Applied Mathematics and Computation 236 (2014) 19–26

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Approximation by Lupas�-Beta integral operators

http://dx.doi.org/10.1016/j.amc.2014.03.0330096-3003/� 2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author.E-mail addresses: vijaygupta2001@hotmail.com (V. Gupta), trassias@math.ntua.gr (T.M. Rassias), raniyadav23@gmail.com (R. Yadav).

Vijay Gupta a,⇑, Themistocles M. Rassias b, Rani Yadav a

a Department of Mathematics, Netaji Subhash Institute of Technology, Sector 3 Dwarka, New Delhi 110078, Indiab Department of Mathematics, National Technical University of Athens, Zografou Campus, 157 80 Athens, Greece

a r t i c l e i n f o

Keywords:Lupas� operatorsStirling numberAsymptotic formulaWeighted approximationRate of convergence

a b s t r a c t

In the present article we introduce certain Lupas�-Beta operators which preserve constantas well as linear functions. We define the operators in terms of hypergeometric seriesand establish moments using such approach. Asymptotic formula error estimation dueto modulus of continuity and weighted approximation are studied. We also establish therate of convergence for those functions whose derivatives are of bounded variation.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

The gamma first kind transform of a function f is given by

ðCðaÞp f ÞðxÞ ¼ 1CðpÞ

Z 1

0e�ttp�1f x

tp

� �a� �dt:

If we consider a ¼ 1 and p ¼ n, we get the Post-Wider’s linear positive operators. Also if we put a ¼ 1 and replace p by nx, weget the Rathore’s linear positive operators [13] given as

Rnðf ; xÞ ¼1

CðnxÞ

Z 1

0e�ttnx�1f

tn

� �dt:

Lupas� [3] introduced the operator Lnðf ; xÞ for f : ½0;1Þ ! R and x P 0 as

Lnðf ; xÞ ¼X1k¼0

ln;kðxÞfkn

� �; ð1:1Þ

where the Lupas� basis function is given by

ln;kðxÞ ¼ 2�nx nxþ k� 1k

� �2�k:

The operators Ln preserve linear functions. It was observed by Mihesan [9] that Lnðf ; xÞ ¼ RnðSnðf ; xÞ; xÞ, where the Szász–Mirakyan operators are defined by

Snðf ; xÞ ¼X1k¼0

e�nx ðnxÞk

k!f

kn

� �:

20 V. Gupta et al. / Applied Mathematics and Computation 236 (2014) 19–26

In order to approximate the integrable functions the Durrmeyer variant of the operators (1.1) as introduced in [1] is definedby

Dnðf ; xÞ ¼X1k¼0

cn;kln;kðxÞZ 1

0ln;kðuÞf ðuÞdu;

where the weights are given by cn;k ¼R1

0 ln;kðtÞdt� ��1 � 1

n2kk!

Pki¼0ð�1Þk�i sk;i :i!

2iþ1 with sk;i representing the Stirling numbers of the

first kind having the form ðxÞk ¼Pk

i¼0ð�1Þk�isk;ixi and ðxÞk ¼ xðxþ 1Þðxþ 2Þ � � � ðxþ k� 1Þ.Many results on different polynomials and operators were studied in the last two decades, we mention here the books

due to Rassias et al. [11,12] and Agarwal–Gupta [7]. Recently Gupta–Yadav [8] proposed an integral modification of theLupas� operators having weights of Beta basis function, but their operators reproduce only the constant functions. Also in[6,5] authors have considered different modifications of the Lupas� operators having weights in general setting and of theSzász basis functions. For f 2 L1½0;1Þ, we now propose a new integral modification of the Lupas� operators having weightsof Beta basis function as

Mnðf ; xÞ ¼Z 1

0Knðx; tÞf ðtÞdt ¼

X1k¼1

ln;kðxÞZ 1

0bn;kðtÞf ðtÞdt þ ln;0ðxÞf ð0Þ; x 2 ½0;1Þ; ð1:2Þ

where ln;kðxÞ is given by (1.1) and

bn;kðtÞ ¼1

Bðnþ 1; kÞtk�1

ð1þ tÞnþkþ1

is the Beta basis function. Also, the kernel function Knðx; tÞ is given by

Knðx; tÞ ¼X1k¼1

ln;kðxÞbn;kðtÞ þ ln;0ðxÞdðtÞ

and dðtÞ being the Dirac delta function.The hypergeometric function is defined as

2F1ða; b; c; xÞ ¼X1k¼0

ðaÞkðbÞkðcÞkk!

xk:

The operators (1.2) can be written in terms of hypergeometric form as

Mnðf ;xÞ¼X1k¼1

ln;kðxÞZ 1

0bn;kðtÞf ðtÞdtþ ln;0ðxÞf ð0Þ¼2�nx

X1k¼1

nxþk�1k

� �2�k

Z 1

0

1Bðnþ1;kÞ

tk�1

ð1þ tÞnþkþ1 f ðtÞdtþ2�nxf ð0Þ

¼2�nxZ 1

0

f ðtÞð1þ tÞnþ2

X1k¼1

ðnxþk�1Þ!2�k

k!ðnx�1Þ!ðnþkÞ!

n!ðk�1Þ!tk�1

ð1þ tÞk�1 dtþ2�nxf ð0Þ

¼2�nxZ 1

0

f ðtÞð1þ tÞnþ2

X1k¼1

ðnxÞðnxþ1Þ� � � ðnxþk�1Þk!2k

�ðnþ1Þ � � �ðnþkÞðk�1Þ!

tk�1

ð1þ tÞk�1 dtþ2�nxf ð0Þ

¼2�nxZ 1

0

f ðtÞð1þ tÞnþ2

X1k¼0

ðnxÞðnxþ1Þkð2Þk2kþ1 �ðnþ1Þðnþ2Þk

k!

tk

ð1þ tÞkdtþ2�nxf ð0Þ

¼nðnþ1Þx2nxþ1

Z 1

0

f ðtÞð1þ tÞnþ2 2F1 nxþ1;nþ2;2;

t2ð1þ tÞ

� �dtþ2�nxf ð0Þ:

The present article deals with approximation properties of the operators Mnðf ; xÞ. We study asymptotic formula, error esti-mate in terms of modulus of continuity, weighted approximation and rate of convergence for the operators (1.2).

2. Moments

In order to prove the main results we need the following basic lemmas.

Lemma 1. For r > 0; Mnðer ; xÞ satisfies the following relation

Mnðer; xÞ ¼r!ðn� rÞ!nx

n!2F1ðnxþ 1;1� r; 2;�1Þ:

Proof. By the definition (1.2), we have

V. Gupta et al. / Applied Mathematics and Computation 236 (2014) 19–26 21

Mnðer; xÞ ¼ 2�nxX1k¼1

nxþ k� 1k

� �2�k

Z 1

0

1Bðnþ 1; kÞ

tkþr�1

ð1þ tÞnþkþ1 dt

¼ 2�nxX1k¼1

ðnxþ k� 1Þ!k!ðnx� 1Þ! 2�k 1

Bðnþ 1; kÞBðkþ r;n� r þ 1Þ ¼ 2�nx�1 ðn� rÞ!n!ðnx� 1Þ!

X1k¼0

ðnxþ kÞ!ðkþ rÞ!ðkþ 1Þ!2kk!

¼ 2�nx�1 ðn� rÞ!n!

X1k¼0

ðnxþ 1ÞkðnxÞðr þ 1ÞkCðr þ 1Þð2Þk2kk!

¼ 2�nx�1 r!ðnxÞðn� rÞ!n!

2F1 nxþ 1; r þ 1; 2;12

� �:

Using 2F1ða; b; c; xÞ ¼ ð1� xÞ�a2F1 a; c � b; c; x

x�1

� �, we get

Mnðer ; xÞ ¼r!ðnxÞðn� rÞ!

n!2F1 nxþ 1;1� r; 2;�1ð Þ: �

Remark 1. By simple computation from Lemma 1, we have

Mnðe0; xÞ ¼ 1; Mnðe1; xÞ ¼ x;

Mnðe2; xÞ ¼xðnxþ 3Þ

n� 1; n > 1;

Mnðe3; xÞ ¼n2x3 þ 9nx2 þ 14xðn� 1Þðn� 2Þ ; n > 2;

Mnðe4; xÞ ¼n3x4 þ 18n2x3 þ 83nx2 þ 90xðn� 1Þðn� 2Þðn� 3Þ ; n > 3:

Remark 2. Let wxðtÞ ¼ t � x; n 2 N then from Remark 1, we have

Mnðwx; xÞ ¼ 0;

Mnðw2x ; xÞ ¼

xðxþ 3Þn� 1

; n > 1;

Mnðjt � xj; xÞ 6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixðxþ 3Þ

n� 1

r; n > 1:

Further, for r ¼ 0;1;2; . . ., we have

Mnðwrx; xÞ ¼ O n�½ðrþ1Þ=2�� �

:

Lemma 2. For n P 1 and for fixed x 2 ð0;1Þ, we have

knðx; yÞ ¼Z y

0Knðx; tÞdt 6

xðxþ 3Þðn� 1Þðx� yÞ2

; 0 6 y < x;

1� knðx; zÞ ¼Z 1

zKnðx; tÞdt 6

xðxþ 3Þðn� 1Þðz� xÞ2

; x < z <1:

The proof of this lemma easily follows by using Remark 2.

3. Main theorems

In this section, we state and prove the main direct estimates.

Theorem 1. If f is a continuous function on ½0;1Þ then the sequence fMnðf Þg converges uniformly to f in ½a; b� � ½0;1Þ.

Proof. For sufficiently large n, it is obvious from Remark 1 that Mnðe0; xÞ; Mnðe1; xÞ and Mnðe2; xÞ converge uniformly to 1; xand x2, respectively on every compact subset of ½0;1Þ. Thus the required result follows from Bohman–Korovkin theorem. h

By CB½0;1Þ, we denote the class of real valued continuous bounded functions f ðxÞ for x 2 ½0;1Þ with the normed definedby

jjf jj ¼ supx2½0;1Þ

jf ðxÞj:

22 V. Gupta et al. / Applied Mathematics and Computation 236 (2014) 19–26

For f 2 CB½0;1Þ and d > 0 the first and second order modulus of continuity are defined as

xðf ; dÞ ¼ sup06h6d

supx2½0;1Þ

jf ðxþ hÞ � f ðxÞj

and

x2ðf ; dÞ ¼ sup06h6d

supx2½0;1Þ

jf ðxþ 2hÞ � 2f ðxþ hÞ þ f ðxÞj;

respectively.The Peetre’s K-functional is defined as

K2ðf ; dÞ ¼ infg2C2

B ½0;1Þjjf � gjj þ djjg00jj : g 2 C2

B½0;1Þn o

;

where

C2B½0;1Þ ¼ fg 2 CB½0;1Þ : g0; g00 2 CB½0;1Þg:

There exists a constant C > 0 due to [2] such that for d > 0, we have

K2ðf ; dÞ 6 Cx2ðf ;ffiffiffidpÞ: ð3:1Þ

Theorem 2. For x 2 ½0;1Þ and f 2 CB½0;1Þ, there exists a constant C > 0 such that

Mnðf ; xÞ � f ðxÞj j 6 Cx2 f ;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixðxþ 3Þ

n� 1

r !:

Proof. Let g 2 C2B½0;1Þ. By using Taylor’s theorem, we have

gðtÞ ¼ gðxÞ þ g0ðt � xÞ þZ t

xðt � uÞg00ðuÞdu:

Therefore

Mnðg; xÞ � gðxÞj j ¼ Mn

Z t

xðt � uÞg00ðuÞdu; x

� ���������:

Since we have

Z t

xðt � uÞg00ðuÞdu

�������� 6 ðt � xÞ2jjg00jj

by Remark 2, we get

Mnðg; xÞ � gðxÞj j 6 xðxþ 3Þn� 1

jjg00jj: ð3:2Þ

Now by Remark 1, we have

Mnðf ; xÞj j 6X1k¼1

ln;kðxÞZ 1

0bn;kðtÞjf ðtÞjdt 6 jjf jj: ð3:3Þ

Finally by using (3.2) and (3.3), we get

Mnðf ; xÞ � f ðxÞj j 6 Mnððf � gÞ; xÞ � ðf � gÞðxÞj j þ Mnðg; xÞ � gðxÞj j 6 2jjf � gjj þ xðxþ 3Þn� 1

jjg00jj;

which by using (3.1) and taking infimum over all C2B½0;1Þ completes the proof of Theorem. h

Theorem 3. Let f be a bounded and integrable function on the interval ½0;1Þ, admitting the second derivative of f at a fixed pointx 2 ½0;1Þ. Then

limn!1

n Mnðf ; xÞ � f ðxÞ½ � ¼ xðxþ 3Þ2

f 00ðxÞ:

Proof. By the Taylor’s formula we may write

V. Gupta et al. / Applied Mathematics and Computation 236 (2014) 19–26 23

f ðtÞ ¼ f ðxÞ þ f 0ðxÞðt � xÞ þ 12

f 00ðxÞðt � xÞ2 þ rðt; xÞðt � xÞ2; ð3:4Þ

where rðt; xÞ is the remainder term and limt!xrðt; xÞ ¼ 0. Operating by Mn to (3.4) we obtain

Mnðf ; xÞ � f ðxÞ ¼ Mnðt � x; xÞf 0ðxÞ þMn t � xð Þ2; x� f 00ðxÞ

2þMn r t; xð Þ t � xð Þ2; x

� :

Applying the Cauchy–Schwarz inequality, we have

Mn r t; xð Þ t � xð Þ2; x�

6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiMn r2 t; xð Þ; xð Þ

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiMn t � xð Þ4; x�

:

rð3:5Þ

As r2 x; xð Þ ¼ 0 and r2 :; xð Þ 2 C�2½0;1Þ, we have from Remark 2, that

limn!1

Mn r2 t; xð Þ; x� �

¼ r2 x; xð Þ ¼ 0 ð3:6Þ

uniformly with respect to x 2 0;A½ �. Now from (3.5) and (3.6) and Remark 1, we have

limn!1

Mn r t; xð Þ t � xð Þ2; x�

¼ 0:

Thus

limn!1

n Mnðf ; xÞ � f ðxÞð Þ ¼ limn!1

12

f 00ðxÞMn t � xð Þ2; x�

¼ x2 þ 3x2

f 00ðxÞ:

This completes the proof of Theorem 3. h

For weighted approximation, we provide the following result. Let C�x2 ½0;1Þ be the space of all continuous functions sat-isfying the condition limx!1

f ðxÞ1þx2 being finite and belonging to Bx2 ½0;1Þ, where

Bx2 ½0;1Þ ¼ ff : for every x 2 ½0;1Þ; jf ðxÞj 6 Mf ð1þ x2Þ;Mf being a constant depending on fg:

The norm on C�x2 0;1½ Þ is fk kx2 ¼ supx2 0;1½ Þf xð Þj j

1þx2.

Theorem 4. For each f 2 C�x2 0;1½ Þ, we have

limn!1

Mnðf Þ � fk kx2 ¼ 0:

Proof. In order to prove the theorem, it is sufficient to show that (see [4]):

limn!1

Mnðem; xÞ � emk kx2 ¼ 0; m ¼ 0;1;2:

Since Mn e0; xð Þ ¼ 1 and Mn e1; xð Þ ¼ x, the above condition holds for m ¼ 0; 1. Next, we can write

Mn e2; xð Þ � x2

x2 6 supx2 0;1½ Þ

x2

ð1þ x2Þðn� 1Þ þ supx2 0;1½ Þ

x1þ x2

3n� 1

;

which implies that

limn!1

Mn e2; xð Þ � x2

x2 ¼ 0:

Thus the result holds for m ¼ 0;1;2. This completes the proof of Theorem. h

Let L denote the class of all Lebesgue measurable functions f on ½0;1Þ as

L ¼ f :

Z 1

0

jf ðtÞjð1þ tÞn

dt <1 for some integer n > 1� �

:

We note that the operators given in (1.2) are defined for the class L which is bigger than CB½0;1Þ.

Theorem 5. Let f 2 L be bounded on every finite sub-interval of ½0;1Þ and f ðtÞ ¼ OðtaÞ as t !1 for some a > 0. If f admits aderivative of order 3 at a fixed point x 2 ð0;1Þ, then we have

limn!1

nd

dwMnðf ;wÞ

� �w¼x

� f 0ðxÞ �

¼ 2xþ 32

f 00ðxÞ þ xðxþ 3Þ2

f 000ðxÞ:

Proof. From the Taylor’s theorem, we may write

24 V. Gupta et al. / Applied Mathematics and Computation 236 (2014) 19–26

f ðtÞ ¼X3

i¼0

ðt � xÞi

i!f ðiÞðxÞ þ wðt; xÞðt � xÞ3; t 2 ½0;1Þ; ð3:7Þ

where the function wðt; xÞ ! 0 as t ! x and wðt; xÞ ¼ Oððt � xÞcÞ as t !1 for some c > 0. From Eq. (3.7), we obtain

ddw

Mnðf ðtÞ;wÞ� �

w¼x¼ f 0ðxÞ d

dwðMnðt;wÞ � xÞ

� �w¼xþ f 00ðxÞ

2d

dwðMnðt2;wÞ � 2xMnðt;wÞ þ x2Þ

� �w¼x

þ f 000ðxÞ3!

ddwðMnðt3;wÞ � 3xMnðt2;wÞ þ 3x2Mnðt;wÞ � x3Þ

� �w¼x

þ ddwðMnðwðt; xÞðt � xÞ3;wÞ

� �w¼x

:

Using Remark 1, we get

ddw

Mnðf ðtÞ;wÞ� �

w¼x¼ f 0ðxÞ þ 2xþ 3

2ðn� 1Þ f00ðxÞ þ f 000ðxÞ

69nxþ 3nx2 þ 6x2 þ 18xþ 14

ðn� 1Þðn� 2Þ

� �

þ ddwðMnðwðt; xÞðt � xÞ3;wÞ

� �w¼x

:

Taking limit as n!1 on both sides of the above equation, we have

limn!1

nd

dwMnðf ;wÞ

� �w¼x

� f 0ðxÞ �

¼ 2xþ 32

f 00ðxÞ þ xðxþ 3Þ2

f 000ðxÞ þ limn!1

nd

dwðMnðwðt; xÞðt � xÞ3;wÞ

� �w¼x

:

Proceeding as in the proof of Theorem 3, we can easily show that

limn!1

nd

dwðMnðwðt; xÞðt � xÞ3; wÞ

� �w¼x

¼ 0:

This completes the proof of Theorem. h

Our last result in this section is the rate of convergence for the operators Mn for functions having derivatives of boundedvariation. First we consider the following class of functions:

By DBVc½0;1Þ; c P 0 we represent the class of all functions f defined on ½0;1Þ, having a derivative of bounded variationon every finite subinterval of ½0;1Þ and jf ðtÞj 6 Mtc; 8t > 0.

For f 2 DBVc½0;1Þ, we write

f ðxÞ ¼Z x

0gðtÞdt þ f ð0Þ;

where gðtÞ is a function of bounded variation on each finite subinterval of ½0;1Þ.

Theorem 6. Let f 2 DBVc½0;1Þ; c P 0. Then, for every x 2 ð0;1Þ; r 2 N ð2r P cÞ and for n sufficiently large, we have

Mnðf ; xÞ � f ðxÞ � f 0ðxþÞ � f 0ðx�Þ2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixðxþ 3Þ

n� 1

r���������� 6 xffiffiffi

np

_xþ xffiffinp

x� xffiffinp

ðf 0xÞ þðxþ 3Þn� 1

X½ ffiffinp �k¼1

_xþxk

x�xk

ðf 0xÞ þC1ðx; rÞ

nrþ jf ðxÞj ðxþ 3Þ

xðn� 1Þ

þ jf 0ðxþÞjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixðxþ 3Þ

n� 1

rþ xðxþ 3Þ

n� 1jf ð2xÞ � f ðxÞ � xf 0ðxþÞj;

where

f 0xðtÞ ¼f 0ðtÞ � f 0ðxþÞ; x < t <1;0; t ¼ x;

f 0ðtÞ � f 0ðx�Þ; 0 6 t < x

8><>: ð3:8Þ

and _dc ðf 0xÞ is the total variation of f 0x on ½c; d� � ð0;1Þ.

Proof. It follows from (3.8) that

f 0ðtÞ ¼ 12ðf 0ðxþÞ þ f 0ðx�ÞÞ þ f 0xðtÞ þ

12ðf 0ðxþÞ þ f 0ðx�ÞÞsgnðt � xÞ þ dxðtÞ f 0ðtÞ � 1

2ðf 0ðxþÞ þ f 0ðx�ÞÞ

� �;

where

dxðtÞ ¼1; x ¼ t;

0; x – t:

�

V. Gupta et al. / Applied Mathematics and Computation 236 (2014) 19–26 25

By a simple computation, we have

Mnðf ; xÞ � f ðxÞ 6 f 0ðxþÞ þ f 0ðx�Þ2

� �Mnððt � xÞ; xÞ þ f 0ðxþÞ � f 0ðx�Þ

2

� �Mnðjt � xj; xÞ þ E1ðn; xÞ þ E2ðn; xÞ;

where

E1ðn; xÞ ¼Z x

0

Z x

tf 0xðuÞdu

� �Knðx; tÞdt

and

E2ðn; xÞ ¼Z 1

x

Z t

xf 0xðuÞdu

� �Knðx; tÞdt:

Applying Remark 2, we get

Mnðf ; xÞ � f ðxÞ � ðf0ðxþÞ � f 0ðx�ÞÞ

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixðxþ 3Þ

n� 1

r���������� 6 jE1ðn; xÞj þ jE2ðn; xÞj: ð3:9Þ

First, we obtain an estimate of E1ðn; xÞ. From the definition of knðx; tÞ given in Lemma 2, we may write

E1ðn; xÞ ¼Z x

0

Z x

tf 0xðuÞdu

� �@

@tknðx; tÞdt:

Applying the integration by parts, we get

jE1ðn; xÞj 6Z x

0jf 0xðtÞjknðx; tÞdt 6

Z x� xffiffinp

0jf 0xðtÞjknðx; tÞdt þ

Z x

x� xffiffinpjf 0xðtÞjknðx; tÞdt ¼ E11ðn; xÞ þ E12ðn; xÞ; say: ð3:10Þ

Since f 0xðxÞ ¼ 0 and knðx; tÞ 6 1, we have

E12ðn; xÞ ¼Z x

x� xffiffinpjf 0xðtÞ � f 0xðxÞjknðx; tÞdt 6

Z x

x� xffiffinp

_xt

ðf 0xÞdt 6_x

x� xffiffinp

ðf 0xÞZ x

x� xffiffinp

dt ¼ xffiffiffinp

_xx� xffiffi

np

ðf 0xÞ: ð3:11Þ

Next, we estimate E11ðn; xÞ. By applying Lemma 2 and putting t ¼ x� xu, we have

E11ðn; xÞ 6xðxþ 3Þ

n� 1

Z x� xffiffinp

0jf 0xðtÞ � f 0xðxÞj

dt

ðx� tÞ26

xðxþ 3Þn� 1

Z x� xffiffinp

x

_xt

ðf 0xÞdt

ðx� tÞ2¼ ðxþ 3Þ

n� 1

Z ffiffinp

1

_xx�x

u

ðf 0xÞdu

6ðxþ 3Þn� 1

X½ ffiffinp �k¼1

_xx�x

u

ðf 0xÞ: ð3:12Þ

Substituting the values of E11ðn; xÞ and E12ðn; xÞ from Eqs. (3.11) and (3.12) in (3.10), we obtain

jE1ðn; xÞj 6xffiffiffinp

_xx� xffiffi

np

ðf 0xÞ þðxþ 3Þn� 1

X½ ffiffinp �k¼1

_xx�x

u

ðf 0xÞ: ð3:13Þ

Now, we find an estimate of E2ðn; xÞ.

E2ðn; xÞ 6Z 1

2x

Z t

xf 0xðuÞdu

� �Knðx; tÞdt

��������þ

Z 2x

x

Z t

xf 0xðuÞdu

� �@

@tð1� knðx; tÞÞdt

��������

6

Z 1

2x½f ðtÞ � f ðxÞ�Knðx; tÞdt

��������þ jf 0ðxþÞj

Z 1

2xðt � xÞKnðx; tÞdt

��������þ

Z 2x

xf 0xðuÞdu

��������j1� knðx;2xÞj

þZ 2x

xf 0xðtÞdtð1� knðx; tÞÞdt

��������:

We note that there exists an integer r ð2r P cÞ, such that f ðtÞ ¼ Oðt2rÞ, for every t > 0. Proceeding in a manner similar to thetreatment of E1ðn; xÞ in (3.10), and using Remark 2 and Lemma 2, we have

jE2ðn; xÞj 6 MZ 1

2xt2rKnðx; tÞdt þ jf ðxÞj

Z 1

2xKnðx; tÞdt þ jf 0ðxþÞj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixðxþ 3Þ

n� 1

rþ ðxþ 3Þ

n� 1jf ð2xÞ � f ðxÞ � xf 0ðxþÞj

þ xffiffiffinp

_xþ xffiffinp

x

ðf 0xÞ þðxþ 3Þn� 1

X½ ffiffinp �k¼1

_xþxk

x

ðf 0xÞ: ð3:14Þ

26 V. Gupta et al. / Applied Mathematics and Computation 236 (2014) 19–26

Since t 6 2ðt � xÞ and x 6 t � x when t P 2x, in view of Hölder’s inequality and Remark 2, we obtain

Z 1

2xt2rKnðx; tÞdt þ jf ðxÞj

Z 1

2xKnðx; tÞdt 6 22r

Z 1

2xðt � xÞ2rKnðx; tÞdt þ jf ðxÞj

x2

Z 1

2xðt � xÞ2Knðx; tÞdt

6 22rZ 1

2xðt � xÞ2rKnðx; tÞdt þ jf ðxÞj

x2 Mnðw2x ; xÞ 6

C1ðx; rÞnr

þ jf ðxÞj ðxþ 3Þxðn� 1Þ : ð3:15Þ

Finally, combining (3.9) and , (3.13)–(3.15), we get the required result. Hence, the proof is complete. h

Remark 3. Very recently Paltanea [10] considered generalized operators, the special case of which provides the well knownPhillips operators. We propose here the mixed operators having Lupas� and Paltanea basis functions in summation and inte-gral respectively using two parameters a > 0; q > 0 in the following way:

Pqaðf ; xÞ ¼

X1k¼1

la;kðxÞZ 1

0hqa;kðtÞf ðtÞdt þ la;0ðxÞf ð0Þ; x 2 ½0;1Þ;

where la;kðxÞ is as defined in (1.2) and

hqa;kðtÞ ¼

aqCðaqÞ e

�aqtðaqtÞkq�1:

It is observed that if q ¼ 1, we get the operators as discussed on p. 200 in [5], but at this moment its not possible to writethese operators in terms of hypergeometric representation.

Acknowledgments

The authors are thankful to the referees for making valuable suggestions leading to the better presentation of the paper.

References

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