Prashantkumar Patel, Vishnu Narayan Mishra

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2010 Mathematics Subject Classification: 41A25, 41A30, 41A36. .... n,γ ppt ´ xqm,xq, m P N. Then. µ α,β n,1,γpxq “ B α,β n,γ pt ´ x, xq “ α ´ βx n ` β. ,. µ α,β.
DEMONSTRATIO MATHEMATICA Vol. XLVIII No 1 2015

Prashantkumar Patel, Vishnu Narayan Mishra APPROXIMATION PROPERTIES OF CERTAIN SUMMATION INTEGRAL TYPE OPERATORS Abstract. In the present paper, we study approximation properties of a family of linear positive operators and establish direct results, asymptotic formula, rate of convergence, weighted approximation theorem, inverse theorem and better approximation for this family of linear positive operators.

1. Introduction In 2005, Finta [4] studied a new type of Baskakov–Durrmeyer operators as follows ż8 8 ÿ (1.1) Bn pf, xq “ pn,k pxq bn,k ptqf ptqdt ` pn,0 pxqf p0q, k“1

where

0

ˆ ˙ n`k´1 xk pn,k pxq “ , k p1 ` xqn`k

1 tk´1 . Bpn ` 1, kq p1 ` tqn`k`1 Gupta et al. [12] studied pointwise convergence, an asymptotic formula and error estimation for the operators Bn pf, ¨q. Recently, Govil and Gupta [7] studied some approximation properties for the operators Bn pf, ¨q and estimated local results in terms of modulus of continuity. Generalization of the operators (1.1) with parameter γ was discussed by Gupta [9], defined as ż8 8 ÿ Bn,γ pf, xq “ pn,k,γ pxq (1.2) bn,k,γ ptqf ptqdt ` pn,0,γ pxqf p0q, bn,k ptq “

k“1

0

2010 Mathematics Subject Classification: 41A25, 41A30, 41A36. Key words and phrases: Bernstein polynomial, Baskakov operators, direct results, asymptotic formula, rate of convergence, weighted approximation theorem, inverse theorem, better approximation. DOI: 10.2478/dema-2015-0008 c Copyright by Faculty of Mathematics and Information Science, Warsaw University of Technology

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P. Patel, V. N. Mishra

where pn,k,γ pxq “ bn,k,γ ptq “

pγxqk Γpn{γ ` kq ¨ , Γpk ` 1qΓpn{γq p1 ` γxqpn{γq`k

γΓpn{γ ` k ` 1q pγtqk´1 . ¨ ΓpkqΓpn{γ ` 1q p1 ` γtqpn{γq`k`1

The operators Bn,γ pf, ¨q are called a certain integral modification of the well known Baskakov operators with the weight function of Beta basis function. As a special case i.e. γ “ 1, the operators Bn,γ pf, ¨q reduce to the operators (1.1). Recently, in [13, 19] authors introduced generalization of Bernstein polynomials with two parameters α and β with 0 ≤ α ≤ β and investigated convergence and approximation properties of these operators. Many other researchers work in this direction and obtain different approximation properties of many operators [20, 21, 15, 16, 11, 1]. Motivated by such type operators, Patel and Mishra [17] introduced Stancu type generalization of integral modification of the well known Baskakov operators with the weight function of Beta basis function as: For x P r0, 8q, γ ą 0, 0 ≤ α ≤ β, ˆ ˙ ż8 8 ÿ nt ` α α,β pn,k,γ pxq bn,k,γ ptqf Bn,γ pf ptq, xq “ (1.3) dt n`β 0 k“1 ˆ ˙ α ` pn,0,γ pxqf , n`β where pn,k,γ pxq and bn,k,γ ptq as defined in (1.2). This present note deals with operators (1.3). We estimate moments pα,βq and recurrence relation for the moments for the operators Bn,γ . Also, we study asymptotic formula, local approximation theorems, rate of convergence, weighted approximation theorem, inverse theorem and better approximation for these operators. 2. Estimation of moments α,β i In this section, we shall obtain Bn,γ pt , xq, i “ 0, 1, 2, . . .. Lemma 2.1. [9] Let the function Tn,m,γ pxq, m, P N Y t0u be defined as (2.1)

Tn,m,γ pxq “ Bn,γ ppt ´ xqm , xq ż8 8 ÿ “ pn,k,γ pxq bn,k,γ ptqpt ´ xqm dt ` p1 ` γxq´n{γ p´xqm . k“1

0

Approximation properties of certain summation integral type operators

Then Tn,0,γ pxq “ 1, Tn,1,γ pxq “ 0 and Tn,2,γ pxq “

79

2xp1 ` γxq , and also the n´γ

following recurrence relation holds: pn ´ γmqTn,m`1,γ pxq p1q “ xp1 ` γxqrTn,m,γ pxq ` 2mTn,m´1,γ pxqs ` mp1 ` 2γxqTn,m,γ pxq.

Lemma 2.2. The following equalities hold: α,β p1, xq “ 1, (1) Bn,γ nx ` α α,β , (2) Bn,γ pt, xq “ n`β n2 pn ` γq 2n pn ` αpn ´ γqq α2 α,β 2 2 (3) Bn,γ pt , xq “ x ` x ` for pn ` βq2 pn ´ γq pn ` βq2 pn ´ γq pn ` βq2 n ą γ. α,β Proof. The operators Bn,γ are well defined on the function 1, t, t2 , we obtain α,β Bn,γ p1, xq “ 1, α,β Bn,γ pt, xq “

n α nx ` α Bn,γ pt, xq ` Bn,γ p1, xq “ . n`β n`β n`β

For n ą γ, we have ˆ ˙2 n 2nα α2 α,β 2 Bn,γ pt , xq “ Bn,γ pt2 , xq ` B pt, xq ` Bn,γ p1, xq n,γ n`β pn`βq2 pn`βq2 ˆ ˙2 ˆ ˙ n 2x x2 pn ` γq 2nαx α2 “ ` ` ` n`β n´γ pn ´ γq pn ` βq2 pn ` βq2 3 2 2 2 n x ´ α γ ` nαpα ´ 2xγq ` n xp2 ` 2α ` xγq “ pn ` βq2 pn ´ γq 2n pn ` αpn ´ γqq α2 n2 pn ` γq 2 x ` x ` . “ pn ` βq2 pn ´ γq pn ` βq2 pn ´ γq pn ` βq2 α,β Lemma 2.3. If we define the central moments as µα,β n,m,γ pxq “ Bn,γ ppt ´ xqm , xq, m P N. Then α ´ βx α,β , µα,β n,1,γ pxq “ Bn,γ pt ´ x, xq “ n`β ` 2 ˘ 2n ´ 2nαβ ` 2αβγ α2 α,β α,β 2 µn,2,γ pxq “ Bn,γ ppt ´ xq , xq “ `x pn ` βq2 pn ` βq2 pn ´ γq ` 2 ˘ 2 2 nβ ` 2n γ ´ β γ ` x2 . pn ` βq2 pn ´ γq Remark 2.4. For all m P N, 0 ≤ α ≤ β; we have the following recursive α,β m relation for the images of the monomials tm under Bn,γ pt , xq in terms of

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P. Patel, V. N. Mishra

Bn,γ ptj , xq; j “ 0, 1, 2, . . . , m as α,β m Bn,γ pt , xq

m ˆ ˙ j m´j ÿ m n α B ptj , xq. “ m n,γ pn ` βq j j“0

Also, α,β Bn,γ ppt ´ xqm , xq “

m ˆ ˙ ÿ m α,β k p´xqm´k Bn,γ pt , xq. k k“0

It is easily verified that for each x P p0, 8q α,β m Bn,γ pt , xq “

nm Γpn{γ ` mqΓpn{γ ´ m ` 1q m x pn ` βqm Γpn{γ ` 1qΓpn{γq mnm´1 Γpn{γ ` m ´ 1qΓpn{γ ´ m ` 1q ` pn ` βqm Γpn{γ ` 1qΓpn{γq ˆ rnpm ´ 1q ` αpn{γ ´ m ` 1qs xm´1 αmpm ´ 1qnm´2 Γpn{γ ` m ´ 2qΓpn{γ ´ m ` 2q ` pn ` βqm Γpn{γ ` 1qΓpn{γq „  αpn{γ ´ m ` 2q m´2 ˆ npm ´ 2q ` x ` Opn´2 q. 2

3. Direct result and asymptotic formula Let the space CB r0, 8q of all continuous and bounded functions be endowed with the norm }f } “ supt|f pxq| : x P r0, 8qu. Further, let us consider the following K-functional: (3.1)

K2 pf, δq “ inf t}f ´ g} ` δ}g 2 }u, gPW 2

where δ ą 0 and W 2 “ tg P CB r0, 8q : g 1 , g 2 P CB r0, 8qu. By the method given in [2, p.177, Theorem 2.4], there exists an absolute constant C ą 0 such that ? (3.2) K2 pf, δq ≤ Cω2 pf, δq, where (3.3)

? ω2 pf, δq “

sup

sup |f px ` 2hq ´ 2f px ` hq ` f pxq|

? 0ăhă δ xPr0,8q

is the second order modulus of smoothness of f P CB r0, 8q. Also, we set ? (3.4) ωpf, δq “ sup? sup |f px ` hq ´ f pxq|. 0ăhă δ xPr0,8q

Approximation properties of certain summation integral type operators

81

Theorem 3.1. For f P CB r0, 8q, we have ˙ ˆ d ˙ ˙ ˆ ˆ α ´ βx 2 |α´βx| α,β α,β `Cω2 f, µn,2,γ pxq ` , |Bn,γ pf, xq´f pxq| ≤ ω1 f, n`β n`β where C is positive constant. Proof. We are introducing the auxiliary operators as follows ˆ ˙ nx ` α α,β α,β ˆn,γ pf, xq “ Bn,γ pf, xq ´ f B ` f pxq, n`β α,β ˆn,γ are linear and preserves the linear for every x P r0, 8q. The operators B functions: ˆ α,β pt ´ x, xq “ 0. (3.5) B n,γ

Let g P

2 W8

and x, t P r0, 8q. By Taylor’s expansion, we have żt gptq “ gpxq ` pt ´ xqg 1 pxq ` pt ´ uqg 2 puqdu. x

α,β ˆn,γ , we get Applying B

ˆ α,β pg, xq ´ gpxq “ g 1 pxqB ˆ α,β pt ´ x, xq ` B ˆ α,β B n,γ n,γ n,γ

ˆż t

˙ pt ´ uqg puqdu, x , 2

x

and applying (3.5), we get ˆ α,β pg, xq ´ gpxq “ B ˆ α,β B n,γ n,γ

ˆż t

˙ pt ´ uqg puqdu, x . 2

x

Hence, ˇ ˙ˇ ˆż t ˇ ˇ α,β 2 α,β ˆ ˇ pt ´ uqg puqdu, x ˇˇ |Bn,γ pg, xq ´ gpxq| ≤ ˇBn,γ x

ˇż ˇ ` ˇˇ

nx`α n`β

ˇ ˙ ˇ nx ` α 2 ´ u g puqduˇˇ n`β x ˇ ˙ ż nx`α ˇˆ ˇ n`β ˇ nx ` α α,β 2 2 2 ˇ ˇ ≤ Bn,γ ppt ´ xq , xq}g } ` ˇ n ` β ´ u g puqˇdu x « ˆ ˙ ff α ´ βx 2 α,β ≤ µn,2,γ pxq ` }g 2 }. n`β ˆ

Since α,β |Bn,γ pf, xq| ˇ ˆ ˇ ˆ ˙ˇ ˙ˇ ż8 8 ÿ ˇ nt`α ˇ ˇ ˇ α ˇ ˇ ˇ ˇ ≤ }f }, ≤ pn,k,γ pxq bn,k,γ ptqˇf dt ` pn,0,γ ˇf ˇ ˇ n`β n`β 0 k“1

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P. Patel, V. N. Mishra

α,β |Bn,γ pf, xq ´ f pxq|

ˇˆ ˇ ˙ ˇ nx ` α ˇ α,β α,β ˆ ˆ ˇ ≤ |Bn,γ pf ´ g, xq ´ pf ´ gqpxq| ` |Bn,γ pg, xq ´ gpxq| ` ˇ ´ f pxqˇˇ n`β « ˆ ˙2 ff ˆ ˙ α ´ βx |α ´ βx| α,β 2 ≤ 4}f ´ g} ` µn,2,γ pxq ` }g } ` ω1 f, . n`β n`β Hence, taking infimum on the right hand side over all g P W 2 , we get ˜ ˆ ˙ ¸ ˆ ˙ α ´ βx 2 |α ´ βx| α,β α,β |Bn,γ pf, xq ´ f pxq| ≤ K f, µn,2,γ pxq ` ` ω1 f, . n`β n`β In view of (3.2), we get ˆ ˆ d ˙ ˙ ˆ ˙ α ´ βx 2 |α ´ βx| α,β pxq ` |Bn,γ pf, xq´f pxq| ≤ Cω2 f, µα,β `ω f, . 1 n,2,γ n`β n`β This completes the proof of the theorem. Our next result in this section is the Voronovskaja type asymptotic formula: Let Bx2 r0, 8q “ tf : for every x P r0, 8q, |f pxq| ≤ Mf p1 ` x2 q}, Mf being a constant depending on f . By Cx2 r0, 8q, we denote the subspace of all continuous functions to Bx2 r0, 8q. Also, Cx˚2 r0, 8q is a subspace of f pxq all functions f P Cx2 r0, 8q, for which lim is finite. The norm on xÑ8 1 ` x2 |f pxq| . Cx˚2 r0, 8q is }f }x2 “ sup 2 xPr0,8q 1 ` x Theorem 3.2. For any function f P Cx2 r0, 8q such that f 1 , f 2 P Cx2 r0, 8q, we have α,β lim nrBn,γ pf, xq´f pxqs “ pα´βxqf 1 pxq`xp1`γxqf 2 pxq,

nÑ8

for every x ≥ 0.

Proof. Let f, f 1 , f 2 P Cx˚2 r0, 8q and x P r0, 8q be fixed. By Taylor expansion, we can write pt ´ xq2 2 f pxq ` rpt, xqpt ´ xq2 , 2! where rpt, xq is the Peano form of the remainder, rpt, xq P CB r0, 8q and α,β lim rpt, xq “ 0. Applying Bn,γ , we get f ptq “ f pxq ` pt ´ xqf 1 pxq `

tÑx

α,β nrBn,γ pf, xq ´ f pxqs α,β “ f 1 pxqnBn,γ pt ´ x, xq `

f 2 pxq α,β α,β nBn,γ ppt ´ xq2 , xq ` nBn,γ prpt, xqpt ´ xq2 , xq, 2!

Approximation properties of certain summation integral type operators

83

α,β lim nrBn,γ pf, xq ´ f pxqs

nÑ8

f 2 pxq α,β lim nBn,γ ppt ´ xq2 , xq nÑ8 2 nÑ8 α,β ` lim nBn,γ prpt, xqpt ´ xq2 , xq

α,β “ f 1 pxq lim nBn,γ ppt ´ xq, xq ` nÑ8

α,β “ pα ´ βxqf 1 pxq ` xp1 ` γxqf 2 pxq ` lim nBn,γ prpt, xqpt ´ xq2 , xq. nÑ8

By Cauchy–Schwarz inequality, we have (3.6)

α,β lim nBn,γ prpt, xqpt ´ xq2 , xq

nÑ8

α,β 2 α,β “ lim nBn,γ pr pt, xq, xq1{2 Bn,γ ppt ´ xq4 , xq1{2 .

Observe that

r2 px, xq

“0

nÑ8 and r2 p¨, xq

P Cx˚2 r0, 8q. Then, it follows that

α,β 2 lim nBn,γ pr pt, xq, xq “ r2 px, xq “ 0,

(3.7)

nÑ8

uniformly with respect to x P r0, As. Now from (3.6) and (3.7), we obtain α,β lim nBn,γ prpt, xqpt ´ xq2 , xq “ 0.

nÑ8

Thus, we have α,β lim nrBn,γ pf, xq ´ f pxqs “ pα ´ βxqf 1 pxq ` xp1 ` γxqf 2 pxq,

nÑ8

which completes the proof. 4. Rate of convergence For any positive a, by ωa pf, δq “ sup

sup |f ptq ´ f pxq|

|t´x|≤δ x,tPr0,as

we denote the usual modulus of continuity of f on the closed interval r0, as. We know that for a function f P Cx2 r0, 8q, the modulus of the continuity ωa pf, δq tends to zero. α,β Now, we give a rate of convergence theorem for the operators Bn,γ . Theorem 4.1. Let f P Cx2 r0, 8q and ωa`1 pf, δq be its modulus of continuity on the finite interval r0, a ` 1s Ă r0, 8q, where a ą 0. Then for every n ą 3, ˘ ` b α,β α,β }Bn,γ pf q ´ f }Cr0,as ≤ 6Mf p1 ` a2 qµα,β pxq ` ω f, µn,2,γ pxq . a`1 n,2,γ Proof. For x P r0, as and t ą a ` 1, since t ´ x ą 1, we have (4.1)

|f ptq ´ f pxq| ≤ Mf p2 ` x2 ` t2 q ` ˘ ≤ Mf 2 ` 3x2 ` 2pt ´ xq2 ≤ 6Mf p1 ` a2 qpt ´ xq2 .

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P. Patel, V. N. Mishra

For x P r0, as and t ≤ a ` 1, we have ˆ (4.2)

|f pxq ´ f ptq| ≤ ωa`1 pf, |t ´ x|q ≤

1`

|t ´ x| δ

˙ ωa`1 pf, δq

with δ ą 0. From (4.1) and (4.2), we can write ˆ

(4.3)

|t ´ x| |f ptq ´ f pxq| ≤ 6Mf p1 ` a qpt ´ xq ` 1 ` δ 2

˙

2

ωa`1 pf, δq,

for x P r0, as and t ≥ 0. Thus α,β α,β |Bn,γ pf, xq ´ f pxq| ≤ Bn,γ p|f ptq ´ f pxq|, xq α,β ≤ 6Mf p1 ` a2 qBn,γ ppt ´ xq2 , xq ˙ ˆ 1 α,β 2 1{2 . `ωa`1 pf, δq 1 ` Bn,γ ppt ´ xq , xq δ

Hence, by Schwarz’s inequality and Lemma 2.3, for x P r0, as α,β |Bn,γ pf, xq ´ f pxq|

ˆ ˙ b 1 α,β ≤ 6Mf p1 ` a2 qµα,β pxq ` ω pf, δq 1 ` µ pxq . a`1 n,2,γ n,2,γ δ b Taking δ “ µα,β n,2,γ pxq, we get α,β }Bn,γ pf q

´ f }Cr0,as ≤ 6Mf p1 ` a

2

qµα,β n,2,γ pxq

ˆ b ˙ α,β ` ωa`1 f, µn,2,γ pxq .

This completes the proof of theorem. Corollary 4.2. If f P LipM1 ρ on r0, a ` 1s, then for n ą γ b α,β }Bn,γ pf q ´ f }Cr0,as ≤ p1 ` 2M1 q µα,β n,2,γ pxq. Proof. For sufficiently large n, µα,β n,2,γ pxq ≤

b µα,β n,2,γ pxq.

Hence, by f P LipM1 ρ, we obtain the assertion of the corollary. 5. Weighted approximation Now we shall discuss the weighted approximation theorem, where the approximation formula holds true on the interval r0, 8q.

Approximation properties of certain summation integral type operators

85

Theorem 5.1. For each f P Cx˚2 r0, 8q, we have α,β lim }Bn,γ pf q ´ f }x2 “ 0.

nÑ8

Proof. Using the theorem in [5], we see that it is sufficient to verify the following three conditions (5.1)

α,β r lim }Bn,γ pt , xq ´ xr }x2 “ 0, r “ 0, 1, 2.

nÑ8

α,β Since Bn,γ p1, xq “ 1, the first condition of (5.1) is fulfilled for r “ 0. By Lemma 2.2, we have α,β |Bn,γ pt, xq ´ x| 1 ` x2 xPr0,8q ˇ ˇ ˇ nx ˇ α βx ` α x ≤ ˇˇ ´ xˇˇ sup ` ≤ 2 n`β n`β n`β xPr0,8q 1 ` x

α,β }Bn,γ pt, xq ´ x}x2 “ sup

and the second condition of (5.1) holds for r “ 1 as n Ñ 8. Similarly, we can write for n ą γ α,β 2 pt , xq ´ x2 | |Bn,γ 1 ` x2 xPr0,8q ˇ ˇ ˇ n2 pn ` γq ˇ x2 ˇ ˇ sup ≤ˇ ´ 1 ˇ 2 pn ` βq2 pn ´ γq xPr0,8q 1 ` x ˇ ˇ ˇ 2n pn ` αpn ´ γqq ˇ α2 x ˇ sup ` ` ˇˇ pn ` βq2 pn ´ γq ˇ xPr0,8q 1 ` x2 pn ` βq2 ˇ ˇ ˇ ´2n2 β ´ nβ 2 ` 2n2 γ ` 2nβγ ` β 2 γ ˇ ˇ ≤ ˇˇ ˇ pn ` βq2 pn ´ γq ˇ ˇ ˇ 2n pn ` αpn ´ γqq ˇ α2 ˇ` ` ˇˇ , pn ` βq2 pn ´ γq ˇ pn ` βq2

α,β 2 }Bn,γ pt , xq ´ x2 }x2 “ sup

which implies that α,β 2 lim }Bn,γ pt , xq ´ x2 }x2 “ 0.

nÑ8

Thus the proof is completed. We give the following to approximate all functions in Cx2 r0, 8q. This type of results are given in [6] for locally integrable functions. Theorem 5.2. For each f P Cx2 r0, 8q and ν ą 0, we have α,β |Bn,γ pf, xq ´ f pxq| “ 0. nÑ8 xPr0,8q p1 ` x2 qν`1

lim

sup

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P. Patel, V. N. Mishra

Proof. For any fixed x0 ą 0, α,β α,β α,β |Bn,γ pf, xq´f pxq| |Bn,γ pf, xq´f pxq| |Bn,γ pf, xq ´ f pxq| sup ≤ sup ` sup 2 ν`1 2 ν`1 p1 ` x q p1 ` x q p1 ` x2 qν`1 x≤x0 x≥x0 xPr0,8q α,β ≤ }Bn,γ pf q ´ f }Cr0,x0 s ` }f }x2 sup

x≥x0

α,β |f pxq| |Bn,γ p1`t2 , xq| ` sup . p1 ` x2 qν`1 p1 ` x2 q1`ν x≥x0

The first term of the above inequality tends to zero from Theorem 4.1. By Lemma 2.2, for any fixed x0 ą 0, it is easily seen that sup x≥x0

α,β |Bn,γ p1 ` t2 , xq| p1 ` x2 qν`1

tends to zero n Ñ 8. We can choose x0 ą 0 so large that the last part of the above inequality can be made small enough. Thus the proof is completed. Remark 5.3. The following theorem is inverse theorem in simultaneous α,β . approximation for the operators Bn,γ Theorem. Let 0 ă ζ ă 2, 0 ă a1 ă a2 ă b2 ă b1 ă 8 and suppose f P Cγ r0, 8q. Then in the following statements (i)ñ(ii) ` ˘ α,β prq (i) }pBn,γ q pf, ¨q ´ f prq }Cra1 ,b1 s “ O n´ζ{2 , (ii) f prq P Lip˚ pα, a2 , b2 q, where Lip˚ pα, a2 , b2 q denotes the Zygmund class satisfying ω2 pf, δ, a2 , b2 q ≤ M2 δ ζ . The proof of the above Theorem follows along line of [12, Theorem 4.1]; thus, we omit the details. 6. Better approximation Many well-known operators preserve the linear as well as constant function for example Bernstein, Baskakov, Szász–Mirakyan and Szász-Beta operators possess these properties i.e. Ln pei , xq “ ei pxq where ei pxq “ xi pi “ 0, 1q. To make the convergence faster, King [14] proposed an approach to modify the classical Bernstein polynomial, so that the sequence preserve test function e0 and e2 . Later this approach was applied to some well-known operators. For detail see [3, 18]. α,β As the operators Bn,γ preserve only the constant function, further modification of these operators is proposed such that the modified operators preserves the constant as well as linear function, for this purpose the modi-

Approximation properties of certain summation integral type operators

87

α,β fication of Bn,γ is as follows:

(6.1)

˚pα,βq Bn,γ pf, xq

8 ÿ



ż8

ˆ bn,k,γ ptqf

pn,k,γ prn pxqq 0

k“1

nt ` α n`β

˙ dt

ˆ

˙ α ` pn,0,γ prn pxqqf , n`β „ ˙ pn ` βqx ´ α α where rn pxq “ and x P In “ ,8 . n n`β Lemma 6.1. For each x P In , we have ˚pα,βq Bn,γ p1, xq “ 1,

˚pα,βq Bn,γ pt, xq “ x, ` ˘ 2 ´nα ` α2 γ 2xpn ´ 2αγq x2 pn ` γq ˚pα,βq 2 ` ` . Bn,γ pt , xq “ n´γ pn ` βqpn ´ γq pn ` βq2 pn ´ γq

Lemma 6.2. For each x P In and n ą γ, we have ˚pα,βq Bn,γ pt ´ x, xq “ 0, ˚pα,βq

˚pα,βq ppt ´ xq2 , xq µn,2,γ pxq “ Bn,γ ` ˘ 2 ´nα ` α2 γ 2x pn ´ 2αγq 2γx2 “ ` ` . pn ` βq2 pn ´ γq pn ` βq2 pn ´ γq pn ´ γq

Theorem 6.3. Let f P CB pIn q and x P In . Then for n ą γ, there exists a positive constant such that ˆ b ˙ ˚pα,βq ˚pα,βq |Bn,γ pf, xq ´ f pxq| ≤ M ω2 f, µn,2,γ pxq . Proof. Let g P CB pIn q and x, t P In . By Taylor’s expansion, we have żt 1 gptq “ gpxq ` pt ´ xqg pxq ` pt ´ uqg 2 puqdu. x

Applying

˚pα,βq Bn,γ ,

we get

˚pα,βq Bn,γ pg, xq ´ gpxq

ˆż t “g

1

˚pα,βq pxqBn,γ ppt

´ xq, xq `

˚pα,βq Bn,γ

˙ pt ´ uqg puqdu, x . 2

x

ˇż t ˇ ˇ ˇ 2 Obviously, we have ˇˇ pt ´ uqg puqduˇˇ ≤ pt ´ xq2 }g 2 }, x ` ˘ ˚pα,βq ˚pα,βq ˚pα,βq |Bn,γ pg, xq ´ gpxq| ≤ Bn,γ pt ´ xq2 , x }g 2 } “ µn,2,γ pxq}g 2 }.

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Since |Bn,γ

pf, xq| ≤ }f }

˚pα,βq ˚pα,βq |Bn,γ pf, xq ´ f pxq| ≤ |Bn,γ pf ´ g, xq ´ pf ´ gqpxq| ˚pα,βq ` |Bn,γ pg, xq ´ gpxq| ˚pα,βq

≤ 2}f ´ g} ` µn,2,γ pxq}g 2 }. Taking infimum over all g P Cx2 pIn q, we obtain ˆ b ˙ ˚pα,βq ˚pα,βq |Bn,γ pf, xq ´ f pxq| ≤ K2 f, µn,2,γ pxq . In view of (3.2), we have ˚pα,βq |Bn,γ pf, xq

˙ ˆ b ˚pα,βq ´ f pxq| ≤ M ω2 f, µn,2,γ pxq ,

which proves the theorem. Theorem 6.4. For f P Cx2 pIn q such that f 1 , f 2 P Cx2 pIn q, we have ” ı ˚pα,βq lim n Bn,γ pf, xq ´ f pxq “ px ` γx2 qf 2 pxq. nÑ8

The proof follows along the lines of Theorem 3.2. Remark 6.5. One can discuss rate of approximation in weighted space for ˚pα,βq the operators Bn,γ . We omit the details as it is similar to Theorem 5.1 and 5.2. Remark 6.6. Very recently Gupta and Aral [10] introduced the q-analogue of original Baskakov-Beta operators [8]. We can now propose the q-analogue of operators (1.2) as ż 8{A 8 ÿ q q (6.2) Bn,γ pf, xq “ pn,k,γ pxq q ´k bqn,k,γ ptqf ptqdt ` pqn,0,γ pxqf p0q, 0

k“1

where pqn,k,γ pxq “ q

k2 2

bqn,k,γ pxq “ γq

k2 2

Γq pn{γ ` kq pqγxqk ¨ , Γq pk ` 1qΓq pn{γq p1 ` qγxqqpn{γq`k Γq pn{γ ` k ` 1q pγtqk´1 ¨ Γq pkqΓq pn{γ ` 1q p1 ` γtqqpn{γq`k`1

and the q improper integral is defined as ˆ n˙ n ż 8{A 8 ÿ q q f pxqdq x “ p1 ´ qq f , A A 0 n“´8

A ą 0.

Approximation properties of certain summation integral type operators

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For further details, we refer the reader to [10, 1]. For the operators (6.2), one can study their local approximation properties and Voronovskaja type asymptotic formula results based on q-integer. Acknowledgment. The authors are thankful to the referee, for his/her critical suggestion, for the overall improvement of the paper. References [1] A. Aral, V. Gupta, R. P. Agarwal, Applications of q-Calculus in Operator Theory, Springer, 2013. [2] R. A. De Vore, G. G. Lorentz, Constructive Approximation, Springer, Berlin, 1993. [3] O. Duman, M. A. Ozarsla¨ n, Sz´ asz–Mirakjan type operators providing a better error estimation, Appl. Math. Lett. 20 (2007), 1184–1188. [4] Z. Finta, On converse approximation theorem, J. Math. Anal. Appl. 312(1) (2005), 159–180. [5] A. D. Gadzhiev, Theorems of the type of P. P. Korovkin type theorems, Math. Zametki 20(5) (1976), 781–786. Math. Notes, English Translation 20(5-6) (1976), 996–998. [6] A. D. Gadjiev, R. O. Efendiyev, E. Ibikli, On Korovkin type theorem in the space of locally integrable functions, Czechoslovak Math. J. 1(128) (2003), 45–53. [7] N. K. Govil, V. Gupta, Direct estimates in simultaneous approximation for Durrmeyer type operators, Math. Ineqaul. Appl. 10(2) (2007), 371–379. [8] V. Gupta, A note on modified Baskakov type operators, Approx. Theory Appl. (N.S.) 10(3) (1994), 74–78. [9] V. Gupta, Approximation for modified Baskakov Durrmeyer type operators, Rocky Mountain J. Math. 39(3) (2009), 1–16. [10] V. Gupta, A. Aral, Approximation by q Baskakov-Beta operators, Acta Math. Appl. Sinica 27(4) (2011), 569–580. [11] V. Gupta, H. Karsli, Some approximation properties by q-Szasz–Mirakyan–Baskakov– Stancu operators, Lobachevskii J. Math. 33(2) (2012), 175–182. [12] V. Gupta, M. A. Noor, M. S. Beniwal, On simultaneous approximation for certain Baskakov Durrmeyer type operators, J. Inequal. Pure Appl. Math. 7 (2006), 1–15. [13] B. Ibrahim, Approximation by Stancu–Chlodowsky polynomials, Comput. Math. Appl. 59 (2010), 274–282. [14] J. P. King, Positive linear operators which preseves x2 , Acta Math. Hungar. 99 (2003), 203–208. [15] V. Mishra, P. Patel, Approximation properties of q-Baskakov–Durrmeyer–Stancu operators, Mathematical Sciences 7(1) (2013), 38. [16] V. Mishra, P. Patel, Some approximation properties of modified Jain-Beta operators, J. Cal. Var. 2013, (2013), 8 pages. [17] V. Mishra, P. Patel, On simultaneous approximation for generalized integral type Baskakov operators, (submitted for publication). [18] M. A. Ozarsla¨ n, O. Duman, MKZ type operators providing a better estimation on [1/2,1), Canad. Math. Bull. 50 (2007), 434–439. [19] D. D. Stancu, Approximation of function by means of a new generalized Bernstein operator, Calcolo 20(2) (1983), 211–229. [20] D. K. Verma, V. Gupta, P. N. Agrawal, Some approximation properties of Baskakov– Durrmeyer–Stancu operators, Appl. Math. Comput. 218 (2012), 6549–6556.

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[21] R. Yadav, Bézier variant of Baskakov-Beta-Stancu operators, Ann. Univ. Ferrara, (2011). DOI 10.1007/s11565-011-0145-1 P. Patel DEPARTMENT OF APPLIED MATHEMATICS & HUMANITIES SARDAR VALLABHBHAI NATIONAL INSTITUTE OF TECHNOLOGY SURAT-395 007 (GUJARAT), INDIA DEPARTMENT OF MATHEMATICS ST. XAVIER’S COLLEGE AHMEDABAD-380 009 (GUJARAT), INDIA E-mail: [email protected] V. N. Mishra DEPARTMENT OF APPLIED MATHEMATICS & HUMANITIES SARDAR VALLABHBHAI NATIONAL INSTITUTE OF TECHNOLOGY SURAT-395 007 (GUJARAT), INDIA L. 1627 AWADH PURI COLONY BENIGANJ PHASE -III, OPP. I.T.I. AYODHYA MAIN ROAD FAIZABAD-224 001 (UTTAR PRADESH), INDIA E-mail: [email protected], [email protected]

Received August 7, 2013; revised version October 28, 2013. Communicated by V. Gupta.