319–336 DOI: 10.18514/MMN.2018.2216 BLENDING TYPE APPROXIMATION BY GENERALIZED BERNSTEIN-DURRMEYER TYPE OPERATORS ARUN KAJLA AND TUNCER ACAR Received 27 January, 2017 Abstract

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Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 19 (2018), No. 1, pp. 319–336 DOI: 10.18514/MMN.2018.2216

BLENDING TYPE APPROXIMATION BY GENERALIZED BERNSTEIN-DURRMEYER TYPE OPERATORS

ARUN KAJLA AND TUNCER ACAR Received 27 January, 2017

Abstract. In this article we construct a Durrmeyer modification of the operators introduced by Chen et al. in [10] based on a non-negative real parameter. We establish local approximation, global approximation, Voronovskaja type asymptotic theorem. The rate of convergence for differentiable functions whose derivatives are of bounded variation is also obtained.

2010Mathematics Subject Classification: 41A25; 26A15

Keywords: local approximation, global approximation, asymptotic formula, bounded variation

1. INTRODUCTION

The approximation theory by linear positive operators investigates how the functions can be best approximated by simpler functions. The most famous basic result for convergence of linear positive operators is due to Weierstrass who introduced an important theorem named Weierstrass’ approximation theorem. At last in 1912 Bern- stein introduced the most famous algebraic polynomialsBn.fIx/in approximation theory in order to give a constructive proof of Weierstrass’ theorem, which are given by

Bn.fIx/D

n

X

kD0

p_n;k.x/f k

n

; x2Œ0; 1;

wherep_n;k.x/D n k

!

x^k.1 x/^{n k} and he proved that iff 2C Œ0; 1thenBn.fIx/

converges uniformly tof .x/inŒ0; 1:

The Bernstein operators have been used in many branches of mathematics and computer science. Due to their useful structure, Bernstein polynomials and their modifications have been intensively studied. Among other papers, we refer the read- ers to [3,7,9,13,23,25].

c 2018 Miskolc University Press

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Forf 2C Œ0; 1;Chen et al. in [10] introduced a generalization of the Bernstein operators based on a non-negative parameter˛ .0˛1/as follows:

T_n^.˛/.fIx/D

n

X

kD0

p_n;k^.˛/.x/f k

n

; x2Œ0; 1 (1.1) where

p_n;k^.˛/.x/D

"

n 2 k

!

.1 ˛/xC n 2 k 2

!

.1 ˛/.1 x/C n k

!

˛x.1 x/

#

x^{k 1}.1 x/^{n k 1} (1.2) andn2:They proved the rate of convergence, Voronovskaja type asymptotic formula and shape preserving properties for these operators. For the special case,˛D1;

these operators reduce the well-known Bernstein operators.

The Durrmeyer type modification of the operators is a method to approximate the Lebesgue integrable functions. For this aim, many researchers have studied in this direction. Gupta and Rassias [18] introduced the Lupas-Durrmeyer type operators based on Polya distribution and established asymptotic approximation, local and global results. Goyal et al. [14] considered a one parameter family of Baskakov- Sz´asz type operators and studied quantitative convergence theorems for these operators. Gupta et al. [16] introduced hybrid operators involving inverse Polya- Eggenberger distribution and studied degree of approximation of these operators which include global approximation and uniform convergence. Very recently, Acu and Gupta [15] defined a summation-integral type operators depending on two para- meters and discussed some approximation results e.g. local approximation, Voro- novskaja type asymtotic theorem and weighted approximation of these operators.

In the literature survey, several authors have studied the approximation behavior of mixed hybrid operators (cf. [1,2,4–6,8,17,19–21]).

Inspired by their work, forf 2C Œ0; 1we define the following Durrmeyer type modification of the operators (1.1) as:

D_n^.˛/.fIx/D.nC1/

n

X

kD0

p_n;k^.˛/.x/

Z 1 0

p_n;k.t /f .t /dt; x2Œ0; 1: (1.3) The purpose of this paper is to study the Voronovskaja type theorem, local approximation, pointwise estimates and global approximation results for these operators (1.3).

The rate of convergence for differentiable functions whose derivatives are of bounded variation is also obtained.

2. BASIC RESULTS

In what follows letjj jjdenote the uniform norm onC Œ0; 1.

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Letei Dtⁱ; i2N[ f0g:By simple computation, we get Z 1

0

pn;k.t /tⁱdt D nŠ.kCi /Š

kŠ.nCiC1/Š: (2.1)

In order to prove the main results, we will show some lemmas in this section. We need the following auxiliary results.

Lemma 1([10]). For the operatorsTn^.˛/.fIx/;we have (i) Tn^.˛/.e0Ix/D1I

(ii) T_n^.˛/.e1Ix/DxI

(iii) Tn^.˛/.e2Ix/Dx²C.nC2.1 ˛//

n² x.1 x/I (iv) Tn^.˛/.e3Ix/Dx³C3.nC2.1 ˛//

n² x².1 x/

C.nC6.1 ˛//

n³ x.1 x/.1 2x/I (v) T_n^.˛/.e4Ix/Dx⁴C6.nC2.1 ˛//

n² x³.1 x/

C4.nC6.1 ˛//

n³ x².1 x/.1 2x/

C..3n.n 2/C12.n 6/.1 ˛// x.1 x/C.nC14.1 ˛///

n⁴ x.1 x/:

Lemma 2. For the operatorsD^.˛/_n .fIx/;we have (i) D^.˛/_n .e0Ix/D1I

(ii) D^.˛/n .e1Ix/DxC 1 2x .nC2/I

(iii) D^.˛/n .e2Ix/Dx²C2x².˛ 3n 4/

.nC2/.nC3/ C2x.2n ˛C1/

.nC2/.nC3/ C 2 .nC2/.nC3/I (iv) D^.˛/_n .e3Ix/Dx³C6x³. n.5C2n ˛/ 2.1C˛//

.nC2/.nC3/.nC4/

C3x².n.3n 2˛ 1/C10.˛ 1//

.nC2/.nC3/.nC4/ C 18x.n ˛C1/

.nC2/.nC3/.nC4/

C 6

.nC2/.nC3/.nC4/I

(v) D^.˛/_n .e4Ix/Dx⁴Cx⁴. 4.nC3/.16Cn.3C5n//C12˛.n 3/.n 2//

.nC2/.nC3/.nC4/.nC5/

C4x³.n 2/ .n.4n 3˛ 1/C33.˛ 1//

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C24x² nC3n²C14.˛ 1/ 4n˛

.nC2/.nC3/.nC4/.nC5/ C 48x.2n 3˛C3/

C 24

.nC2/.nC3/.nC4/.nC5/:

Proof. This lemma follows easily applying Lemma1and relation (2.1). Hence the

details are omitted.

Lemma 3. From Lemma2, we get (i) D^.˛/n .t xIx/D1 2x

nC2I

(ii) D^.˛/n ..t x/²Ix/D2x.1 x/.n ˛ 2/

.nC2/.nC3/ C 2

.nC2/.nC3/I (iii) D^.˛/n ..t x/⁴Ix/D12x³.x 2/ .n.n 2˛ 19/C46˛ 36/

C12x².n.n 2˛ 25/C58˛ 38/

.nC2/.nC3/.nC4/.nC5/ C 24x.3n 6˛C1/

C 24

.nC2/.nC3/.nC4/.nC5/:

Lemma 4. Forf 2C Œ0; 1;we havekD_n^.˛/.fI /k kfk: Proof. From definition (1.3) and Lemma2, we have

kD_n^.˛/.fI /k .nC1/

n

X

kD0

p^.˛/_n;k.x/

Z 1 0

p_n;k.t /jf .t /jdt kfkD^.˛/_n .e0Ix/D kfk:

Lemma 5. Forn2N, we have

D_n^.˛/..t x/²Ix/ 2_n².x/

.nC2/; where_n².x/D'².x/C_.n_C¹_2/ and'².x/Dx.1 x/:

Proof. This result is obtained by straightforward computation, but the details are

omitted.

Remark1. Letn^˛;mWDDn^.˛/..t x/^mIx/; mD1; 2; 4be the central moments of D_n^.˛/, we get

nlim!1n _n^˛;1.x/D1 2x;

nlim!1n _n^˛;2.x/D2x.1 x/;

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nlim!1n²_n^˛;4.x/D12x².1 x/²: 3. BASIC CONVERGENCE THEOREM

Theorem 1. Suppose thatf 2C Œ0; 1and˛ 2Œ0; 1: Then lim

n!1D_n^.˛/.fIx/D f .x/;uniformly inŒ0; 1.

Proof. Applying Lemma2,Dn^.˛/.e0Ix/D1; Dn^.˛/.e1Ix/!x; Dn^.˛/.e2Ix/!x² asn! 1; uniformly inŒ0; 1. By the well-known Bohman-Korovkin theorem, it follows thatDn^.˛/.fIx/!f .x/asn! 1;uniformly inŒ0; 1.

4. LOCALAPPROXIMATION

TheK-functional is given by :

K2.f; ı/Dinffjjf gjj Cıjjg⁰⁰jj Wg2W²g.ı > 0/;

whereW²D fgWg⁰⁰2C Œ0; 1gandjj:jjis the uniform norm onC Œ0; 1:By [11] there exists a positive constantM > 0such that

K2.f; ı/M!2.f;p

ı/; (4.1)

where the second order modulus of continuity forf 2C Œ0; 1is defined as

!2.f;p

ı/D sup

0<hp ı

sup

x;xC2h2Œ0;1jf .xC2h/ 2f .xCh/Cf .x/j: We define the usual modulus of continuity forf 2C Œ0; 1as

!.f; ı/D sup

0<hı

sup

x;xCh2Œ0;1

jf .xCh/ f .x/j:

Theorem 2. For the operatorsD_n^.˛/;there exists a constantM > 0such that jD_n^.˛/.fIx/ f .x/jM!2

f; .nC2/ ¹⁼²n.x/

C!

f;

ˇ ˇ ˇ ˇ

1 2x nC2 ˇ ˇ ˇ ˇ

;

wheref 2C Œ0; 1,˛2Œ0; 1; _n².x/D'².x/C_.n_C¹_2/ andx2Œ0; 1:

Proof. We define the auxiliary operators as follows:

D^.˛/_n .fIx/DD^.˛/_n .fIx/Cf .x/ f

nxC1 nC2

:

Then, we can easily check that

D^.˛/_n .1Ix/D1 and D^.˛/_n .tIx/Dx:

Letg2W²andt2Œ0; 1. By Taylor’s expansion we have g.t /Dg.x/C.t x/g⁰.x/C

Z t x

.t u/g⁰⁰.u/du:

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Applying the operatorD^.˛/_n on both sides of the above relation, we may write D^.˛/_n .gIx/Dg.x/CD^.˛/_n

Z t x

.t u/g⁰⁰.u/du

Dg.x/CD_n^.˛/

Z t x

.t u/g⁰⁰.u/du; x

Z ^nx_.nC2/^C¹

x

nxC1 nC2 u

g⁰⁰.u/du:

Hence

jD^.˛/_n .gIx/ g.x/j D_n^.˛/

ˇ ˇ ˇ ˇ

Z t

x jt ujjg⁰⁰.u/jdu ˇ ˇ ˇ ˇ

; x

C ˇ ˇ ˇ ˇ

Z _.n^nx^C¹

C2/

x

ˇ ˇ ˇ ˇ

nxC1 nC2 u

ˇ ˇ ˇ

ˇjg⁰⁰.u/jdu ˇ ˇ ˇ ˇ

D_n^.˛/..t x/²Ix/C

nxC1 nC2 x

2 jjg⁰⁰jj D

D_n^.˛/..t x/²Ix/C

1 2x nC2

2

jjg⁰⁰jj: (4.2) From Lemma5, we have

D_n^.˛/..t x/²Ix/C

1 2x nC2

2

.nC2/_n².x/C

1 2x nC2

2

.nC2/_n².x/C 1 .nC2/²

3

.nC2/_n².x/: (4.3)

Thus, by (4.2) we have

jD^.˛/_n .gIx/ g.x/j 3

.nC2/_n².x/jjg⁰⁰jj; (4.4) wherex2Œ0; 1:Furthermore, by Lemma4, we have

jD^.˛/_n .fIx/j 3jjfjj; (4.5) for allf 2C Œ0; 1andx2Œ0; 1.

Now, forf 2C Œ0; 1andg2W², using (4.4) and (4.5) we obtain that jD_n^.˛/.fIx/ f .x/j

ˇ ˇ ˇ ˇ

D^.˛/_n .fIx/ f .x/Cf

nxC1 nC2

f .x/

ˇ ˇ ˇ ˇ

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jD^.˛/_n .f gIx/j C jD^.˛/_n .gIx/ g.x/j C jg.x/ f .x/j C

ˇ ˇ ˇ ˇ f

nxC1 nC2

f .x/

ˇ ˇ ˇ ˇ 4jjf gjj C 3

.nC2/_n².x/jjg⁰⁰jj C!

f;

ˇ ˇ ˇ ˇ

1 2x nC2

ˇ ˇ ˇ ˇ

:

Taking the infimum on the right hand side over allg2W²;we get jD_n^.˛/.fIx/ f .x/j4K2

f; 1

.nC2/_n².x/

C!

f;

ˇ ˇ ˇ ˇ

:

Now considering the relation (4.1), we obtain jD_n^.˛/.fIx/ f .x/jM!2

f; .nC2/ ¹⁼²n.x/

C!

f;

ˇ ˇ ˇ ˇ

;

which completes the proof.

Leta10,a2> 0and let us now consider the Lipschitz-type space [24]:

Lip_M ./WD

f 2C Œ0; 1W jf .t / f .x/j M jt xj

.tCa1x²Ca2x/²Ix; t2.0; 1/

;

where2.0; 1:

Theorem 3. Letf 2Lip_M ./:Then, for allx2.0; 1;we have jD_n^.˛/.fIx/ f .x/j M n^˛;2.x/

a1x²Ca2x

!₂

;

where_n^˛;2.x/DD_n^.˛/..t x/²Ix/:

Proof. First, we show the result for the caseD1:We may write jD_n^.˛/.fIx/ f .x/j .nC1/

n

X

kD0

p_n;k^.˛/.x/

Z 1

0 jf .t / f .x/jdt M.nC1/

n

X

kD0

p_n;k^.˛/.x/

Z 1 0

p_n;k.t / jt xj

p.tCa1x²Ca2x/dt:

Using the fact that 1

p.tCa1x²Ca2x/< 1

p.a1x²Ca2x/ and the Cauchy-Schwarz inequality, we have

jD^.˛/_n .fIx/ f .x/j M.nC1/

p.a1x²Ca2x/

n

X

kD0

p_n;k^.˛/.x/

Z 1 0

pn;k.t /jt xjdt

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D M

p.a1x²Ca2x/D^.˛/_n .jt xjIx/M 0

@ s

n^˛;2.x/

a1x²Ca2x 1 A;

hence the result is obtained for D1: Now, we prove the theorem for the case 0 < < 1:By the H¨older’s inequality withpD¹ andqD₁¹ ;we get

jD^.˛/_n .fIx/ f .x/j .nC1/

n

X

kD0

p_n;k^.˛/.x/

Z 1 0

p_n;k.t /jf .t / f .x/jdt

ⁿ

X

kD0

p_n;k^.˛/.x/

.nC1/

Z 1 0

pn;k.t /jf .t / f .x/jdt

1

.nC1/

n

X

kD0

p^.˛/_n;k.x/

Z 1 0

p_n;k.t /jf .t / f .x/j¹dt

M

.nC1/

n

X

kD0

p_n;k^.˛/.x/

Z 1 0

pn;k.t / jt xj

p.tCa1x²Ca2x/dt

M

.a1x²Ca2x/²

.nC1/

n

X

kD0

p^.˛/_n;k.x/

Z 1 0

p_n;k.t /jt xjdt

M

.a1x²Ca2x/².D_n^.˛/.jt xjIx//M ^˛;2_n .x/

a1x²Ca2x

!₂ :

Thus, the proof is completed.

Next, we study the local direct estimate of the operators defined in (1.3) applying the Lipschitz-type maximal function of ordergiven by Lenze [22] as

e!.f; x/D sup

t¤x; t2Œ0;1

jf .t / f .x/j

jt xj ; x2Œ0; 1 and 2.0; 1: (4.6) Theorem 4. Letf 2C Œ0; 1and0 < 1:Then, for allx2Œ0; 1;we have

jD_n^.˛/.fIx/ f .x/j e!.f; x/

2

.nC2/_n².x/

₂ :

Proof. In view of (4.6), we have

jf .t / f .x/j e!.f; x/jt xj and

jD_n^.˛/.fIx/ f .x/j D_n^.˛/.jf .t / f .x/jIx/e!.f; x/D_n^.˛/.jt xjIx/:

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Now, applying the H¨older’s inequality withpD2 and 1

q D1 1

p;we have jD_n^.˛/.fIx/ f .x/j e!.f; x/D_n^.˛/..t x/²Ix/²

e!.f; x/

2

.nC2/_n².x/

₂ :

5. GLOBAL APPROXIMATION

Letf 2C Œ0; 1and'.x/Dp

x.1 x/; x2Œ0; 1:The second order Ditzian-Totik modulus of smoothness and correspondingK-functional are given by, respectively,

!₂^'.f;p

ı/D sup

0<hp ı

sup

x˙h'.x/2Œ0;1

jf .xCh'.x// 2f .x/Cf .x h'.x//j; KQ_2;'.x/.f; ı/Dinffjjf gjj Cıjj'²g⁰⁰jj Cı²jjg⁰⁰jj Wg2W².'/g; .ı > 0/;

whereW².'/D fg2C Œ0; 1Wg⁰2AC_locŒ0; 1; '²g⁰⁰2C Œ0; 1gandg⁰2AC_locŒ0; 1

means that g is differentiable andg⁰is absolutely continuous on every closed interval Œa; b.0; 1/:It is known ([12], Theorem 1.3.1) that there exists a positive constant M > 0;such that

KQ_2;'.x/.f; ı/M!₂^'.f;p

ı/: (5.1)

Also, the Ditzian-Totik modulus of first order is given by

!!.f; ı/D sup

0<hı

sup

x˙^h₂ .x/2Œ0;1

ˇ ˇ ˇ ˇ f

xCh

2 .x/

f

x h

2 .x/

ˇ ˇ ˇ ˇ

;

where WŒ0; 1!Ris an admissible step-weight function.

Now we state our next main result.

Theorem 5. Letf 2C Œ0; 1and0˛1:Then, forx2Œ0; 1;

jjD_n^.˛/f fjj M!₂^'.f; .nC2/ ¹⁼²/C !! f; .nC2/ ¹

C! fI.nC2/ ¹

;

where'².x/Dx.1 x/and .x/D

1 2x x2Œ0; 1=2

2x 1 x2Œ1=2; 1 : Proof. We consider the auxiliary operators as follows:

D^.˛/_n .fIx/DD^.˛/_n .fIx/Cf .x/ f

nxC1 nC2

:

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Letg2W².'/then by using Taylor’s expansion ofg;on proceeding as in the proof of Theorem2, we obtain that

jD^.˛/_n .gIx/ g.x/j D_n^.˛/

ˇ ˇ ˇ ˇ

Z t

x jt ujjg⁰⁰.u/jdu ˇ ˇ ˇ ˇ

; x

C Z _.n^nxC1

C2/

x

ˇ ˇ ˇ ˇ

nxC1 nC2 u

ˇ ˇ ˇ

ˇjg⁰⁰.u/jdu: (5.2) SettinguDˇxC.1 ˇ/t; ˇ2Œ0; 1;and also applying the concavity of_n², we have

jt uj

_n².u/ D ˇjt xj

_n².ˇxC.1 ˇ/t / ˇjt xj

_n².x/ˇC_n².t /.1 ˇ/ jt xj

_n².x/ : (5.3) Thus, inequality (5.2), in view of (5.3) leads us to

jD^.˛/_n .gIx/ g.x/j D_n^.˛/

ˇ ˇ ˇ ˇ

Z t x

jt uj _n².u/du

ˇ ˇ ˇ ˇ

; x

jj_n²g⁰⁰jj

C 0 B B

@ Z _.n^nxC1

C2/

x

ˇ ˇ ˇ ˇ

nxC1 nC2 u

ˇ ˇ ˇ ˇ _n².u/ du

1 C C

Ajj_n²g⁰⁰jj: 1

_n².x/jj_n²g⁰⁰jj

D^.˛/_n ..t x/²Ix/C

1 2x nC2

2

: (5.4) Now, using inequality (4.3), we get

jD^.˛/_n .gIx/ g.x/j 3

.nC2/jj_n²g⁰⁰jj

3

.nC2/

jj'²g⁰⁰jj C 1

.nC2/jjg⁰⁰jj

:

Applying (4.5) and (5.4), we have forf 2C Œ0; 1;

jD_n^.˛/.fIx/ f .x/j jD^.˛/_n .f g; x/j C jD^.˛/_n .gIx/ g.x/j C jg.x/ f .x/j C

ˇ ˇ ˇ ˇ f

nxC1 nC2

f .x/

ˇ ˇ ˇ ˇ 4jjf gjj C 3

.nC2/jj'²g⁰⁰jj C 3

.nC2/²jjg⁰⁰jj C

ˇ ˇ ˇ ˇ f

nxC1 nC2

f .x/

ˇ ˇ ˇ ˇ

Taking the infimum on the right hand side over allg2W².'/;we get jD^.˛/_n .fIx/ f .x/j4KQ2;'

f; 1

nC2

C ˇ ˇ ˇ ˇ f

nxC1 nC2

f .x/

ˇ ˇ ˇ ˇ

: (5.5)

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Also, ˇ ˇ ˇ ˇ f

nxC1 nC2

f .x/

ˇ ˇ ˇ ˇD

ˇ ˇ ˇ ˇ f

xC1 2x nC2

f .x/

ˇ ˇ ˇ ˇ

ˇ ˇ ˇ ˇ f

xC.1 2x/

nC2

f

x .1 2x/

nC2 ˇ

ˇ ˇ ˇ

(5.6) C

ˇ ˇ ˇ ˇ f

x .1 2x/

nC2

f .x/

ˇ ˇ ˇ ˇ !! f; .nC2/ ¹

C! fI.nC2/ ¹

: (5.7)

Hence, combining (5.1), (5.5) and (5.6), the desired relation is immediate.

6. POINTWISE CONVERGENCE OFDn^.˛/

Now we present a Voronovskaja type asymptotic formula for the operatorsD_n^.˛/: Theorem 6. Letf 2C Œ0; 1and˛ 2Œ0; 1: Iff⁰; f⁰⁰ exists at a pointx2Œ0; 1

then

nlim!1n

D_n^.˛/.fIx/ f .x/

D.1 2x/f⁰.x/Cx.1 x/f⁰⁰.x/; (6.1) Further, iff⁰⁰2C Œ0; 1then (6.1) holds uniformly inŒ0; 1.

Proof. By Taylor’s formula, we can write f .t /Df .x/C.t x/f⁰.x/C1

2.t x/²f⁰⁰.x/C.t; x/.t x/²; (6.2) where.t; x/!0ast!xand is a continuous function onŒ0; 1. OperatingDn^.˛/to (6.2) and Remark1, we get

D_n^.˛/.fIx/ f .x/Df⁰.x/D_n^.˛/..t x/Ix/C1

2f⁰⁰.x/D_n^.˛/..t x/²Ix/

CD_n^.˛/..t; x/.t x/²Ix/;

nlim!1n

D_n^.˛/.fIx/ f .x/

D.1 2x/f⁰.x/Cx.1 x/f⁰⁰.x/

C lim

n!1nD_n^.˛/..t; x/.t x/²Ix/:

Since.t; x/!0ast!x;for a given > 0;there exists aı > 0such thatj.t; x/j<

wheneverjt xj< ı:Forjt xj ı;we havej.t; x/j M^{.t x/}_ı₂ ²;for someM >

0:Let _ı.t / denote the characteristic function of the interval .x ı; xCı/: From Lemma3, we get

jD_n^.˛/..t; x/.t x/²Ix/j D_n^.˛/.j.t; x/j.t x/²_ı.t /Ix/

CD_n^.˛/.j.t; x/j.t x/².1 _ı.t //Ix/

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< D_n^.˛/..t x/²Ix/CM

ı²D_n^.˛/..t x/⁴Ix/

D O 1

n

CO 1

n²

;

which implies that lim

n!1nD_n^.1=n/..t; x/.t x/²Ix/D0;due to the arbitrariness of > 0:This proves the first assertion of the theorem.

To prove the uniformity assertion, due to the uniform continuity off inŒ0; 1, the ıin the above proof can be chosen independent ofxand all the other estimates hold

uniformly inx2Œ0; 1:

7. RATE OF CONVERGENCE

DBV Œ0; 1denotes the class of all absolutely continuous functionsf defined on Œ0; 1, having onŒ0; 1a derivativef⁰ equivalent to a function of bounded variation onŒ0; 1. We notice that the functionsf 2DBV Œ0; 1possess a representation

f .x/D Z x

0

g.t /dtCf .0/

whereg2BV Œ0; 1, i.e.,gis a function of bounded variation onŒ0; 1.

The operatorsDn^.˛/.fIx/also admit the integral representation D_n^.˛/.fIx/D

Z 1 0

N_n^.˛/.x; t /f .t /dt; x2Œ0; 1; (7.1) where the kernelN_n^.˛/.x; t /is given by

N_n^.˛/.x; t /D.nC1/

n

X

kD0

p_n;k^.˛/.x/pn;k.t /:

Lemma 6. For a fixedx2.0; 1/and sufficiently largen, we have (i) #n.x; y/D

Z y 0

N_n^.˛/.x; t /dt 2 .nC2/

_n².x/

.x y/²; 0y < x;

(ii) 1 #n.x; ´/D Z 1

´

N_n^.˛/.x; t /dt 2 .nC2/

_n².x/

.´ x/²; x < ´ < 1:

Proof. (i) Using Lemma5we get

#n.x; y/D Z y

0

N_n^.˛/.x; t /dt Z y

0

x t x y

2

N_n^.˛/.x; t /dt

DD_n^.˛/..t x/²Ix/.x y/ ² 2 .nC2/

_n².x/

.x y/²:

The proof of (ii) is similar hence the details are omitted.

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Theorem 7. Letf 2DBV Œ0; 1: Then for everyx2.0; 1/and sufficiently large n, we have

jD^.˛/_n .fIx/ f .x/j ˇ ˇ ˇ ˇ

.1 2x/

nC2 ˇ ˇ ˇ ˇ

jf⁰.xC/Cf⁰.x /j 2

C s

2

.nC2/n.x/jf⁰.xC/ f⁰.x /j 2

C2_n².x/

.nC2/x ¹

Œp n

X

kD1 x

_

x .x=k/

.f_x⁰/C x pn

x

_

x .x=p n/

.f_x⁰/

C 2_n².x/

.1 x/.nC2/

Œp n

X

kD1

xC..1 x/=k/

_

x

.f_x⁰/

C.1 x/

pn

xC..1 x/=p n/

_

x

.f_x⁰/;

whereWb

a.f_x⁰/denotes the total variation off_x⁰onŒa; bandf_x⁰is defined by

f_x⁰.t /D 8

<

:

f⁰.t / f⁰.x /; 0t < x

0; tDx

f⁰.t / f⁰.xC/ x < t < 1:

(7.2)

Proof. SinceD_n^.˛/.e0Ix/D1;by applying (7.1), for everyx2.0; 1/we obtain D_n^.˛/.fIx/ f .x/D

Z 1 0

N_n^.˛/.x; t /.f .t / f .x//dt

D Z 1

0

N_n^.˛/.x; t / Z t

x

f⁰.u/dudt: (7.3) For anyf 2DBV Œ0; 1;by (7.2) we may write

f⁰.u/Df_x⁰.u/C1

2.f⁰.xC/Cf⁰.x //C1

2.f⁰.xC/ f⁰.x //sg n.u x/

Cıx.u/Œf⁰.u/ 1

2.f⁰.xC/Cf⁰.x //; (7.4) where

ı_x.u/D

1 ; uDx 0 ; u¤x:

Obviously, Z 1

0

Z t x

f⁰.u/ 1

2.f⁰.xC/Cf⁰.x //

ıx.u/du

N_n^.˛/.x; t /dtD0:

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By (7.1) and simple calculations we have Z 1

0

Z t x

1

2.f⁰.xC/Cf⁰.x //du

N_n^.˛/.x; t /dt

D1

2.f⁰.xC/Cf⁰.x //

Z 1 0

.t x/N_n^.˛/.x; t /dt

D1

2.f⁰.xC/Cf⁰.x //D^.˛/_n ..t x/Ix/

andˇ ˇ ˇ ˇ

Z 1 0

N_n^.˛/.x; t / Z t

x

1

2.f⁰.xC/ f⁰.x //sg n.u x/du

dt ˇ ˇ ˇ ˇ

1

2 jf⁰.xC/ f⁰.x /j Z 1

0 jt xjN_n^.˛/.x; t /dt 1

2 jf⁰.xC/ f⁰.x /jD_n^.˛/.jt xjIx/

1

2 jf⁰.xC/ f⁰.x /j

D_n^.˛/..t x/²Ix/

1=2

:

Considering Lemmas3and5and using (7.3), (7.4) we obtain the following estimate

jD_n^.˛/.fIx/ f .x/j 1

2jf⁰.xC/Cf⁰.x /j ˇ ˇ ˇ ˇ

1 2x .nC2/

ˇ ˇ ˇ ˇ

C1

2jf⁰.xC/ f⁰.x /j s 2

.nC2/n.x/

C ˇ ˇ ˇ ˇ

Z x 0

Z t x

f_x⁰.u/du

N_n^.˛/.x; t /dt

C Z 1

x

Z t x

f_x⁰.u/du

N_n^.˛/.x; t /dt ˇ ˇ ˇ ˇ

: (7.5)

Let

F_n^.˛/.f_x⁰; x/D Z x

0

Z t x

f_x⁰.u/du

N_n^.˛/.x; t /dt;

G_n^.˛/.f_x⁰; x/D Z 1

x

Z t x

f_x⁰.u/du

N_n^.˛/.x; t /dt:

To complete the proof, it is sufficient to estimate the terms F_n^.˛/.f_x⁰; x/ and Gn^.˛/.f_x⁰; x/:SinceRb

a dt#n.x; t /1for allŒa; bŒ0; 1;using integration by parts

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and applying Lemma6withyDx .x=p

n/;we have jF_n^.˛/.f_x⁰; x/j D

ˇ ˇ ˇ ˇ

Z x 0

Z t x

f_x⁰.u/du

dt#n.x; t / ˇ ˇ ˇ ˇD

ˇ ˇ ˇ ˇ

Z x 0

#n.x; t /f_x⁰.t /dt ˇ ˇ ˇ ˇ

Z y 0 C

Z x y

jf_x⁰.t /j j#n.x; t /jdt 2_n².x/

.nC2/

Z y 0

x

_

t

.f_x⁰/.x t / ²dtC Z x

y x

_

t

.f_x⁰/dt

2_n².x/

.nC2/

Z y 0

x

_

t

.f_x⁰/.x t / ²dtC x pn

x

_

x .x=p n/

.f_x⁰/:

By the substitution ofuDx=.x t /;we obtain 2_n².x/

.nC2/

Z x .x=p n/

0

.x t / ²

x

_

t

.f_x⁰/dtD2_n².x/

.nC2/x ¹ Z ^pn

1 x

_

x .x=u/

.f_x⁰/du

2_n².x/

.nC2/x ¹

Œp n

X

kD1

Z kC1 k

x

_

x .x=k/

.f_x⁰/du

2_n².x/

.nC2/x ¹

Œp n

X

kD1 x

_

x .x=k/

.f_x⁰/:

Thus,

jF_n^.˛/.f_x⁰; x/j 2_n².x/

.nC2/x ¹

Œp n

X

kD1 x

_

x .x=k/

.f_x⁰/C x pn

x

_

x .x=p n/

.f_x⁰/: (7.6)

Using integration by parts and applying Lemma6with ´DxC..1 x/=p n/;we have

jG_n^.˛/.f_x⁰; x/j D ˇ ˇ ˇ ˇ

Z 1 x

Z t x

f_x⁰.u/du

N_n^.˛/.x; t /dt ˇ ˇ ˇ ˇ D

ˇ ˇ ˇ ˇ

Z ´ x

Z t x

f_x⁰.u/du

dt.1 #n.x; t //

C Z 1

´

Z t x

f_x⁰.u/du

dt.1 #n.x; t //

ˇ ˇ ˇ ˇ D

ˇ ˇ ˇ ˇ

Z t x

f_x⁰.u/.1 #n.x; t //du ´

x

Z ´ x

f_x⁰.t /.1 #n.x; t //dt

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C Z 1

´

Z t x

f_x⁰.u/du

dt.1 #n.x; t //

ˇ ˇ ˇ ˇ D

ˇ ˇ ˇ ˇ

Z ´ x

f_x⁰.u/du.1 #n.x; ´//

Z ´ x

f_x⁰.t /.1 #n.x; t //dt

C Z t

x

f_x⁰.u/du.1 #n.x; t //

1

´

Z 1

´

f_x⁰.t /.1 #n.x; t //dt ˇ ˇ ˇ ˇ D

ˇ ˇ ˇ ˇ

Z ´ x

f_x⁰.t /.1 #n.x; t //dtC Z 1

´

f_x⁰.t /.1 #n.x; t //dt ˇ ˇ ˇ ˇ

2_n².x/

.nC2/

Z 1

´ t

_

x

.f_x⁰/.t x/ ²dtC Z ´

x t

_

x

.f_x⁰/dt

D2_n².x/

.nC2/

Z 1

xC..1 x/=p n/

t

_

x

.f_x⁰/.t x/ ²dt

C.1 x/

pn

xC..1 x/=p n/

_

x

.f_x⁰/:

By the substitution ofvD.1 x/=.t x/;we get jG_n^.˛/.f_x⁰; x/j 2_n².x/

.nC2/

Z ^pn 1

xC..1 x/=v/

_

x

.f_x⁰/.1 x/ ¹dv (7.7)

C.1 x/

pn

xC..1 x/=p n/

_

x

.f_x⁰/

2_n².x/

.1 x/.nC2/

Œp n

X

kD1

Z kC1 k

xC..1 x/=v/

_

x

.f_x⁰/dv (7.8)

C.1 x/

pn

xC..1 x/=p n/

_

x

.f_x⁰/

D 2_n².x/

.1 x/.nC2/

Œp n

X

kD1

xC..1 x/=k/

_

x

.f_x⁰/ (7.9)

C.1 x/

pn

xC..1 x//=p n

_

x

.f_x⁰/: (7.10)

Combining the estimates (7.5)-(7.7), we get the required result.

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REFERENCES

[1] U. Abel, V. Gupta, and M. Ivan, “Asymptotic approximation of functions and their derivatives by generalized Baskakov-Sz´asz-Durrmeyer operators,”Anal. Theory Appl., vol. 21, no. 1, pp. 15–26, 2005, doi:10.1007/BF02835246. [Online]. Available:http://dx.doi.org/10.1007/BF02835246 [2] T. Acar, “Asymptotic formulas for generalized Sz´asz-Mirakyan operators,” Appl. Math.

Comput., vol. 263, pp. 233–239, 2015, doi: 10.1016/j.amc.2015.04.060. [Online]. Available:

http://dx.doi.org/10.1016/j.amc.2015.04.060

[3] T. Acar and A. Aral, “On pointwise convergence of q-Bernstein operators and their q-derivatives,” Numer. Funct. Anal. Optim., vol. 36, no. 3, pp. 287–304, 2015, doi:

10.1080/01630563.2014.970646. [Online]. Available: http://dx.doi.org/10.1080/01630563.2014.

970646

[4] T. Acar, A. Aral, and I. Ras¸a, “Modified Bernstein-Durrmeyer operators,”General Mathematics, vol. 22, pp. 27–41, 2014.

[5] T. Acar, V. Gupta, and A. Aral, “Rate of convergence for generalized Sz´asz operators,”Bull.

Math. Sci., vol. 1, no. 1, pp. 99–113, 2011, doi:10.1007/s13373-011-0005-4. [Online]. Available:

http://dx.doi.org/10.1007/s13373-011-0005-4

[6] T. Acar and G. Ulusoy, “Approximation by modified Sz´asz-Durrmeyer operators,”Period. Math.

Hungar., vol. 72, no. 1, pp. 64–75, 2016, doi:10.1007/s10998-015-0091-2. [Online]. Available:

http://dx.doi.org/10.1007/s10998-015-0091-2

[7] P. N. Agrawal, M. Goyal, and A. Kajla, “q-Berntsein-Schurer-Kantorovich type operators,”Boll.

Unione Mat. Ital., vol. 8, pp. 169–180, 2015.

[8] P. N. Agrawal, V. Gupta, A. Sathish Kumar, and A. Kajla, “Generalized Baskakov-Sz´asz type operators,”Appl. Math. Comput., vol. 236, pp. 311–324, 2014, doi: 10.1016/j.amc.2014.03.084.

[Online]. Available:http://dx.doi.org/10.1016/j.amc.2014.03.084

[9] D. C´ardenas-Morales, P. Garrancho, and I. Ras¸a, “Asymptotic formulae via a Korovkin-type result,”Abstr. Appl. Anal., pp. Art. ID 217 464, 12, 2012, doi: 10.1155/2012/217464. [Online].

Available:http://dx.doi.org/10.1155/2012/217464

[10] X. Chen, J. Tan, Z. Liu, and J. Xie, “Approximation of functions by a new family of generalized Bernstein operators,”J. Math. Anal. Appl., vol. 450, no. 1, pp. 244–261, 2017, doi:

10.1215/20088752-3764507. [Online]. Available:http://dx.doi.org/10.1016/j.jmaa.2016.12.075 [11] R. A. DeVore and G. G. Lorentz, Constructive Approximation, ser. Grundlehren der

Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer- Verlag, Berlin, 1993, vol. 303. [Online]. Available:http://dx.doi.org/10.1007/978-3-662-02888-9.

doi:10.1007/978-3-662-02888-9

[12] Z. Ditzian and V. Totik, Moduli of smoothness, ser. Springer Series in Computational Mathematics. Springer-Verlag, New York, 1987, vol. 9. [Online]. Available: http:

//dx.doi.org/10.1007/978-1-4612-4778-4. doi:10.1007/978-1-4612-4778-4

[13] A. D. Gadjiev and A. M. Ghorbanalizadeh, “Approximation properties of a new type Bernstein-Stancu polynomials of one and two variables,” Appl. Math. Comput., vol.

216, no. 3, pp. 890–901, 2010, doi: 10.1016/j.amc.2010.01.099. [Online]. Available:

http://dx.doi.org/10.1016/j.amc.2010.01.099

[14] M. Goyal, V. Gupta, and P. N. Agrawal, “Quantitative convergence results for a family of hybrid operators,”Appl. Math. Comput., vol. 271, pp. 893–904, 2015, doi: 10.1016/j.amc.2015.08.122.

[15] V. Gupta and A. M. Acu, “Direct results for certain summation-integral type Baskakov-Sz´asz operators,”Results. Math., pp. 10.1007/s00 025–016–0603–2., 2017.

(18)

[16] V. Gupta, A. M. Acu, and D. F. Sofonea, “Approximation of Baskakov type P´olya-Durrmeyer operators,”Appl. Math. Comput., vol. 294, pp. 318–331, 2017, doi: 10.1016/j.amc.2016.09.012.

[17] V. Gupta and R. P. Agarwal,Convergence Estimates in Approximation Theory. Springer, Cham, 2014. [Online]. Available: http://dx.doi.org/10.1007/978-3-319-02765-4. doi: 10.1007/978-3- 319-02765-4

[18] V. Gupta and T. M. Rassias, “Lupas¸-Durrmeyer operators based on Polya distribution,”

Banach J. Math. Anal., vol. 8, no. 2, pp. 146–155, 2014. [Online]. Available: http:

//projecteuclid.org/euclid.bjma/1396640060

[19] A. Kajla and T. Acar, “A new modification of Durrmeyer type mixed hybrid operators,”Carpath- ian J. Math., vol. 34, no. 1, pp. 47–56, 2018.

[20] A. Kajla, A. M. Acu, and P. N. Agrawal, “Baskakov-Sz´asz-type operators based on inverse P´olya-Eggenberger distribution,” Ann. Funct. Anal., vol. 8, no. 1, pp. 106–123, 2017, doi:

10.1215/20088752-3764507. [Online]. Available:http://dx.doi.org/10.1215/20088752-3764507 [21] A. Kajla and P. N. Agrawal, “Sz´asz-Durrmeyer type operators based on Charlier polynomials,”

Appl. Math. Comput., vol. 268, pp. 1001–1014, 2015, doi:10.1016/j.amc.2015.06.126. [Online].

Available:http://dx.doi.org/10.1016/j.amc.2015.06.126

[22] B. Lenze, “On Lipschitz-type maximal functions and their smoothness spaces,” Nederl. Akad.

Wetensch. Indag. Math., vol. 50, no. 1, pp. 53–63, 1988.

[23] M. Mursaleen, K. J. Ansari, and A. Khan, “On .p; q/-analogue of Bernstein operators,”

Appl. Math. Comput., vol. 266, pp. 874–882, 2015, doi: 10.1016/j.amc.2015.04.090. [Online].

[24] M. A. ¨Ozarslan and H. Aktu˘ glu, “Local approximation properties for certain King type operators,” Filomat, vol. 27, no. 1, pp. 173–181, 2013, doi: 10.2298/FIL1301173O. [Online].

Available:http://dx.doi.org/10.2298/FIL1301173O

[25] M. Wang, D. Yu, and P. Zhou, “On the approximation by operators of Bernstein-Stancu types,”

Appl. Math. Comput., vol. 246, pp. 79–87, 2014, doi: 10.1016/j.amc.2014.08.015. [Online].

Authors’ addresses

Arun Kajla

Department of Mathematics, Central University of Haryana, Haryana-123031, India E-mail address:rachitkajla47@gmail.com

Tuncer Acar

Department of Mathematics,Faculty of Science, Selcuk University, Selcuklu, Konya, 42003, Turkey E-mail address:tunceracar@ymail.com