610038161003816100381610038161003816
le (int
119910
0
+int
119909
119910
)
10038161003816100381610038161003816(1198911015840
)119909
(119905)
10038161003816100381610038161003816
10038161003816100381610038161003816120601119899120574(119909 119905)
10038161003816100381610038161003816119889119905
le
2119909 (1 + 120574119909)
119899 minus 120574
int
119910
0
119909
⋁
119905
((1198911015840
)119909
)
1
(119909 minus 119905)2119889119905
+ int
119909
119910
119909
⋁
119905
((1198911015840
)119909
) 119889119905
le
2119909 (1 + 120574119909)
119899 minus 120574
int
119910
0
119909
⋁
119905
((1198911015840
)119909
)
1
(119909 minus 119905)2119889119905
+
119909
radic119899
119909
⋁
119909minus119909radic119899
((1198911015840
)119909
)
=
2119909 (1 + 120574119909)
119899 minus 120574
int
radic119899
1
119909
⋁
119909minus119909119906
((1198911015840
)119909
) 119889119906
+
119909
radic119899
119909
⋁
119909minus119909radic119899
((1198911015840
)119909
)
le
2119909 (1 + 120574119909)
119899 minus 120574
[radic119899]
sum
119896=1
119909
⋁
119909minus119909119896
((1198911015840
)119909
)
+
119909
radic119899
119909
⋁
119909minus119909radic119899
((1198911015840
)119909
)
(18)
where 119906 = 119909(119909 minus 119905)On the other hand we have10038161003816100381610038161003816119875119899120574(119891 119909)
10038161003816100381610038161003816
=
10038161003816100381610038161003816100381610038161003816
int
infin
119909
(int
119905
119909
(1198911015840
)119909
(119906) 119889119906)119882119899120574(119909 119905) 119889119905
10038161003816100381610038161003816100381610038161003816
=
10038161003816100381610038161003816100381610038161003816
int
infin
2119909
(int
119905
119909
(1198911015840
)119909
(119906) 119889119906)119882119899120574(119909 119905) 119889119905
+int
2119909
119909
(int
119905
119909
(1198911015840
)119909
(119906) 119889119906)119889119905 (1 minus 120601119899120574(119909 119905))
100381610038161003816100381610038161003816100381610038161003816
le
10038161003816100381610038161003816100381610038161003816
int
infin
2119909
(119891 (119905) minus 119891 (119909))119882119899120574(119909 119905) 119889119905
10038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161198911015840
(119909+
)
10038161003816100381610038161003816
10038161003816100381610038161003816100381610038161003816
int
infin
2119909
(119905 minus 119909)119882119899120574(119909 119905) 119889119905
10038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
int
2119909
119909
(1198911015840
)119909
(119906) 119889119906
100381610038161003816100381610038161003816100381610038161003816
100381610038161003816100381610038161 minus 120601119909120574(119909 2119909)
10038161003816100381610038161003816
4 Journal of Difference Equations
+ int
2119909
119909
10038161003816100381610038161003816(1198911015840
)119909
(119905)
10038161003816100381610038161003816
100381610038161003816100381610038161 minus 120601119909120574(119909 119905)
10038161003816100381610038161003816119889119905
le [
119872
119909
int
infin
2119909
119882119899120574(119909 119905) 119905
119903
|119905 minus 119909| 119889119905
+
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
1199092
int
infin
2119909
119882119899120574(119909 119905) (119905 minus 119909)
2
119889119905]
+ [
100381610038161003816100381610038161198911015840
(119909+
)
10038161003816100381610038161003816int
infin
2119909
119882119899120574(119909 119905) |119905 minus 119909| 119889119905]
+
2 (1 + 120574119909)
(119899 minus 120574) 119909
10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891
1015840
(119909+
)
10038161003816100381610038161003816
+
2 (1 + 120574119909)
119899 minus 120574
[radic119899]
sum
119896=1
119909+119909119896
⋁
119909
((1198911015840
)119909
)
+
119909
radic119899
119909+119909radic119899
⋁
119909
((1198911015840
)119909
)
= 119877119899120574(119891 119909) + 119878
119899120574(119891 119909)
+
2 (1 + 120574119909)
(119899 minus 120574) 119909
10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891
1015840
(119909+
)
10038161003816100381610038161003816
+
2 (1 + 120574119909)
119899 minus 120574
[radic119899]
sum
119896=1
119909+119909119896
⋁
119909
((1198911015840
)119909
)
+
119909
radic119899
119909+119909radic119899
⋁
119909
((1198911015840
)119909
)
(19)
Applying Holderrsquos inequality Remark 2 and Lemma 1 wehave
119877119899120574(119891 119909) le
119872
119909
(int
infin
2119909
119882119899120574(119909 119905) 119905
2119903
119889119905)
12
times(int
infin
0
119882119899120574(119909 119905) (119905 minus 119909)
2
119889119905)
12
+
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
1199092
int
infin
2119909
119882119899120574(119909 119905) (119905 minus 119909)
2
119889119905
le 1198722119903
119874(119899minus1199032
)radic
2119909 (1 + 120574119909)
119899 minus 120574
+1003816100381610038161003816119891 (119909)
1003816100381610038161003816
2 (1 + 120574119909)
(119899 minus 120574) 119909
(20)
Also
119878119899120574(119891 119909) le
1003816100381610038161003816119891 (119909+
)1003816100381610038161003816int
infin
0
119882119899(119905 119909) |119905 minus 119909| 119889119905
le1003816100381610038161003816119891 (119909+
)1003816100381610038161003816radic
2119909 (1 + 120574119909)
119899 minus 120574
(21)
Combining the estimates (17)ndash(21) we get the desired resultsThis completes the proof of Theorem
3 Rate of Convergence for Stancu TypeGeneralization of 119861
119899120574
In 1968 Stancu introduces Bernstein-Stancu operators in[10] a sequence of the linear positive operators depending ontwo parameters 120572 and 120573 satisfying the condition 0 le 120572 le 120573Recently many researchers applied this approach to manyoperators for details see [11ndash17] For 119891 isin 119862[0infin) Stancugeneralization of operators (1) is as follows
119861120572120573
119899120574(119891 (119905) 119909) =
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) 119891(
119899119905 + 120572
119899 + 120573
)119889119905
+ 1199011198990120574
(119909) 119891(
120572
119899 + 120573
)
= int
infin
0
119882119899120574(119909 119905) 119891(
119899119905 + 120572
119899 + 120573
)119889119905
(22)
where 119901119899119896120574
(119909) 119887119899119896120574
(119909) and119882119899120574(119909 119905)are as defined in (1)
Lemma5 (see [18]) If we define the centralmoments for every119898 isin N as
120583120572120573
119899119898120574(119909) = 119861
120572120573
119899120574((119905 minus 119909)
119898
119909)
=
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (
119899119905 + 120572
119899 + 120573
minus 119909)
119898
119889119905
+ 1199011198990120574
(119909) (
120572
119899 + 120573
minus 119909)
119898
(23)
then 1205831205721205731198990120574
(119909) = 1 1205831205721205731198991120574
(119909) = (120572 minus 120573119909)(119899 + 120573) and
120583120572120573
1198992120574(119909) =
1205722
(119899 + 120573)2+ 119909
(21198992
minus 2119899120572120573 + 2120572120573120574)
(119899 + 120573)2
(119899 minus 120574)
+ 1199092(1198991205732
+ 21198992
120574 minus 1205732
120574)
(119899 + 120573)2
(119899 minus 120574)
(24)
For 119899 gt 119898 we have the following recurrence relation
(119899 minus 120574119898) (119899 + 120573) 120583120572120573
119899119898+1120574(119909)
= 119899119909 (1 + 120574119909) [(120583120572120573
119899119898120574)
(1)
(119909) + 119898120583120572120573
119899119898minus1120574(119909)]
+ [119898119899 + 1198992
119909 minus (2120574119898 minus 119899) (120572 minus (119899 + 120573) 119909)] 120583120572120573
119899119898120574(119909)
+ [119898120574 (119899 + 120573) (
120572
119899 + 120573
minus 119909)
2
minus 119898119899(
120572
119899 + 120573
minus 119909)]
times 120583120572120573
119899119898minus1120574(119909)
(25)
Journal of Difference Equations 5
From the recurrence relation it can be easily verified that forall 119909 isin [0infin) we have 120583120572120573
119899119898120574(119909) = 119874(119899
minus[(119898+1)2]
)
Remark 6 Observe that 119861120572120573119899120574
preserve constant functions butnot linear functions If 120572 = 120573 = 0 these operators reduce tothe operators defined in (1) Notice that
120583120572120573
1198992120574(119909) = [
1205732
120574(119899 + 120573)2+
21198992
(119899 + 120573)2
(119899 minus 120574)
] 119909 (1 + 120574119909)
+
120573 [(2120572 + 120573) 120574 minus 119899 (120573 + 2120572120574)]
120574(119899 + 120573)2
(119899 minus 120574)
119909 +
1205722
(119899 + 120573)2
(26)
Remark 7 From Lemma 3 taking 119899 to be sufficiently largeand 119909 isin (0infin) we observe that
120583120572120573
1198992120574(119909) le
119862119909 (1 + 120574119909)
119899 minus 120574
(27)
where 119862 is positive constant
Remark 8 Applying the Cauchy-Schwarz inequality andkeeping the same condition as in Remark 7 for 119909 119899 and 119862we derive from Lemma 5 that
119861120572120573
119899120574(|119905 minus 119909| 119909) le [119861
120572120573
119899120574((119905 minus 119909)
2
119909)]
12
le radic
119862119909 (1 + 120574119909)
119899 minus 120574
(28)
Lemma 9 Let 119909 isin (0infin) and119882119899120574(119909 119905) be the kernel defined
in (1) Then for 119899 being sufficiently large we have
(c) 120601119899120574(119909 119910) = int
119910
0
119882119899120574(119909 119905)119889119905 = 119862119909(1 + 120574119909)(119899 minus
120574)(119909 minus 119910)2
0 le 119910 lt 119909
(d) 1 minus 120601119899120574(119909 119911) = int
infin
119911
119882119899120574(119909 119905)119889119905 = 119862119909(1 + 120574119909)(119899 minus
120574)(119911 minus 119909)2
119909 lt 119911 lt infin
The proof is the same as Lemma 3 thus we omit the details
Theorem 10 Let 119891 isin 119863119861119903(0infin) 119903 isin N and 119909 isin (0infin)
Then for 119899 being sufficiently large we have
10038161003816100381610038161003816119861120572120573
119899120574(119891 119909) minus 119891 (119909)
10038161003816100381610038161003816
le
119862 (1 + 120574119909)
119899 minus 120574
[radic119899]
sum
119896=1
119909+119909119896
⋁
119909minus119909119896
((1198911015840
)119909
)
+
119909
radic119899
119909+119909radic119899
⋁
119909minus119909radic119899
((1198911015840
)119909
) +
119862 (1 + 120574119909)
(119899 minus 120574) 119909
times [
10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891
1015840
(119909+
)
10038161003816100381610038161003816+1003816100381610038161003816119891 (119909)
1003816100381610038161003816]
+ radic
119862119909 (1 + 120574119909)
119899 minus 120574
times [11987212119903
119874(119899minus1199032
) +
100381610038161003816100381610038161198911015840
(119909+
)
10038161003816100381610038161003816]
+
1
2
radic
119862119909 (1 + 120574119909)
(119899 minus 120574)
100381610038161003816100381610038161198911015840
(119909+
) minus 1198911015840
(119909minus
)
10038161003816100381610038161003816
+
1
2
100381610038161003816100381610038161198911015840
(119909+
) + 1198911015840
(119909minus
)
10038161003816100381610038161003816
1003816100381610038161003816120572 minus 120573119909
1003816100381610038161003816
119899 + 120573
(29)
where the auxiliary functions 119891119909and ⋁119887
119886119891(119909) were defined in
Theorem 4The proof of the above theorem follows along the lines of
Theorem 4 thus we omit the details
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are thankful to the anonymous referee formaking valuable comments leading to the better presen-tation of the paper Special thanks are due to ProfessorDr Abdelalim A Elsadany Editor of Journal of DifferenceEquations for kind cooperation and smooth behavior duringcommunication and for his efforts to send the reports of thepaper timely
References
[1] V Gupta ldquoApproximation for modified Baskakov Durrmeyertype operatorsrdquo The Rocky Mountain Journal of Mathematicsvol 39 no 3 pp 825ndash841 2009
[2] V Gupta M A Noor M S Beniwal and M K Gupta ldquoOnsimultaneous approximation for certain Baskakov Durrmeyertype operatorsrdquo Journal of Inequalities in Pure and AppliedMathematics vol 7 pp 1ndash15 2006
[3] V Gupta and P N Agrawal ldquoRate of convergence for certainBaskakov Durrmeyer type operatorsrdquoAnnals of Oradea Univer-sity Fascicola Matematica vol 14 pp 33ndash39 2007
[4] HKarsli andE Ibikli ldquoConvergence rate of a newBezier variantof Chlodowsky operators to bounded variation functionsrdquoJournal of Computational and AppliedMathematics vol 212 no2 pp 431ndash443 2008
[5] V N Mishra H H Khan K Khatri and L N Mishra ldquoHyper-geometric representation for Baskakov-Durrmeyer-Stancu typeoperatorsrdquo Bulletin of Mathematical Analysis and Applicationsvol 5 no 3 pp 18ndash26 2013
[6] V N Mishra K Khatri and L N Mishra ldquoOn simultaneousapproximation for Baskakov-Durrmeyer-Stancu type opera-torsrdquo Journal of Ultra Scientist of Physical Sciences vol 24 no3 pp 567ndash577 2012
[7] H H Khan Approximation of classes of function [PhD thesis]AMU Aligarh India 1974
6 Journal of Difference Equations
[8] X M Zeng and F C Cheng ldquoOn the rates of approximation ofBernstein type operatorsrdquo Journal of ApproximationTheory vol109 no 2 pp 242ndash256 2001
[9] M Mursaleen V Karakaya M Erturk and F GursoyldquoWeighted statistical convergence and its application toKorovkin type approximation theoremrdquo Applied Mathematicsand Computation vol 218 no 18 pp 9132ndash9137 2012
[10] D D Stancu ldquoApproximation of function by a new class ofpolynomial operatorsrdquoRevue Roumaine deMathematique Pureset Appliquees vol 3 no 8 pp 1173ndash1194 1968
[11] R Yang J Xiong and F Cao ldquoMultivariate Stancu operatorsdefined on a simplexrdquo Applied Mathematics and Computationvol 138 no 2-3 pp 189ndash198 2003
[12] H M Srivastava M Mursaleen and A Khan ldquoGeneralizedequi-statistical convergence of positive linear operators andassociated approximation theoremsrdquo Mathematical and Com-puter Modelling vol 55 no 9-10 pp 2040ndash2051 2012
[13] M Mursaleen A Khan H M Srivastava and K S NisarldquoOperators constructed by means of q-Lagrange polynomialsand A-statistical approximationrdquo Applied Mathematics andComputation vol 219 no 12 pp 6911ndash6918 2013
[14] V N Mishra and P Patel ldquoApproximation by the Durrmeyer-Baskakov-Stancu operatorsrdquo Lobachevskii Journal ofMathemat-ics vol 34 no 3 pp 272ndash281 2013
[15] V N Mishra K Khatri and L N Mishra ldquoInverse result insimultaneous approximation by Baskakov-Durrmeyer-Stancuoperatorsrdquo Journal of Inequalities and Applications vol 2013 no586 2013
[16] V N Mishra and P Patel ldquoSome approximation properties ofmodified jain-beta operatorsrdquo Journal of Calculus of Variationsvol 2013 Article ID 489249 8 pages 2013
[17] V N Mishra K Khatri and L N Mishra ldquoStatisticalapproximation by Kantorovich-type discrete q-Beta operatorsrdquoAdvances in Difference Equations vol 345 no 1 2013
[18] V N Mishra and P Patel ldquoOn simultaneous approximationfor generalized integral typerdquo Baskakov Operator Submitted forpublication
Submit your manuscripts athttpwwwhindawicom
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
2 Journal of Difference Equations
We denote 120601119899120574(119909 119905) = int
119905
0
119882119899120574(119909 119904)119889119904 then in particular
we have
120601119899120574(119909infin) = int
infin
0
119882119899120574(119909 119904) 119889119904 = 1 (3)
By 119863119861119903(0infin) 119903 ge 0 we denote the class of absolutely
continuous functions 119891 defined on the interval (0infin) suchthat
(i) 119891(119905) = 119874(119905119903) 119905 rarr infin(ii) having a derivative 1198911015840 on the interval (0infin) coincid-
ing ae with a function which is of bounded variationon every finite subinterval of (0infin)
It can be observed that all function 119891 isin 119861119863119903(0infin) possess
for each 119888 gt 0 a representation
119891 (119909) = 119891 (119888) + int
119909
119888
120595 (119905) 119889119905 119909 ge 119888 (4)
2 Rate of Convergence for 119861119899120574
Lemma 1 (see [1]) Let the function 119879119899119898120574
(119909)119898 isin N cup 0 bedefined as
119879119899119898120574
(119909) = 119861119899120574((119905 minus 119909)
119898
119909)
=
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (119905 minus 119909)119898
119889119905
+ (1 + 120574119909)minus119899120574
(minus119909)119898
(5)
Then it is easily verified that for each 119909 isin (0infin) 1198791198990120574
(119909) = 11198791198991120574
(119909) = 0 and 1198791198992120574
(119909) = 2119909(1 + 120574119909)(119899 minus 120574) and also thefollowing recurrence relation holds
(119899 minus 120574119898)119879119899119898+1120574
(119909) = 119909 (1 + 120574119909)
times [119879(1)
119899119898120574(119909) + 2119898119879
119899119898minus1120574(119909)]
+ 119898 (1 + 2120574119909) 119879119899119898120574
(119909)
(6)
From the recurrence relation it can be easily be verified that forall 119909 isin [0infin) we have 119879
119899119898120574(119909) = 119874(119899
minus[(119898+1)2]
)
Remark 2 FromLemma 1 usingCauchy-Schwarz inequalityit follows that
119861119899120574(|119905 minus 119909| 119909) le [119861
119899120574((119905 minus 119909)
2
119909)]
12
le radic
2119909 (1 + 120574119909)
119899 minus 120574
(7)
Lemma 3 Let 119909 isin (0infin) and119882119899120574(119909 119905) be the kernel defined
in (1) Then for 119899 being sufficiently large one has
(a) 120601119899120574(119909 119910) = int
119910
0
119882119899120574(119909 119905)119889119905 le 2119909(1 + 120574119909)(119899 minus
120574)(119909 minus 119910)2
0 le 119910 lt 119909
(b) 1 minus 120601119899120574(119909 119911) = int
infin
119911
119882119899120574(119909 119905)119889119905 le 2119909(1 + 120574119909)(119899 minus
120574)(119911 minus 119909)2
119909 lt 119911 lt infin
Proof First we prove (a) by using Lemma 1 we have
int
119910
0
119882119899120574(119909 119905) 119889119905 le int
119910
0
(119909 minus 119905)2
(119909 minus 119910)2119882119899120574(119909 119905) 119889119905
le (119909 minus 119910)minus2
1198791198992120574
(119909)
le
2119909 (1 + 120574119909)
(119899 minus 120574) (119909 minus 119910)2
(8)
The proof of (b) is similar we omit the details
Theorem 4 Let 119891 isin 119863119861119903(0infin) 119903 isin N and 119909 isin (0infin) Then
for 119899 being sufficiently large we have10038161003816100381610038161003816119861119899120574(119891 119909) minus 119891 (119909)
10038161003816100381610038161003816
le
2 (1 + 120574119909)
119899 minus 120574
[radic119899]
sum
119896=1
119909+119909119896
⋁
119909minus119909119896
((1198911015840
)119909
)
+
119909
radic119899
119909+119909radic119899
⋁
119909minus119909radic119899
((1198911015840
)119909
) +
2 (1 + 120574119909)
(119899 minus 120574) 119909
times [
10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891
1015840
(119909+
)
10038161003816100381610038161003816+1003816100381610038161003816119891 (119909)
1003816100381610038161003816]
+ radic
2119909 (1 + 120574119909)
119899 minus 120574
[1198722119903
119874(119899minus1199032
) +
100381610038161003816100381610038161198911015840
(119909+
)
10038161003816100381610038161003816]
+ radic
119909 (1 + 120574119909)
2 (119899 minus 120574)
100381610038161003816100381610038161198911015840
(119909+
) minus 1198911015840
(119909minus
)
10038161003816100381610038161003816
(9)
where the auxiliary function 119891119909is given by
119891119909(119905) =
119891 (119905) minus 119891 (119909minus
) 0 le 119905 lt 119909
0 119905 = 119909
119891 (119905) minus 119891 (119909+
) 119909 lt 119905 lt infin
(10)
⋁119887
119886119891(119909) denotes the total variation of 119891
119909on [119886 119887]
Proof By the application of mean value theorem we have
119861119899120574(119891 119909) minus 119891 (119909) = int
infin
0
119882119899120574(119909 119905) (119891 (119905) minus 119891 (119909)) 119889119905
= int
infin
0
int
119905
119909
119882119899120574(119909 119905) (119891
1015840
(119906) 119889119906) 119889119905
(11)
Also using the identity
1198911015840
(119906) =
1198911015840
(119909+
) + 1198911015840
(119909minus
)
2
+ (1198911015840
)119909
(119906)
+
1198911015840
(119909+
) minus 1198911015840
(119909minus
)
2
sgn (119906 minus 119909)
+ [1198911015840
(119909) minus
1198911015840
(119909+
) + 1198911015840
(119909minus
)
2
] 120594119909(119906)
(12)
Journal of Difference Equations 3
where
120594119909(119906) =
1 119906 = 119909
0 119906 = 119909
(13)
we can see that
int
infin
0
(int
119905
119909
(1198911015840
(119909) minus
1198911015840
(119909+
) + 1198911015840
(119909minus
)
2
)120594119909(119906) 119889119906)
times119882119899120574(119905 119909) 119889119905 = 0
(14)
Also
int
infin
0
(int
119905
119909
(
1198911015840
(119909+
) minus 1198911015840
(119909minus
)
2
) sgn (119906 minus 119909) 119889119906)119882119899120574(119905 119909) 119889119905
=
1198911015840
(119909+
) minus 1198911015840
(119909minus
)
2
119861119899120574(|119905 minus 119909| 119909)
int
infin
0
(int
119905
119909
(
1198911015840
(119909+
) + 1198911015840
(119909minus
)
2
) 119889119906)119882119899120574(119905 119909) 119889119905
=
1198911015840
(119909+
) + 1198911015840
(119909minus
)
2
119861119899120574(119905 minus 119909 119909)
(15)
Substitute value of 1198911015840(119906) from (12) in (11) and using (14) and(15) we get
10038161003816100381610038161003816119861119899120574(119891 119909) minus 119891 (119909)
10038161003816100381610038161003816le
10038161003816100381610038161003816100381610038161003816
int
infin
119909
(int
119905
119909
(1198911015840
)119909
(119906) 119889119906)119882119899120574(119909 119905) 119889119905
10038161003816100381610038161003816100381610038161003816
+
10038161003816100381610038161003816100381610038161003816
int
119909
0
(int
119905
119909
(1198911015840
)119909
(119906) 119889119906)119882119899120574(119909 119905) 119889119905
10038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161198911015840
(119909+
) minus 1198911015840
(119909minus
)
10038161003816100381610038161003816
2
119861119899120574(|119905 minus 119909| 119909)
+
100381610038161003816100381610038161198911015840
(119909+
) + 1198911015840
(119909minus
)
10038161003816100381610038161003816
2
119861119899120574(119905 minus 119909 119909)
(16)
Using Lemma 1 and Remark 2 we obtain
10038161003816100381610038161003816119861119899120574(119891 119909) minus 119891 (119909)
10038161003816100381610038161003816le
10038161003816100381610038161003816100381610038161003816
int
infin
119909
(int
119905
119909
(1198911015840
)119909
(119906) 119889119906)119882119899120574(119909 119905) 119889119905
10038161003816100381610038161003816100381610038161003816
+
10038161003816100381610038161003816100381610038161003816
int
119909
0
(int
119905
119909
(1198911015840
)119909
(119906) 119889119906)119882119899120574(119909 119905) 119889119905
10038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161198911015840
(119909+
) minus 1198911015840
(119909minus
)
10038161003816100381610038161003816
2
radic
2119909 (1 + 120574119909)
119899 minus 120574
=
10038161003816100381610038161003816119875119899120574(119891 119909)
10038161003816100381610038161003816+
10038161003816100381610038161003816119876119899120574(119891 119909)
10038161003816100381610038161003816
+
100381610038161003816100381610038161198911015840
(119909+
) minus 1198911015840
(119909minus
)
10038161003816100381610038161003816
2
radic
2119909 (1 + 120574119909)
119899 minus 120574
(17)
On applying Lemma 3 with 119910 = 119909 minus 119909radic119899 and integrating byparts we have
10038161003816100381610038161003816119876119899120574(119891 119909)
10038161003816100381610038161003816=
10038161003816100381610038161003816100381610038161003816
int
119909
0
int
119905
119909
(1198911015840
)119909
(119906) 119889119906 119889119905 (120601119899120574(119909 119910))
10038161003816100381610038161003816100381610038161003816
=
10038161003816100381610038161003816100381610038161003816
int
119909
0
120601119899120574(119909 119910) (119891
1015840
)119909
(119905) 119889119905
10038161003816100381610038161003816100381610038161003816
le (int
119910
0
+int
119909
119910
)
10038161003816100381610038161003816(1198911015840
)119909
(119905)
10038161003816100381610038161003816
10038161003816100381610038161003816120601119899120574(119909 119905)
10038161003816100381610038161003816119889119905
le
2119909 (1 + 120574119909)
119899 minus 120574
int
119910
0
119909
⋁
119905
((1198911015840
)119909
)
1
(119909 minus 119905)2119889119905
+ int
119909
119910
119909
⋁
119905
((1198911015840
)119909
) 119889119905
le
2119909 (1 + 120574119909)
119899 minus 120574
int
119910
0
119909
⋁
119905
((1198911015840
)119909
)
1
(119909 minus 119905)2119889119905
+
119909
radic119899
119909
⋁
119909minus119909radic119899
((1198911015840
)119909
)
=
2119909 (1 + 120574119909)
119899 minus 120574
int
radic119899
1
119909
⋁
119909minus119909119906
((1198911015840
)119909
) 119889119906
+
119909
radic119899
119909
⋁
119909minus119909radic119899
((1198911015840
)119909
)
le
2119909 (1 + 120574119909)
119899 minus 120574
[radic119899]
sum
119896=1
119909
⋁
119909minus119909119896
((1198911015840
)119909
)
+
119909
radic119899
119909
⋁
119909minus119909radic119899
((1198911015840
)119909
)
(18)
where 119906 = 119909(119909 minus 119905)On the other hand we have10038161003816100381610038161003816119875119899120574(119891 119909)
10038161003816100381610038161003816
=
10038161003816100381610038161003816100381610038161003816
int
infin
119909
(int
119905
119909
(1198911015840
)119909
(119906) 119889119906)119882119899120574(119909 119905) 119889119905
10038161003816100381610038161003816100381610038161003816
=
10038161003816100381610038161003816100381610038161003816
int
infin
2119909
(int
119905
119909
(1198911015840
)119909
(119906) 119889119906)119882119899120574(119909 119905) 119889119905
+int
2119909
119909
(int
119905
119909
(1198911015840
)119909
(119906) 119889119906)119889119905 (1 minus 120601119899120574(119909 119905))
100381610038161003816100381610038161003816100381610038161003816
le
10038161003816100381610038161003816100381610038161003816
int
infin
2119909
(119891 (119905) minus 119891 (119909))119882119899120574(119909 119905) 119889119905
10038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161198911015840
(119909+
)
10038161003816100381610038161003816
10038161003816100381610038161003816100381610038161003816
int
infin
2119909
(119905 minus 119909)119882119899120574(119909 119905) 119889119905
10038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
int
2119909
119909
(1198911015840
)119909
(119906) 119889119906
100381610038161003816100381610038161003816100381610038161003816
100381610038161003816100381610038161 minus 120601119909120574(119909 2119909)
10038161003816100381610038161003816
4 Journal of Difference Equations
+ int
2119909
119909
10038161003816100381610038161003816(1198911015840
)119909
(119905)
10038161003816100381610038161003816
100381610038161003816100381610038161 minus 120601119909120574(119909 119905)
10038161003816100381610038161003816119889119905
le [
119872
119909
int
infin
2119909
119882119899120574(119909 119905) 119905
119903
|119905 minus 119909| 119889119905
+
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
1199092
int
infin
2119909
119882119899120574(119909 119905) (119905 minus 119909)
2
119889119905]
+ [
100381610038161003816100381610038161198911015840
(119909+
)
10038161003816100381610038161003816int
infin
2119909
119882119899120574(119909 119905) |119905 minus 119909| 119889119905]
+
2 (1 + 120574119909)
(119899 minus 120574) 119909
10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891
1015840
(119909+
)
10038161003816100381610038161003816
+
2 (1 + 120574119909)
119899 minus 120574
[radic119899]
sum
119896=1
119909+119909119896
⋁
119909
((1198911015840
)119909
)
+
119909
radic119899
119909+119909radic119899
⋁
119909
((1198911015840
)119909
)
= 119877119899120574(119891 119909) + 119878
119899120574(119891 119909)
+
2 (1 + 120574119909)
(119899 minus 120574) 119909
10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891
1015840
(119909+
)
10038161003816100381610038161003816
+
2 (1 + 120574119909)
119899 minus 120574
[radic119899]
sum
119896=1
119909+119909119896
⋁
119909
((1198911015840
)119909
)
+
119909
radic119899
119909+119909radic119899
⋁
119909
((1198911015840
)119909
)
(19)
Applying Holderrsquos inequality Remark 2 and Lemma 1 wehave
119877119899120574(119891 119909) le
119872
119909
(int
infin
2119909
119882119899120574(119909 119905) 119905
2119903
119889119905)
12
times(int
infin
0
119882119899120574(119909 119905) (119905 minus 119909)
2
119889119905)
12
+
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
1199092
int
infin
2119909
119882119899120574(119909 119905) (119905 minus 119909)
2
119889119905
le 1198722119903
119874(119899minus1199032
)radic
2119909 (1 + 120574119909)
119899 minus 120574
+1003816100381610038161003816119891 (119909)
1003816100381610038161003816
2 (1 + 120574119909)
(119899 minus 120574) 119909
(20)
Also
119878119899120574(119891 119909) le
1003816100381610038161003816119891 (119909+
)1003816100381610038161003816int
infin
0
119882119899(119905 119909) |119905 minus 119909| 119889119905
le1003816100381610038161003816119891 (119909+
)1003816100381610038161003816radic
2119909 (1 + 120574119909)
119899 minus 120574
(21)
Combining the estimates (17)ndash(21) we get the desired resultsThis completes the proof of Theorem
3 Rate of Convergence for Stancu TypeGeneralization of 119861
119899120574
In 1968 Stancu introduces Bernstein-Stancu operators in[10] a sequence of the linear positive operators depending ontwo parameters 120572 and 120573 satisfying the condition 0 le 120572 le 120573Recently many researchers applied this approach to manyoperators for details see [11ndash17] For 119891 isin 119862[0infin) Stancugeneralization of operators (1) is as follows
119861120572120573
119899120574(119891 (119905) 119909) =
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) 119891(
119899119905 + 120572
119899 + 120573
)119889119905
+ 1199011198990120574
(119909) 119891(
120572
119899 + 120573
)
= int
infin
0
119882119899120574(119909 119905) 119891(
119899119905 + 120572
119899 + 120573
)119889119905
(22)
where 119901119899119896120574
(119909) 119887119899119896120574
(119909) and119882119899120574(119909 119905)are as defined in (1)
Lemma5 (see [18]) If we define the centralmoments for every119898 isin N as
120583120572120573
119899119898120574(119909) = 119861
120572120573
119899120574((119905 minus 119909)
119898
119909)
=
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (
119899119905 + 120572
119899 + 120573
minus 119909)
119898
119889119905
+ 1199011198990120574
(119909) (
120572
119899 + 120573
minus 119909)
119898
(23)
then 1205831205721205731198990120574
(119909) = 1 1205831205721205731198991120574
(119909) = (120572 minus 120573119909)(119899 + 120573) and
120583120572120573
1198992120574(119909) =
1205722
(119899 + 120573)2+ 119909
(21198992
minus 2119899120572120573 + 2120572120573120574)
(119899 + 120573)2
(119899 minus 120574)
+ 1199092(1198991205732
+ 21198992
120574 minus 1205732
120574)
(119899 + 120573)2
(119899 minus 120574)
(24)
For 119899 gt 119898 we have the following recurrence relation
(119899 minus 120574119898) (119899 + 120573) 120583120572120573
119899119898+1120574(119909)
= 119899119909 (1 + 120574119909) [(120583120572120573
119899119898120574)
(1)
(119909) + 119898120583120572120573
119899119898minus1120574(119909)]
+ [119898119899 + 1198992
119909 minus (2120574119898 minus 119899) (120572 minus (119899 + 120573) 119909)] 120583120572120573
119899119898120574(119909)
+ [119898120574 (119899 + 120573) (
120572
119899 + 120573
minus 119909)
2
minus 119898119899(
120572
119899 + 120573
minus 119909)]
times 120583120572120573
119899119898minus1120574(119909)
(25)
Journal of Difference Equations 5
From the recurrence relation it can be easily verified that forall 119909 isin [0infin) we have 120583120572120573
119899119898120574(119909) = 119874(119899
minus[(119898+1)2]
)
Remark 6 Observe that 119861120572120573119899120574
preserve constant functions butnot linear functions If 120572 = 120573 = 0 these operators reduce tothe operators defined in (1) Notice that
120583120572120573
1198992120574(119909) = [
1205732
120574(119899 + 120573)2+
21198992
(119899 + 120573)2
(119899 minus 120574)
] 119909 (1 + 120574119909)
+
120573 [(2120572 + 120573) 120574 minus 119899 (120573 + 2120572120574)]
120574(119899 + 120573)2
(119899 minus 120574)
119909 +
1205722
(119899 + 120573)2
(26)
Remark 7 From Lemma 3 taking 119899 to be sufficiently largeand 119909 isin (0infin) we observe that
120583120572120573
1198992120574(119909) le
119862119909 (1 + 120574119909)
119899 minus 120574
(27)
where 119862 is positive constant
Remark 8 Applying the Cauchy-Schwarz inequality andkeeping the same condition as in Remark 7 for 119909 119899 and 119862we derive from Lemma 5 that
119861120572120573
119899120574(|119905 minus 119909| 119909) le [119861
120572120573
119899120574((119905 minus 119909)
2
119909)]
12
le radic
119862119909 (1 + 120574119909)
119899 minus 120574
(28)
Lemma 9 Let 119909 isin (0infin) and119882119899120574(119909 119905) be the kernel defined
in (1) Then for 119899 being sufficiently large we have
(c) 120601119899120574(119909 119910) = int
119910
0
119882119899120574(119909 119905)119889119905 = 119862119909(1 + 120574119909)(119899 minus
120574)(119909 minus 119910)2
0 le 119910 lt 119909
(d) 1 minus 120601119899120574(119909 119911) = int
infin
119911
119882119899120574(119909 119905)119889119905 = 119862119909(1 + 120574119909)(119899 minus
120574)(119911 minus 119909)2
119909 lt 119911 lt infin
The proof is the same as Lemma 3 thus we omit the details
Theorem 10 Let 119891 isin 119863119861119903(0infin) 119903 isin N and 119909 isin (0infin)
Then for 119899 being sufficiently large we have
10038161003816100381610038161003816119861120572120573
119899120574(119891 119909) minus 119891 (119909)
10038161003816100381610038161003816
le
119862 (1 + 120574119909)
119899 minus 120574
[radic119899]
sum
119896=1
119909+119909119896
⋁
119909minus119909119896
((1198911015840
)119909
)
+
119909
radic119899
119909+119909radic119899
⋁
119909minus119909radic119899
((1198911015840
)119909
) +
119862 (1 + 120574119909)
(119899 minus 120574) 119909
times [
10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891
1015840
(119909+
)
10038161003816100381610038161003816+1003816100381610038161003816119891 (119909)
1003816100381610038161003816]
+ radic
119862119909 (1 + 120574119909)
119899 minus 120574
times [11987212119903
119874(119899minus1199032
) +
100381610038161003816100381610038161198911015840
(119909+
)
10038161003816100381610038161003816]
+
1
2
radic
119862119909 (1 + 120574119909)
(119899 minus 120574)
100381610038161003816100381610038161198911015840
(119909+
) minus 1198911015840
(119909minus
)
10038161003816100381610038161003816
+
1
2
100381610038161003816100381610038161198911015840
(119909+
) + 1198911015840
(119909minus
)
10038161003816100381610038161003816
1003816100381610038161003816120572 minus 120573119909
1003816100381610038161003816
119899 + 120573
(29)
where the auxiliary functions 119891119909and ⋁119887
119886119891(119909) were defined in
Theorem 4The proof of the above theorem follows along the lines of
Theorem 4 thus we omit the details
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are thankful to the anonymous referee formaking valuable comments leading to the better presen-tation of the paper Special thanks are due to ProfessorDr Abdelalim A Elsadany Editor of Journal of DifferenceEquations for kind cooperation and smooth behavior duringcommunication and for his efforts to send the reports of thepaper timely
References
[1] V Gupta ldquoApproximation for modified Baskakov Durrmeyertype operatorsrdquo The Rocky Mountain Journal of Mathematicsvol 39 no 3 pp 825ndash841 2009
[2] V Gupta M A Noor M S Beniwal and M K Gupta ldquoOnsimultaneous approximation for certain Baskakov Durrmeyertype operatorsrdquo Journal of Inequalities in Pure and AppliedMathematics vol 7 pp 1ndash15 2006
[3] V Gupta and P N Agrawal ldquoRate of convergence for certainBaskakov Durrmeyer type operatorsrdquoAnnals of Oradea Univer-sity Fascicola Matematica vol 14 pp 33ndash39 2007
[4] HKarsli andE Ibikli ldquoConvergence rate of a newBezier variantof Chlodowsky operators to bounded variation functionsrdquoJournal of Computational and AppliedMathematics vol 212 no2 pp 431ndash443 2008
[5] V N Mishra H H Khan K Khatri and L N Mishra ldquoHyper-geometric representation for Baskakov-Durrmeyer-Stancu typeoperatorsrdquo Bulletin of Mathematical Analysis and Applicationsvol 5 no 3 pp 18ndash26 2013
[6] V N Mishra K Khatri and L N Mishra ldquoOn simultaneousapproximation for Baskakov-Durrmeyer-Stancu type opera-torsrdquo Journal of Ultra Scientist of Physical Sciences vol 24 no3 pp 567ndash577 2012
[7] H H Khan Approximation of classes of function [PhD thesis]AMU Aligarh India 1974
6 Journal of Difference Equations
[8] X M Zeng and F C Cheng ldquoOn the rates of approximation ofBernstein type operatorsrdquo Journal of ApproximationTheory vol109 no 2 pp 242ndash256 2001
[9] M Mursaleen V Karakaya M Erturk and F GursoyldquoWeighted statistical convergence and its application toKorovkin type approximation theoremrdquo Applied Mathematicsand Computation vol 218 no 18 pp 9132ndash9137 2012
[10] D D Stancu ldquoApproximation of function by a new class ofpolynomial operatorsrdquoRevue Roumaine deMathematique Pureset Appliquees vol 3 no 8 pp 1173ndash1194 1968
[11] R Yang J Xiong and F Cao ldquoMultivariate Stancu operatorsdefined on a simplexrdquo Applied Mathematics and Computationvol 138 no 2-3 pp 189ndash198 2003
[12] H M Srivastava M Mursaleen and A Khan ldquoGeneralizedequi-statistical convergence of positive linear operators andassociated approximation theoremsrdquo Mathematical and Com-puter Modelling vol 55 no 9-10 pp 2040ndash2051 2012
[13] M Mursaleen A Khan H M Srivastava and K S NisarldquoOperators constructed by means of q-Lagrange polynomialsand A-statistical approximationrdquo Applied Mathematics andComputation vol 219 no 12 pp 6911ndash6918 2013
[14] V N Mishra and P Patel ldquoApproximation by the Durrmeyer-Baskakov-Stancu operatorsrdquo Lobachevskii Journal ofMathemat-ics vol 34 no 3 pp 272ndash281 2013
[15] V N Mishra K Khatri and L N Mishra ldquoInverse result insimultaneous approximation by Baskakov-Durrmeyer-Stancuoperatorsrdquo Journal of Inequalities and Applications vol 2013 no586 2013
[16] V N Mishra and P Patel ldquoSome approximation properties ofmodified jain-beta operatorsrdquo Journal of Calculus of Variationsvol 2013 Article ID 489249 8 pages 2013
[17] V N Mishra K Khatri and L N Mishra ldquoStatisticalapproximation by Kantorovich-type discrete q-Beta operatorsrdquoAdvances in Difference Equations vol 345 no 1 2013
[18] V N Mishra and P Patel ldquoOn simultaneous approximationfor generalized integral typerdquo Baskakov Operator Submitted forpublication
Submit your manuscripts athttpwwwhindawicom
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
Journal of Difference Equations 3
where
120594119909(119906) =
1 119906 = 119909
0 119906 = 119909
(13)
we can see that
int
infin
0
(int
119905
119909
(1198911015840
(119909) minus
1198911015840
(119909+
) + 1198911015840
(119909minus
)
2
)120594119909(119906) 119889119906)
times119882119899120574(119905 119909) 119889119905 = 0
(14)
Also
int
infin
0
(int
119905
119909
(
1198911015840
(119909+
) minus 1198911015840
(119909minus
)
2
) sgn (119906 minus 119909) 119889119906)119882119899120574(119905 119909) 119889119905
=
1198911015840
(119909+
) minus 1198911015840
(119909minus
)
2
119861119899120574(|119905 minus 119909| 119909)
int
infin
0
(int
119905
119909
(
1198911015840
(119909+
) + 1198911015840
(119909minus
)
2
) 119889119906)119882119899120574(119905 119909) 119889119905
=
1198911015840
(119909+
) + 1198911015840
(119909minus
)
2
119861119899120574(119905 minus 119909 119909)
(15)
Substitute value of 1198911015840(119906) from (12) in (11) and using (14) and(15) we get
10038161003816100381610038161003816119861119899120574(119891 119909) minus 119891 (119909)
10038161003816100381610038161003816le
10038161003816100381610038161003816100381610038161003816
int
infin
119909
(int
119905
119909
(1198911015840
)119909
(119906) 119889119906)119882119899120574(119909 119905) 119889119905
10038161003816100381610038161003816100381610038161003816
+
10038161003816100381610038161003816100381610038161003816
int
119909
0
(int
119905
119909
(1198911015840
)119909
(119906) 119889119906)119882119899120574(119909 119905) 119889119905
10038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161198911015840
(119909+
) minus 1198911015840
(119909minus
)
10038161003816100381610038161003816
2
119861119899120574(|119905 minus 119909| 119909)
+
100381610038161003816100381610038161198911015840
(119909+
) + 1198911015840
(119909minus
)
10038161003816100381610038161003816
2
119861119899120574(119905 minus 119909 119909)
(16)
Using Lemma 1 and Remark 2 we obtain
10038161003816100381610038161003816119861119899120574(119891 119909) minus 119891 (119909)
10038161003816100381610038161003816le
10038161003816100381610038161003816100381610038161003816
int
infin
119909
(int
119905
119909
(1198911015840
)119909
(119906) 119889119906)119882119899120574(119909 119905) 119889119905
10038161003816100381610038161003816100381610038161003816
+
10038161003816100381610038161003816100381610038161003816
int
119909
0
(int
119905
119909
(1198911015840
)119909
(119906) 119889119906)119882119899120574(119909 119905) 119889119905
10038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161198911015840
(119909+
) minus 1198911015840
(119909minus
)
10038161003816100381610038161003816
2
radic
2119909 (1 + 120574119909)
119899 minus 120574
=
10038161003816100381610038161003816119875119899120574(119891 119909)
10038161003816100381610038161003816+
10038161003816100381610038161003816119876119899120574(119891 119909)
10038161003816100381610038161003816
+
100381610038161003816100381610038161198911015840
(119909+
) minus 1198911015840
(119909minus
)
10038161003816100381610038161003816
2
radic
2119909 (1 + 120574119909)
119899 minus 120574
(17)
On applying Lemma 3 with 119910 = 119909 minus 119909radic119899 and integrating byparts we have
10038161003816100381610038161003816119876119899120574(119891 119909)
10038161003816100381610038161003816=
10038161003816100381610038161003816100381610038161003816
int
119909
0
int
119905
119909
(1198911015840
)119909
(119906) 119889119906 119889119905 (120601119899120574(119909 119910))
10038161003816100381610038161003816100381610038161003816
=
10038161003816100381610038161003816100381610038161003816
int
119909
0
120601119899120574(119909 119910) (119891
1015840
)119909
(119905) 119889119905
10038161003816100381610038161003816100381610038161003816
le (int
119910
0
+int
119909
119910
)
10038161003816100381610038161003816(1198911015840
)119909
(119905)
10038161003816100381610038161003816
10038161003816100381610038161003816120601119899120574(119909 119905)
10038161003816100381610038161003816119889119905
le
2119909 (1 + 120574119909)
119899 minus 120574
int
119910
0
119909
⋁
119905
((1198911015840
)119909
)
1
(119909 minus 119905)2119889119905
+ int
119909
119910
119909
⋁
119905
((1198911015840
)119909
) 119889119905
le
2119909 (1 + 120574119909)
119899 minus 120574
int
119910
0
119909
⋁
119905
((1198911015840
)119909
)
1
(119909 minus 119905)2119889119905
+
119909
radic119899
119909
⋁
119909minus119909radic119899
((1198911015840
)119909
)
=
2119909 (1 + 120574119909)
119899 minus 120574
int
radic119899
1
119909
⋁
119909minus119909119906
((1198911015840
)119909
) 119889119906
+
119909
radic119899
119909
⋁
119909minus119909radic119899
((1198911015840
)119909
)
le
2119909 (1 + 120574119909)
119899 minus 120574
[radic119899]
sum
119896=1
119909
⋁
119909minus119909119896
((1198911015840
)119909
)
+
119909
radic119899
119909
⋁
119909minus119909radic119899
((1198911015840
)119909
)
(18)
where 119906 = 119909(119909 minus 119905)On the other hand we have10038161003816100381610038161003816119875119899120574(119891 119909)
10038161003816100381610038161003816
=
10038161003816100381610038161003816100381610038161003816
int
infin
119909
(int
119905
119909
(1198911015840
)119909
(119906) 119889119906)119882119899120574(119909 119905) 119889119905
10038161003816100381610038161003816100381610038161003816
=
10038161003816100381610038161003816100381610038161003816
int
infin
2119909
(int
119905
119909
(1198911015840
)119909
(119906) 119889119906)119882119899120574(119909 119905) 119889119905
+int
2119909
119909
(int
119905
119909
(1198911015840
)119909
(119906) 119889119906)119889119905 (1 minus 120601119899120574(119909 119905))
100381610038161003816100381610038161003816100381610038161003816
le
10038161003816100381610038161003816100381610038161003816
int
infin
2119909
(119891 (119905) minus 119891 (119909))119882119899120574(119909 119905) 119889119905
10038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161198911015840
(119909+
)
10038161003816100381610038161003816
10038161003816100381610038161003816100381610038161003816
int
infin
2119909
(119905 minus 119909)119882119899120574(119909 119905) 119889119905
10038161003816100381610038161003816100381610038161003816
+
100381610038161003816100381610038161003816100381610038161003816
int
2119909
119909
(1198911015840
)119909
(119906) 119889119906
100381610038161003816100381610038161003816100381610038161003816
100381610038161003816100381610038161 minus 120601119909120574(119909 2119909)
10038161003816100381610038161003816
4 Journal of Difference Equations
+ int
2119909
119909
10038161003816100381610038161003816(1198911015840
)119909
(119905)
10038161003816100381610038161003816
100381610038161003816100381610038161 minus 120601119909120574(119909 119905)
10038161003816100381610038161003816119889119905
le [
119872
119909
int
infin
2119909
119882119899120574(119909 119905) 119905
119903
|119905 minus 119909| 119889119905
+
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
1199092
int
infin
2119909
119882119899120574(119909 119905) (119905 minus 119909)
2
119889119905]
+ [
100381610038161003816100381610038161198911015840
(119909+
)
10038161003816100381610038161003816int
infin
2119909
119882119899120574(119909 119905) |119905 minus 119909| 119889119905]
+
2 (1 + 120574119909)
(119899 minus 120574) 119909
10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891
1015840
(119909+
)
10038161003816100381610038161003816
+
2 (1 + 120574119909)
119899 minus 120574
[radic119899]
sum
119896=1
119909+119909119896
⋁
119909
((1198911015840
)119909
)
+
119909
radic119899
119909+119909radic119899
⋁
119909
((1198911015840
)119909
)
= 119877119899120574(119891 119909) + 119878
119899120574(119891 119909)
+
2 (1 + 120574119909)
(119899 minus 120574) 119909
10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891
1015840
(119909+
)
10038161003816100381610038161003816
+
2 (1 + 120574119909)
119899 minus 120574
[radic119899]
sum
119896=1
119909+119909119896
⋁
119909
((1198911015840
)119909
)
+
119909
radic119899
119909+119909radic119899
⋁
119909
((1198911015840
)119909
)
(19)
Applying Holderrsquos inequality Remark 2 and Lemma 1 wehave
119877119899120574(119891 119909) le
119872
119909
(int
infin
2119909
119882119899120574(119909 119905) 119905
2119903
119889119905)
12
times(int
infin
0
119882119899120574(119909 119905) (119905 minus 119909)
2
119889119905)
12
+
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
1199092
int
infin
2119909
119882119899120574(119909 119905) (119905 minus 119909)
2
119889119905
le 1198722119903
119874(119899minus1199032
)radic
2119909 (1 + 120574119909)
119899 minus 120574
+1003816100381610038161003816119891 (119909)
1003816100381610038161003816
2 (1 + 120574119909)
(119899 minus 120574) 119909
(20)
Also
119878119899120574(119891 119909) le
1003816100381610038161003816119891 (119909+
)1003816100381610038161003816int
infin
0
119882119899(119905 119909) |119905 minus 119909| 119889119905
le1003816100381610038161003816119891 (119909+
)1003816100381610038161003816radic
2119909 (1 + 120574119909)
119899 minus 120574
(21)
Combining the estimates (17)ndash(21) we get the desired resultsThis completes the proof of Theorem
3 Rate of Convergence for Stancu TypeGeneralization of 119861
119899120574
In 1968 Stancu introduces Bernstein-Stancu operators in[10] a sequence of the linear positive operators depending ontwo parameters 120572 and 120573 satisfying the condition 0 le 120572 le 120573Recently many researchers applied this approach to manyoperators for details see [11ndash17] For 119891 isin 119862[0infin) Stancugeneralization of operators (1) is as follows
119861120572120573
119899120574(119891 (119905) 119909) =
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) 119891(
119899119905 + 120572
119899 + 120573
)119889119905
+ 1199011198990120574
(119909) 119891(
120572
119899 + 120573
)
= int
infin
0
119882119899120574(119909 119905) 119891(
119899119905 + 120572
119899 + 120573
)119889119905
(22)
where 119901119899119896120574
(119909) 119887119899119896120574
(119909) and119882119899120574(119909 119905)are as defined in (1)
Lemma5 (see [18]) If we define the centralmoments for every119898 isin N as
120583120572120573
119899119898120574(119909) = 119861
120572120573
119899120574((119905 minus 119909)
119898
119909)
=
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (
119899119905 + 120572
119899 + 120573
minus 119909)
119898
119889119905
+ 1199011198990120574
(119909) (
120572
119899 + 120573
minus 119909)
119898
(23)
then 1205831205721205731198990120574
(119909) = 1 1205831205721205731198991120574
(119909) = (120572 minus 120573119909)(119899 + 120573) and
120583120572120573
1198992120574(119909) =
1205722
(119899 + 120573)2+ 119909
(21198992
minus 2119899120572120573 + 2120572120573120574)
(119899 + 120573)2
(119899 minus 120574)
+ 1199092(1198991205732
+ 21198992
120574 minus 1205732
120574)
(119899 + 120573)2
(119899 minus 120574)
(24)
For 119899 gt 119898 we have the following recurrence relation
(119899 minus 120574119898) (119899 + 120573) 120583120572120573
119899119898+1120574(119909)
= 119899119909 (1 + 120574119909) [(120583120572120573
119899119898120574)
(1)
(119909) + 119898120583120572120573
119899119898minus1120574(119909)]
+ [119898119899 + 1198992
119909 minus (2120574119898 minus 119899) (120572 minus (119899 + 120573) 119909)] 120583120572120573
119899119898120574(119909)
+ [119898120574 (119899 + 120573) (
120572
119899 + 120573
minus 119909)
2
minus 119898119899(
120572
119899 + 120573
minus 119909)]
times 120583120572120573
119899119898minus1120574(119909)
(25)
Journal of Difference Equations 5
From the recurrence relation it can be easily verified that forall 119909 isin [0infin) we have 120583120572120573
119899119898120574(119909) = 119874(119899
minus[(119898+1)2]
)
Remark 6 Observe that 119861120572120573119899120574
preserve constant functions butnot linear functions If 120572 = 120573 = 0 these operators reduce tothe operators defined in (1) Notice that
120583120572120573
1198992120574(119909) = [
1205732
120574(119899 + 120573)2+
21198992
(119899 + 120573)2
(119899 minus 120574)
] 119909 (1 + 120574119909)
+
120573 [(2120572 + 120573) 120574 minus 119899 (120573 + 2120572120574)]
120574(119899 + 120573)2
(119899 minus 120574)
119909 +
1205722
(119899 + 120573)2
(26)
Remark 7 From Lemma 3 taking 119899 to be sufficiently largeand 119909 isin (0infin) we observe that
120583120572120573
1198992120574(119909) le
119862119909 (1 + 120574119909)
119899 minus 120574
(27)
where 119862 is positive constant
Remark 8 Applying the Cauchy-Schwarz inequality andkeeping the same condition as in Remark 7 for 119909 119899 and 119862we derive from Lemma 5 that
119861120572120573
119899120574(|119905 minus 119909| 119909) le [119861
120572120573
119899120574((119905 minus 119909)
2
119909)]
12
le radic
119862119909 (1 + 120574119909)
119899 minus 120574
(28)
Lemma 9 Let 119909 isin (0infin) and119882119899120574(119909 119905) be the kernel defined
in (1) Then for 119899 being sufficiently large we have
(c) 120601119899120574(119909 119910) = int
119910
0
119882119899120574(119909 119905)119889119905 = 119862119909(1 + 120574119909)(119899 minus
120574)(119909 minus 119910)2
0 le 119910 lt 119909
(d) 1 minus 120601119899120574(119909 119911) = int
infin
119911
119882119899120574(119909 119905)119889119905 = 119862119909(1 + 120574119909)(119899 minus
120574)(119911 minus 119909)2
119909 lt 119911 lt infin
The proof is the same as Lemma 3 thus we omit the details
Theorem 10 Let 119891 isin 119863119861119903(0infin) 119903 isin N and 119909 isin (0infin)
Then for 119899 being sufficiently large we have
10038161003816100381610038161003816119861120572120573
119899120574(119891 119909) minus 119891 (119909)
10038161003816100381610038161003816
le
119862 (1 + 120574119909)
119899 minus 120574
[radic119899]
sum
119896=1
119909+119909119896
⋁
119909minus119909119896
((1198911015840
)119909
)
+
119909
radic119899
119909+119909radic119899
⋁
119909minus119909radic119899
((1198911015840
)119909
) +
119862 (1 + 120574119909)
(119899 minus 120574) 119909
times [
10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891
1015840
(119909+
)
10038161003816100381610038161003816+1003816100381610038161003816119891 (119909)
1003816100381610038161003816]
+ radic
119862119909 (1 + 120574119909)
119899 minus 120574
times [11987212119903
119874(119899minus1199032
) +
100381610038161003816100381610038161198911015840
(119909+
)
10038161003816100381610038161003816]
+
1
2
radic
119862119909 (1 + 120574119909)
(119899 minus 120574)
100381610038161003816100381610038161198911015840
(119909+
) minus 1198911015840
(119909minus
)
10038161003816100381610038161003816
+
1
2
100381610038161003816100381610038161198911015840
(119909+
) + 1198911015840
(119909minus
)
10038161003816100381610038161003816
1003816100381610038161003816120572 minus 120573119909
1003816100381610038161003816
119899 + 120573
(29)
where the auxiliary functions 119891119909and ⋁119887
119886119891(119909) were defined in
Theorem 4The proof of the above theorem follows along the lines of
Theorem 4 thus we omit the details
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are thankful to the anonymous referee formaking valuable comments leading to the better presen-tation of the paper Special thanks are due to ProfessorDr Abdelalim A Elsadany Editor of Journal of DifferenceEquations for kind cooperation and smooth behavior duringcommunication and for his efforts to send the reports of thepaper timely
References
[1] V Gupta ldquoApproximation for modified Baskakov Durrmeyertype operatorsrdquo The Rocky Mountain Journal of Mathematicsvol 39 no 3 pp 825ndash841 2009
[2] V Gupta M A Noor M S Beniwal and M K Gupta ldquoOnsimultaneous approximation for certain Baskakov Durrmeyertype operatorsrdquo Journal of Inequalities in Pure and AppliedMathematics vol 7 pp 1ndash15 2006
[3] V Gupta and P N Agrawal ldquoRate of convergence for certainBaskakov Durrmeyer type operatorsrdquoAnnals of Oradea Univer-sity Fascicola Matematica vol 14 pp 33ndash39 2007
[4] HKarsli andE Ibikli ldquoConvergence rate of a newBezier variantof Chlodowsky operators to bounded variation functionsrdquoJournal of Computational and AppliedMathematics vol 212 no2 pp 431ndash443 2008
[5] V N Mishra H H Khan K Khatri and L N Mishra ldquoHyper-geometric representation for Baskakov-Durrmeyer-Stancu typeoperatorsrdquo Bulletin of Mathematical Analysis and Applicationsvol 5 no 3 pp 18ndash26 2013
[6] V N Mishra K Khatri and L N Mishra ldquoOn simultaneousapproximation for Baskakov-Durrmeyer-Stancu type opera-torsrdquo Journal of Ultra Scientist of Physical Sciences vol 24 no3 pp 567ndash577 2012
[7] H H Khan Approximation of classes of function [PhD thesis]AMU Aligarh India 1974
6 Journal of Difference Equations
[8] X M Zeng and F C Cheng ldquoOn the rates of approximation ofBernstein type operatorsrdquo Journal of ApproximationTheory vol109 no 2 pp 242ndash256 2001
[9] M Mursaleen V Karakaya M Erturk and F GursoyldquoWeighted statistical convergence and its application toKorovkin type approximation theoremrdquo Applied Mathematicsand Computation vol 218 no 18 pp 9132ndash9137 2012
[10] D D Stancu ldquoApproximation of function by a new class ofpolynomial operatorsrdquoRevue Roumaine deMathematique Pureset Appliquees vol 3 no 8 pp 1173ndash1194 1968
[11] R Yang J Xiong and F Cao ldquoMultivariate Stancu operatorsdefined on a simplexrdquo Applied Mathematics and Computationvol 138 no 2-3 pp 189ndash198 2003
[12] H M Srivastava M Mursaleen and A Khan ldquoGeneralizedequi-statistical convergence of positive linear operators andassociated approximation theoremsrdquo Mathematical and Com-puter Modelling vol 55 no 9-10 pp 2040ndash2051 2012
[13] M Mursaleen A Khan H M Srivastava and K S NisarldquoOperators constructed by means of q-Lagrange polynomialsand A-statistical approximationrdquo Applied Mathematics andComputation vol 219 no 12 pp 6911ndash6918 2013
[14] V N Mishra and P Patel ldquoApproximation by the Durrmeyer-Baskakov-Stancu operatorsrdquo Lobachevskii Journal ofMathemat-ics vol 34 no 3 pp 272ndash281 2013
[15] V N Mishra K Khatri and L N Mishra ldquoInverse result insimultaneous approximation by Baskakov-Durrmeyer-Stancuoperatorsrdquo Journal of Inequalities and Applications vol 2013 no586 2013
[16] V N Mishra and P Patel ldquoSome approximation properties ofmodified jain-beta operatorsrdquo Journal of Calculus of Variationsvol 2013 Article ID 489249 8 pages 2013
[17] V N Mishra K Khatri and L N Mishra ldquoStatisticalapproximation by Kantorovich-type discrete q-Beta operatorsrdquoAdvances in Difference Equations vol 345 no 1 2013
[18] V N Mishra and P Patel ldquoOn simultaneous approximationfor generalized integral typerdquo Baskakov Operator Submitted forpublication
Submit your manuscripts athttpwwwhindawicom
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
4 Journal of Difference Equations
+ int
2119909
119909
10038161003816100381610038161003816(1198911015840
)119909
(119905)
10038161003816100381610038161003816
100381610038161003816100381610038161 minus 120601119909120574(119909 119905)
10038161003816100381610038161003816119889119905
le [
119872
119909
int
infin
2119909
119882119899120574(119909 119905) 119905
119903
|119905 minus 119909| 119889119905
+
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
1199092
int
infin
2119909
119882119899120574(119909 119905) (119905 minus 119909)
2
119889119905]
+ [
100381610038161003816100381610038161198911015840
(119909+
)
10038161003816100381610038161003816int
infin
2119909
119882119899120574(119909 119905) |119905 minus 119909| 119889119905]
+
2 (1 + 120574119909)
(119899 minus 120574) 119909
10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891
1015840
(119909+
)
10038161003816100381610038161003816
+
2 (1 + 120574119909)
119899 minus 120574
[radic119899]
sum
119896=1
119909+119909119896
⋁
119909
((1198911015840
)119909
)
+
119909
radic119899
119909+119909radic119899
⋁
119909
((1198911015840
)119909
)
= 119877119899120574(119891 119909) + 119878
119899120574(119891 119909)
+
2 (1 + 120574119909)
(119899 minus 120574) 119909
10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891
1015840
(119909+
)
10038161003816100381610038161003816
+
2 (1 + 120574119909)
119899 minus 120574
[radic119899]
sum
119896=1
119909+119909119896
⋁
119909
((1198911015840
)119909
)
+
119909
radic119899
119909+119909radic119899
⋁
119909
((1198911015840
)119909
)
(19)
Applying Holderrsquos inequality Remark 2 and Lemma 1 wehave
119877119899120574(119891 119909) le
119872
119909
(int
infin
2119909
119882119899120574(119909 119905) 119905
2119903
119889119905)
12
times(int
infin
0
119882119899120574(119909 119905) (119905 minus 119909)
2
119889119905)
12
+
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
1199092
int
infin
2119909
119882119899120574(119909 119905) (119905 minus 119909)
2
119889119905
le 1198722119903
119874(119899minus1199032
)radic
2119909 (1 + 120574119909)
119899 minus 120574
+1003816100381610038161003816119891 (119909)
1003816100381610038161003816
2 (1 + 120574119909)
(119899 minus 120574) 119909
(20)
Also
119878119899120574(119891 119909) le
1003816100381610038161003816119891 (119909+
)1003816100381610038161003816int
infin
0
119882119899(119905 119909) |119905 minus 119909| 119889119905
le1003816100381610038161003816119891 (119909+
)1003816100381610038161003816radic
2119909 (1 + 120574119909)
119899 minus 120574
(21)
Combining the estimates (17)ndash(21) we get the desired resultsThis completes the proof of Theorem
3 Rate of Convergence for Stancu TypeGeneralization of 119861
119899120574
In 1968 Stancu introduces Bernstein-Stancu operators in[10] a sequence of the linear positive operators depending ontwo parameters 120572 and 120573 satisfying the condition 0 le 120572 le 120573Recently many researchers applied this approach to manyoperators for details see [11ndash17] For 119891 isin 119862[0infin) Stancugeneralization of operators (1) is as follows
119861120572120573
119899120574(119891 (119905) 119909) =
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) 119891(
119899119905 + 120572
119899 + 120573
)119889119905
+ 1199011198990120574
(119909) 119891(
120572
119899 + 120573
)
= int
infin
0
119882119899120574(119909 119905) 119891(
119899119905 + 120572
119899 + 120573
)119889119905
(22)
where 119901119899119896120574
(119909) 119887119899119896120574
(119909) and119882119899120574(119909 119905)are as defined in (1)
Lemma5 (see [18]) If we define the centralmoments for every119898 isin N as
120583120572120573
119899119898120574(119909) = 119861
120572120573
119899120574((119905 minus 119909)
119898
119909)
=
infin
sum
119896=1
119901119899119896120574
(119909) int
infin
0
119887119899119896120574
(119905) (
119899119905 + 120572
119899 + 120573
minus 119909)
119898
119889119905
+ 1199011198990120574
(119909) (
120572
119899 + 120573
minus 119909)
119898
(23)
then 1205831205721205731198990120574
(119909) = 1 1205831205721205731198991120574
(119909) = (120572 minus 120573119909)(119899 + 120573) and
120583120572120573
1198992120574(119909) =
1205722
(119899 + 120573)2+ 119909
(21198992
minus 2119899120572120573 + 2120572120573120574)
(119899 + 120573)2
(119899 minus 120574)
+ 1199092(1198991205732
+ 21198992
120574 minus 1205732
120574)
(119899 + 120573)2
(119899 minus 120574)
(24)
For 119899 gt 119898 we have the following recurrence relation
(119899 minus 120574119898) (119899 + 120573) 120583120572120573
119899119898+1120574(119909)
= 119899119909 (1 + 120574119909) [(120583120572120573
119899119898120574)
(1)
(119909) + 119898120583120572120573
119899119898minus1120574(119909)]
+ [119898119899 + 1198992
119909 minus (2120574119898 minus 119899) (120572 minus (119899 + 120573) 119909)] 120583120572120573
119899119898120574(119909)
+ [119898120574 (119899 + 120573) (
120572
119899 + 120573
minus 119909)
2
minus 119898119899(
120572
119899 + 120573
minus 119909)]
times 120583120572120573
119899119898minus1120574(119909)
(25)
Journal of Difference Equations 5
From the recurrence relation it can be easily verified that forall 119909 isin [0infin) we have 120583120572120573
119899119898120574(119909) = 119874(119899
minus[(119898+1)2]
)
Remark 6 Observe that 119861120572120573119899120574
preserve constant functions butnot linear functions If 120572 = 120573 = 0 these operators reduce tothe operators defined in (1) Notice that
120583120572120573
1198992120574(119909) = [
1205732
120574(119899 + 120573)2+
21198992
(119899 + 120573)2
(119899 minus 120574)
] 119909 (1 + 120574119909)
+
120573 [(2120572 + 120573) 120574 minus 119899 (120573 + 2120572120574)]
120574(119899 + 120573)2
(119899 minus 120574)
119909 +
1205722
(119899 + 120573)2
(26)
Remark 7 From Lemma 3 taking 119899 to be sufficiently largeand 119909 isin (0infin) we observe that
120583120572120573
1198992120574(119909) le
119862119909 (1 + 120574119909)
119899 minus 120574
(27)
where 119862 is positive constant
Remark 8 Applying the Cauchy-Schwarz inequality andkeeping the same condition as in Remark 7 for 119909 119899 and 119862we derive from Lemma 5 that
119861120572120573
119899120574(|119905 minus 119909| 119909) le [119861
120572120573
119899120574((119905 minus 119909)
2
119909)]
12
le radic
119862119909 (1 + 120574119909)
119899 minus 120574
(28)
Lemma 9 Let 119909 isin (0infin) and119882119899120574(119909 119905) be the kernel defined
in (1) Then for 119899 being sufficiently large we have
(c) 120601119899120574(119909 119910) = int
119910
0
119882119899120574(119909 119905)119889119905 = 119862119909(1 + 120574119909)(119899 minus
120574)(119909 minus 119910)2
0 le 119910 lt 119909
(d) 1 minus 120601119899120574(119909 119911) = int
infin
119911
119882119899120574(119909 119905)119889119905 = 119862119909(1 + 120574119909)(119899 minus
120574)(119911 minus 119909)2
119909 lt 119911 lt infin
The proof is the same as Lemma 3 thus we omit the details
Theorem 10 Let 119891 isin 119863119861119903(0infin) 119903 isin N and 119909 isin (0infin)
Then for 119899 being sufficiently large we have
10038161003816100381610038161003816119861120572120573
119899120574(119891 119909) minus 119891 (119909)
10038161003816100381610038161003816
le
119862 (1 + 120574119909)
119899 minus 120574
[radic119899]
sum
119896=1
119909+119909119896
⋁
119909minus119909119896
((1198911015840
)119909
)
+
119909
radic119899
119909+119909radic119899
⋁
119909minus119909radic119899
((1198911015840
)119909
) +
119862 (1 + 120574119909)
(119899 minus 120574) 119909
times [
10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891
1015840
(119909+
)
10038161003816100381610038161003816+1003816100381610038161003816119891 (119909)
1003816100381610038161003816]
+ radic
119862119909 (1 + 120574119909)
119899 minus 120574
times [11987212119903
119874(119899minus1199032
) +
100381610038161003816100381610038161198911015840
(119909+
)
10038161003816100381610038161003816]
+
1
2
radic
119862119909 (1 + 120574119909)
(119899 minus 120574)
100381610038161003816100381610038161198911015840
(119909+
) minus 1198911015840
(119909minus
)
10038161003816100381610038161003816
+
1
2
100381610038161003816100381610038161198911015840
(119909+
) + 1198911015840
(119909minus
)
10038161003816100381610038161003816
1003816100381610038161003816120572 minus 120573119909
1003816100381610038161003816
119899 + 120573
(29)
where the auxiliary functions 119891119909and ⋁119887
119886119891(119909) were defined in
Theorem 4The proof of the above theorem follows along the lines of
Theorem 4 thus we omit the details
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are thankful to the anonymous referee formaking valuable comments leading to the better presen-tation of the paper Special thanks are due to ProfessorDr Abdelalim A Elsadany Editor of Journal of DifferenceEquations for kind cooperation and smooth behavior duringcommunication and for his efforts to send the reports of thepaper timely
References
[1] V Gupta ldquoApproximation for modified Baskakov Durrmeyertype operatorsrdquo The Rocky Mountain Journal of Mathematicsvol 39 no 3 pp 825ndash841 2009
[2] V Gupta M A Noor M S Beniwal and M K Gupta ldquoOnsimultaneous approximation for certain Baskakov Durrmeyertype operatorsrdquo Journal of Inequalities in Pure and AppliedMathematics vol 7 pp 1ndash15 2006
[3] V Gupta and P N Agrawal ldquoRate of convergence for certainBaskakov Durrmeyer type operatorsrdquoAnnals of Oradea Univer-sity Fascicola Matematica vol 14 pp 33ndash39 2007
[4] HKarsli andE Ibikli ldquoConvergence rate of a newBezier variantof Chlodowsky operators to bounded variation functionsrdquoJournal of Computational and AppliedMathematics vol 212 no2 pp 431ndash443 2008
[5] V N Mishra H H Khan K Khatri and L N Mishra ldquoHyper-geometric representation for Baskakov-Durrmeyer-Stancu typeoperatorsrdquo Bulletin of Mathematical Analysis and Applicationsvol 5 no 3 pp 18ndash26 2013
[6] V N Mishra K Khatri and L N Mishra ldquoOn simultaneousapproximation for Baskakov-Durrmeyer-Stancu type opera-torsrdquo Journal of Ultra Scientist of Physical Sciences vol 24 no3 pp 567ndash577 2012
[7] H H Khan Approximation of classes of function [PhD thesis]AMU Aligarh India 1974
6 Journal of Difference Equations
[8] X M Zeng and F C Cheng ldquoOn the rates of approximation ofBernstein type operatorsrdquo Journal of ApproximationTheory vol109 no 2 pp 242ndash256 2001
[9] M Mursaleen V Karakaya M Erturk and F GursoyldquoWeighted statistical convergence and its application toKorovkin type approximation theoremrdquo Applied Mathematicsand Computation vol 218 no 18 pp 9132ndash9137 2012
[10] D D Stancu ldquoApproximation of function by a new class ofpolynomial operatorsrdquoRevue Roumaine deMathematique Pureset Appliquees vol 3 no 8 pp 1173ndash1194 1968
[11] R Yang J Xiong and F Cao ldquoMultivariate Stancu operatorsdefined on a simplexrdquo Applied Mathematics and Computationvol 138 no 2-3 pp 189ndash198 2003
[12] H M Srivastava M Mursaleen and A Khan ldquoGeneralizedequi-statistical convergence of positive linear operators andassociated approximation theoremsrdquo Mathematical and Com-puter Modelling vol 55 no 9-10 pp 2040ndash2051 2012
[13] M Mursaleen A Khan H M Srivastava and K S NisarldquoOperators constructed by means of q-Lagrange polynomialsand A-statistical approximationrdquo Applied Mathematics andComputation vol 219 no 12 pp 6911ndash6918 2013
[14] V N Mishra and P Patel ldquoApproximation by the Durrmeyer-Baskakov-Stancu operatorsrdquo Lobachevskii Journal ofMathemat-ics vol 34 no 3 pp 272ndash281 2013
[15] V N Mishra K Khatri and L N Mishra ldquoInverse result insimultaneous approximation by Baskakov-Durrmeyer-Stancuoperatorsrdquo Journal of Inequalities and Applications vol 2013 no586 2013
[16] V N Mishra and P Patel ldquoSome approximation properties ofmodified jain-beta operatorsrdquo Journal of Calculus of Variationsvol 2013 Article ID 489249 8 pages 2013
[17] V N Mishra K Khatri and L N Mishra ldquoStatisticalapproximation by Kantorovich-type discrete q-Beta operatorsrdquoAdvances in Difference Equations vol 345 no 1 2013
[18] V N Mishra and P Patel ldquoOn simultaneous approximationfor generalized integral typerdquo Baskakov Operator Submitted forpublication
Submit your manuscripts athttpwwwhindawicom
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
Journal of Difference Equations 5
From the recurrence relation it can be easily verified that forall 119909 isin [0infin) we have 120583120572120573
119899119898120574(119909) = 119874(119899
minus[(119898+1)2]
)
Remark 6 Observe that 119861120572120573119899120574
preserve constant functions butnot linear functions If 120572 = 120573 = 0 these operators reduce tothe operators defined in (1) Notice that
120583120572120573
1198992120574(119909) = [
1205732
120574(119899 + 120573)2+
21198992
(119899 + 120573)2
(119899 minus 120574)
] 119909 (1 + 120574119909)
+
120573 [(2120572 + 120573) 120574 minus 119899 (120573 + 2120572120574)]
120574(119899 + 120573)2
(119899 minus 120574)
119909 +
1205722
(119899 + 120573)2
(26)
Remark 7 From Lemma 3 taking 119899 to be sufficiently largeand 119909 isin (0infin) we observe that
120583120572120573
1198992120574(119909) le
119862119909 (1 + 120574119909)
119899 minus 120574
(27)
where 119862 is positive constant
Remark 8 Applying the Cauchy-Schwarz inequality andkeeping the same condition as in Remark 7 for 119909 119899 and 119862we derive from Lemma 5 that
119861120572120573
119899120574(|119905 minus 119909| 119909) le [119861
120572120573
119899120574((119905 minus 119909)
2
119909)]
12
le radic
119862119909 (1 + 120574119909)
119899 minus 120574
(28)
Lemma 9 Let 119909 isin (0infin) and119882119899120574(119909 119905) be the kernel defined
in (1) Then for 119899 being sufficiently large we have
(c) 120601119899120574(119909 119910) = int
119910
0
119882119899120574(119909 119905)119889119905 = 119862119909(1 + 120574119909)(119899 minus
120574)(119909 minus 119910)2
0 le 119910 lt 119909
(d) 1 minus 120601119899120574(119909 119911) = int
infin
119911
119882119899120574(119909 119905)119889119905 = 119862119909(1 + 120574119909)(119899 minus
120574)(119911 minus 119909)2
119909 lt 119911 lt infin
The proof is the same as Lemma 3 thus we omit the details
Theorem 10 Let 119891 isin 119863119861119903(0infin) 119903 isin N and 119909 isin (0infin)
Then for 119899 being sufficiently large we have
10038161003816100381610038161003816119861120572120573
119899120574(119891 119909) minus 119891 (119909)
10038161003816100381610038161003816
le
119862 (1 + 120574119909)
119899 minus 120574
[radic119899]
sum
119896=1
119909+119909119896
⋁
119909minus119909119896
((1198911015840
)119909
)
+
119909
radic119899
119909+119909radic119899
⋁
119909minus119909radic119899
((1198911015840
)119909
) +
119862 (1 + 120574119909)
(119899 minus 120574) 119909
times [
10038161003816100381610038161003816119891 (2119909) minus 119891 (119909) minus 119909119891
1015840
(119909+
)
10038161003816100381610038161003816+1003816100381610038161003816119891 (119909)
1003816100381610038161003816]
+ radic
119862119909 (1 + 120574119909)
119899 minus 120574
times [11987212119903
119874(119899minus1199032
) +
100381610038161003816100381610038161198911015840
(119909+
)
10038161003816100381610038161003816]
+
1
2
radic
119862119909 (1 + 120574119909)
(119899 minus 120574)
100381610038161003816100381610038161198911015840
(119909+
) minus 1198911015840
(119909minus
)
10038161003816100381610038161003816
+
1
2
100381610038161003816100381610038161198911015840
(119909+
) + 1198911015840
(119909minus
)
10038161003816100381610038161003816
1003816100381610038161003816120572 minus 120573119909
1003816100381610038161003816
119899 + 120573
(29)
where the auxiliary functions 119891119909and ⋁119887
119886119891(119909) were defined in
Theorem 4The proof of the above theorem follows along the lines of
Theorem 4 thus we omit the details
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are thankful to the anonymous referee formaking valuable comments leading to the better presen-tation of the paper Special thanks are due to ProfessorDr Abdelalim A Elsadany Editor of Journal of DifferenceEquations for kind cooperation and smooth behavior duringcommunication and for his efforts to send the reports of thepaper timely
References
[1] V Gupta ldquoApproximation for modified Baskakov Durrmeyertype operatorsrdquo The Rocky Mountain Journal of Mathematicsvol 39 no 3 pp 825ndash841 2009
[2] V Gupta M A Noor M S Beniwal and M K Gupta ldquoOnsimultaneous approximation for certain Baskakov Durrmeyertype operatorsrdquo Journal of Inequalities in Pure and AppliedMathematics vol 7 pp 1ndash15 2006
[3] V Gupta and P N Agrawal ldquoRate of convergence for certainBaskakov Durrmeyer type operatorsrdquoAnnals of Oradea Univer-sity Fascicola Matematica vol 14 pp 33ndash39 2007
[4] HKarsli andE Ibikli ldquoConvergence rate of a newBezier variantof Chlodowsky operators to bounded variation functionsrdquoJournal of Computational and AppliedMathematics vol 212 no2 pp 431ndash443 2008
[5] V N Mishra H H Khan K Khatri and L N Mishra ldquoHyper-geometric representation for Baskakov-Durrmeyer-Stancu typeoperatorsrdquo Bulletin of Mathematical Analysis and Applicationsvol 5 no 3 pp 18ndash26 2013
[6] V N Mishra K Khatri and L N Mishra ldquoOn simultaneousapproximation for Baskakov-Durrmeyer-Stancu type opera-torsrdquo Journal of Ultra Scientist of Physical Sciences vol 24 no3 pp 567ndash577 2012
[7] H H Khan Approximation of classes of function [PhD thesis]AMU Aligarh India 1974
6 Journal of Difference Equations
[8] X M Zeng and F C Cheng ldquoOn the rates of approximation ofBernstein type operatorsrdquo Journal of ApproximationTheory vol109 no 2 pp 242ndash256 2001
[9] M Mursaleen V Karakaya M Erturk and F GursoyldquoWeighted statistical convergence and its application toKorovkin type approximation theoremrdquo Applied Mathematicsand Computation vol 218 no 18 pp 9132ndash9137 2012
[10] D D Stancu ldquoApproximation of function by a new class ofpolynomial operatorsrdquoRevue Roumaine deMathematique Pureset Appliquees vol 3 no 8 pp 1173ndash1194 1968
[11] R Yang J Xiong and F Cao ldquoMultivariate Stancu operatorsdefined on a simplexrdquo Applied Mathematics and Computationvol 138 no 2-3 pp 189ndash198 2003
[12] H M Srivastava M Mursaleen and A Khan ldquoGeneralizedequi-statistical convergence of positive linear operators andassociated approximation theoremsrdquo Mathematical and Com-puter Modelling vol 55 no 9-10 pp 2040ndash2051 2012
[13] M Mursaleen A Khan H M Srivastava and K S NisarldquoOperators constructed by means of q-Lagrange polynomialsand A-statistical approximationrdquo Applied Mathematics andComputation vol 219 no 12 pp 6911ndash6918 2013
[14] V N Mishra and P Patel ldquoApproximation by the Durrmeyer-Baskakov-Stancu operatorsrdquo Lobachevskii Journal ofMathemat-ics vol 34 no 3 pp 272ndash281 2013
[15] V N Mishra K Khatri and L N Mishra ldquoInverse result insimultaneous approximation by Baskakov-Durrmeyer-Stancuoperatorsrdquo Journal of Inequalities and Applications vol 2013 no586 2013
[16] V N Mishra and P Patel ldquoSome approximation properties ofmodified jain-beta operatorsrdquo Journal of Calculus of Variationsvol 2013 Article ID 489249 8 pages 2013
[17] V N Mishra K Khatri and L N Mishra ldquoStatisticalapproximation by Kantorovich-type discrete q-Beta operatorsrdquoAdvances in Difference Equations vol 345 no 1 2013
[18] V N Mishra and P Patel ldquoOn simultaneous approximationfor generalized integral typerdquo Baskakov Operator Submitted forpublication
Submit your manuscripts athttpwwwhindawicom
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
6 Journal of Difference Equations
[8] X M Zeng and F C Cheng ldquoOn the rates of approximation ofBernstein type operatorsrdquo Journal of ApproximationTheory vol109 no 2 pp 242ndash256 2001
[9] M Mursaleen V Karakaya M Erturk and F GursoyldquoWeighted statistical convergence and its application toKorovkin type approximation theoremrdquo Applied Mathematicsand Computation vol 218 no 18 pp 9132ndash9137 2012
[10] D D Stancu ldquoApproximation of function by a new class ofpolynomial operatorsrdquoRevue Roumaine deMathematique Pureset Appliquees vol 3 no 8 pp 1173ndash1194 1968
[11] R Yang J Xiong and F Cao ldquoMultivariate Stancu operatorsdefined on a simplexrdquo Applied Mathematics and Computationvol 138 no 2-3 pp 189ndash198 2003
[12] H M Srivastava M Mursaleen and A Khan ldquoGeneralizedequi-statistical convergence of positive linear operators andassociated approximation theoremsrdquo Mathematical and Com-puter Modelling vol 55 no 9-10 pp 2040ndash2051 2012
[13] M Mursaleen A Khan H M Srivastava and K S NisarldquoOperators constructed by means of q-Lagrange polynomialsand A-statistical approximationrdquo Applied Mathematics andComputation vol 219 no 12 pp 6911ndash6918 2013
[14] V N Mishra and P Patel ldquoApproximation by the Durrmeyer-Baskakov-Stancu operatorsrdquo Lobachevskii Journal ofMathemat-ics vol 34 no 3 pp 272ndash281 2013
[15] V N Mishra K Khatri and L N Mishra ldquoInverse result insimultaneous approximation by Baskakov-Durrmeyer-Stancuoperatorsrdquo Journal of Inequalities and Applications vol 2013 no586 2013
[16] V N Mishra and P Patel ldquoSome approximation properties ofmodified jain-beta operatorsrdquo Journal of Calculus of Variationsvol 2013 Article ID 489249 8 pages 2013
[17] V N Mishra K Khatri and L N Mishra ldquoStatisticalapproximation by Kantorovich-type discrete q-Beta operatorsrdquoAdvances in Difference Equations vol 345 no 1 2013
[18] V N Mishra and P Patel ldquoOn simultaneous approximationfor generalized integral typerdquo Baskakov Operator Submitted forpublication
Submit your manuscripts athttpwwwhindawicom
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
top related