RATE OF CONVERGENCE OF q ANALOGUE OF A CLASS OF NEW BERNSTEIN TYPE OPERATORS

Vol. 19 (2018), No. 1, pp. 211–234 DOI: 10.18514/MMN.2018.2265

RATE OF CONVERGENCE OF q ANALOGUE OF A CLASS OF NEW BERNSTEIN TYPE OPERATORS

SHEETAL DESHWAL, ANA MARIA ACU, AND P. N. AGRAWAL Received 04 March, 2017

Abstract. Sharma [32] introduced aq-analogue of a new sequence of classical Bernstein type operators defined by Deo et al. [14] for functions defined in the intervalŒ0;_nCⁿ₁:The purpose of this paper is to study the rate of convergence of these operators with the aid of the modulus of continuity and a Lipschitz type space. Subsequently, we define the bivariate case of these operators and discuss the approximation properties by means of the complete and partial modulus of continuity, Lipschitz class and the Peetre’sK- functional. Some numerical results which show the rate of convergence of these operators to certain functions using Maple algorithms are given. Lastly, we construct the associated GBS operators and study the approximation of B¨ogel continuous and B¨ogel differentiable functions. The comparison of convergence of the bivariate operator and its GBS type operator is made considering numerical examples.

2010Mathematics Subject Classification: 41A25; 26A15; 41A28

Keywords: q-Bernstein operators, complete modulus of continuity, Lipschitz class function, GBS operators

1. INTRODUCTION

Forf 2C Œ0; 1, Bernstein [8] constructed a sequence of polynomials Bn.fIx/D

n

X

kD0

bn;k.x/f k

n

;

where

b_n;k.x/D n k

!

x^k.1 x/^{n k}; nD1; 2 ; x2Œ0; 1;

and proved that the sequenceBn.fIx/converges tof .x/;asn! 1;uniformly in x2Œ0; 1. These polynomials are called Bernstein polynomials and possess many remarkable properties. Forf 2L1Œ0; 1, the space of Lebesgue integrable functions inŒ0; 1, Durrmeyer [18] introduced an integral modification of Bernstein operators

c 2018 Miskolc University Press

as

Dn.fIx/D.nC1/

n

X

kD0

b_nk.x/

Z 1 0

b_nk.t /f .t /dt;

which was extensively studied by Derriennic [15].

In recent years, the applications ofq-calculus in the area of approximation theory is one of the main areas of research (see [1], [2], [13]). Also, the reader should consult the monographies of A. Aral et al. [5], Gupta et al. [23] and G. Tachev et al. [24]. In 1987, forf 2C Œ0; 1and0 < q < 1, Lupas [27] introduced aq-analogue of Bernstein polynomials. After a decade, anotherq-generalization of Bernstein polynomials was introduced by Phillips [31]. The q-analogue of Bernstein polynomials due to Phil- lips was studied by several researchers e.g Ostrovska [29,30], Kim [26] and Wang [38] etc. In 2005, Derriennic [16] introduced theq-analogue of Bernstein-Durrmeyer polynomials with Jacobi weights and studied some approximation properties. Later, Gupta [21] introduced the q-analogue of the Bernstein-Durrmeyer operators which was investigated later by Finta and Gupta ([20], [22]) and several other research- ers. Dalmanoglu [13] introduced the Kantorovich type modification ofq-Bernstein polynomials and established some approximation results. Muraru [28] introduced Bernstein-Schurer polynomials based onq-integers and established the rate of con- vergence in terms of the modulus of continuity. Agrawal et al. [2] considered the Stancu variant of these operators and obtained some local and global direct results.

Later, Agrawal et al. [4] proposed the Durrmeyer type modification of these operators and discussed some local direct results and the rate of convergence of the modified limitq-Bernstein-Schurer type operators.

Deo et al. [14] introduced a new sequence of Bernstein type operatorsVnas

.Vnf /.x/D

n

X

kD0

pn;k.x/f k

n

; (1.1)

where

p_n;k.x/D

1C1 n

n

n k

! x^k

n nC1 x

n k

; x2

0; n nC1

:

In the same paper, to approximate Lebesgue integrable functions on the interval Œ0; 1;the authors defined an integral modification of the operators (1.1) as

.Lnf /.x/D.nC1/² n

n

X

kD0

pn;k.x/

Z _nⁿ

C1

0

pn;k.t /f .t /dt;

and studied some approximation properties.

From [25, page 69], the definiteq-integral in the intervalŒ0; a; a > 0is defined as:

Z a 0

f .x/dqxD.1 q/a

1

X

jD0

q^jf .q^ja/:

Later, Sharma [32] investigated theq-analogue of these operators given by .Ln;qf /.x/DŒnC1²_q

Œnq n

X

kD0

q ^kp_n;k;q.x/

Z _Œn^Œnq

C1q

0

p_n;k;q.qt /f .t /dqt; (1.2) where

p_n;k;q.x/WDŒnC1ⁿ_q Œnⁿ_q

"

n k

#

q

x^k

Œnq

ŒnC1q

x n k

q

; x2

0; Œnq

ŒnC1q

: In the present paper, first we obtain the order of approximation of the operators defined by (1.2) by means of the modulus of continuity and the Lipschitz class. Then we proceed to define the bivariate generalization of these operators and investigate their rate of convergence with the help of the moduli of continuity and the K- func- tional. Lastly, we introduce the associated GBS operators and discuss their degree of approximation by means of the mixed modulus of smoothness.

2. DIRECT RESULTS

Lemma 1([32]). For the operators given by (1.2), the following equalities hold:

i) .Ln;q1/.x/D1;

ii) .Ln;qt /.x/D Œnq

ŒnC1qŒnC2qC qŒnqx ŒnC2q

;

iii) .Ln;qt²/.x/D Œnq

ŒnC2qŒnC3q

Œn 1qq⁴x²Cq³xŒnqCq.1C2q/xŒnq

ŒnC1q

C^Œ2_ŒnC^qŒn_12^q q

. Consequently,

i) .Ln;q.t x//.x/D Œnq

ŒnC1qŒnC2qCqŒnq ŒnC2q

ŒnC2q

x;

ii) .Ln;q.t x/²/.x/D Œ2qŒn²_q

ŒnC1²_qŒnC2_qŒnC3_q C

q³Œn²_qCq.1C2q/Œn²_q 2ŒnqŒnC3q

ŒnC1qŒnC2qŒnC3q

x C

q⁴Œn 1qŒnq 2qŒnqŒnC3qCŒnC2qŒnC3q

ŒnC2qŒnC3q

x²:

Lemma 2. For the operators given by (1.2), the following equality holds

.Ln;q.t x/⁴/.x/D 1

ŒnC1⁴qŒnC2qŒnC3qŒnC4qŒnC5qn

x⁴q¹⁴.1 q/⁴Œn⁸_q

Cq⁹x³.1 q/².8xq⁶ 11xq⁵ q⁴C3xq⁴C4q³ 4xq³C6q² 6xq²C4q 4xqC1/Œn⁷_q q⁵x². 1 3qCq¹²x³C3q¹¹x³C3q¹⁰x³C12xq² 36xq⁸C4xqC15xq³C8xq⁶ C6xq⁵C4xq⁴ 19q⁵x² 17q⁶x²C74q¹⁰x²C53q⁸x² 26q¹¹x² 74q⁹x² 8q³x²

6q²x²C9q⁴x²C2xq⁹ 3xq¹⁰ 7q²C4q⁸ 7q³C3q⁶C6q⁵ 2q⁴C2q⁷/Œn⁶_qCO.Œn⁵_q/o :

In what follows, let.qn/n; 0 < qn< 1be a sequence satisfying the following con- dition

nlim!1qnD1:

Remark1. Let.qn/n; 0 < qn< 1be a sequence satisfying the following conditions

nlim!1qnD1; lim

n!1q_nⁿDc; c2Œ0; 1/:

By simple computations, we have

nlim!1Œnqn.Ln;qn.t x//.x/D1 .cC1/x;

nlim!1Œnqn.Ln;qn.t x/²/.x/D2x.1 x/;

nlim!1Œn²_q

n.Ln;q_n.t x/⁴/.x/Dx².1 x/.7x² 7xC5/:

3. MAIN RESULTS

Next we assume thatC Œ0;_Œn^Œn_C1^q

qis the class of all real valued continuous func- tions onŒ0;_Œn^Œn_C1^q

qendowed with the normjj:jjdefined as jjfjj D sup

x2h

0;_ŒnC1q^Œnq i

jf .x/j:

Lemma 3. For eachf 2C Œ0;_Œn^Œn_C1^q

q, we havejjLn;q.f /jj jjfjj:

Proof. Using the definition (1.2) and Lemma 1, the proof of this lemma easily

follows. Hence, the details are omitted.

Forf 2C Œ0;_Œn^Œn_C1^q

q, the Steklov mean is defined as f_h.x/D 4

h² Z ^h₂

0

Z ^h₂

0

Œ2f .xCuCv/ f .xC2.uCv//dudv: (3.1) The first and second order modulus of continuity are respectively defined as

!.f; ı/D sup

x;u;v2h 0;_Œn^Œnq

C1q

i

;ju vjı

jf .xCu/ f .xCv/j

and

!2.f; ı/D sup

x;u;v2 h

0;_Œn^Œnq

C1q

i

;ju vjı

jf .xC2u/ 2f .xCuCv/Cf .xC2v/j; ı > 0:

Lemma 4. The Steklov meanf_h.x/satisfies the following properties:

i) jjf_h fjj !2.f; h/;

ii) Iff is continuous, thenf_h⁰; f_h⁰⁰2Ch 0;_Œn^Œn_C1^q

q

i and

jjf_h⁰jj_C^h_0; ^Œnq

ŒnC1q

i 5

h!.f; h/; jjf_h⁰⁰jj_C^h_0; ^Œnq

ŒnC1q

i 9

h²!2.f; h/:

Theorem 1. Letf 2C Œ0;_Œn^Œn_C1^qn

qn. Then for eachx2Œ0;_Œn^Œn_C1^qn

qn;we have j.Ln;qnf /.x/ f .x/j 5!

f; 1

pŒnC2qn

1C2x pŒnC2qn

C!2

f; 1

pŒnC2qn

2C9

2

qn4Œn 1qnŒnqn 2qnŒnqnŒnC3qnCŒnC2qnŒnC3qn

ŒnC3qn

x²

C

q_n³Œn²_qnCqn.1C2qn/Œn²_qn 2ŒnqnŒnC3qn

ŒnC1qnŒnC3qn

xC Œ2qnŒn²_qn ŒnC1²_qnŒnC3qn

: Proof. Using the Steklov meanfhdefined by (3.1), we may write

j.Ln;qnf /.x/ f .x/j

j.Ln;qn.f fh//.x/j C j.Ln;qn.fh fh.x///.x/j C jfh.x/ f .x/j: (3.2) Using Lemma3and Lemma4, we have

j.Ln;qn.f fh//.x/j jjf fhjj !2.f; h/:

Now, by Taylor’s expansion, we have

f_h.t /Df_h.x/C.t x/f_h⁰.x/C Z t

x

.t u/f_h⁰⁰.u/du j.Ln;qn.fh fh.x///.x/j j.Ln;qn..t x/f_h⁰.x///.x/j

C ˇ ˇ ˇ ˇ

Ln;q_n

Z t x

.t u/f_h⁰⁰.u/du

.x/

ˇ ˇ ˇ ˇ jjf_h⁰jjj.Ln;qn.t x//.x/j

C jjf_h⁰⁰jj.L_n;qn.j Z t

x jt uj/duj/.x/

D jjf_h⁰jjj.Ln;qn.t x//.x/j C1

2jjf_h⁰⁰jj.Ln;qn.t x/²/.x/:

Applying Lemma 4, Lemma 1 and choosing hDq ₁

ŒnC2 qn

;we get the required

result.

Let us assume thatın;qn.x/Dp

Ln;qn..t x/²/.x/.

Theorem 2. Iff has a continuous derivativef⁰ and!.f⁰; ı/is the modulus of continuity off⁰onŒ0;_Œn^Œn_C1^qn

qn, then

j.Ln;qnf /.x/ f .x/j Mjn;qn.x/j C!.f⁰; ı²_n;qn.x//

1Cın;qn.x/

;

where M is a positive constant such thatjf⁰.x/j M; x2

0;_Œn^Œn_C1^qn

qn

and

n;qn.x/D

qnŒnqn ŒnC2qn

ŒnC2qn

xC Œnqn

ŒnC1qnŒnC2qn

: (3.3)

Proof. On applying the mean value theorem, we get f .t / f .x/D.t x/f⁰./

D.t x/f⁰.x/C.t x/.f⁰./ f⁰.x//;

where lies betweenx andt. Using the definition (1.2) and the Cauchy-Schwarz inequality, we have

j.Ln;qnf /.x/ f .x/j jf⁰.x/jj.Ln;qn.t x//.x/j CŒnC1²_qn Œnqn

n

X

kD0

q_n^kp_n;k;qn.x/

Z _Œn^Œnqn_C1qn

0

p_n;k;qn.qnt /jt xjjf⁰./ f⁰.x/jdqnt Mjn;qn.x/j CŒnC1²_qn

Œnqn

n

X

kD0

q_n^kp_n;k;qn.x/

Z _Œn^Œnqn_C1qn

0

p_n;k;qn.qnt /!.f⁰; ı/

jt xj ı C1

jt xjdqnt Mjn;qn.x/j C!.f⁰; ı/

ı .Ln;qn.t x/²/.x/

C!.f⁰; ı/

q

.Ln;qn.t x/²/.x/:

On choosingıWDı_n;q² _n.x/, we get the desired result.

Sz´asz [36], considered the Lipschitz type space defined as:

Lip_M ./WD

f 2C Œ0;1/W jf .t / f .x/j Mf jt xj

.tCx/²IwhereMf

is a constant which depends on f; t2Œ0;1/; x2.0;1/

; where 0 < 1 andr2.0; 1; to establish the uniform convergence of the SzaszK operators for functions in this space.

For r 2.0; 1 and M > 0; we define an analogue of this space in our case as follows:

Lip_M .r/WD

f 2C

0; Œnq

ŒnC1q

W jf .t / f .x/j M jt xj^r .tCx/^r=2I x2

0; Œn_qn ŒnC1qn

; t2

0; Œn_qn ŒnC1qn

:

We observe that we get only pointwise approximation due to the presence of x in the error estimate of.Ln;q_nf /.x/ f .x/;while in the case of Szasz operators [36], itK turns out that this x gets cancelled leading to the uniform convergence of the operat- ors.

Theorem 3. Letf 2Lip_M .r/. Then, for allx2

0;_Œn^Œn_C1^qn

qn

i

we have

j.Ln;qnf /.x/ f .x/j M

ı²_n;qn.x/

x

r=2

:

Proof. Letr2.0; 1;applying the H¨older’s inequality for integration and then for summation withuD2=randvD2=.2 r/, and Lemma 1, we have,

j.Ln;qnf /.x/ f .x/j ŒnC1²_qn

Œnqn

n

X

kD0

q_n^kp_n;k;qn.x/

Z _Œn^Œnqn

C1qn

0

p_n;k;qn.qnt /jf .t / f .x/jdqn.t /

ŒnC1²_qn Œnqn

n

X

kD0

q_n^kp_n;k;qn.x/

Z _Œn^Œnqn_C1qn

0 jf .t / f .x/j^up_n;k;qn.qnt /dqnt ¹_u

Z _Œn^Œnqn_C1qn

0

pn;k;qn.qnt /dqnt ¹_v

ⁿ

X

kD0

ŒnC1²_qn Œnq_n

q_n^kpn;k;q_n.x/

Z _Œn^Œnqn

C1qn

0 jf .t / f .x/j^upn;k;q_n.qnt /dqnt 1=u

ⁿ

X

kD0

ŒnC1²_qn Œnqn

q_n^kp_n;k;qn.x/

Z _Œn^Œnqn

C1qn

0

p_n;k;qn.qnt /dqnt 1=v

M

.x/^{r=2 n}

X

kD0

ŒnC1²_q

n

Œn_qn q_n^kp_n;k;qn.x/

Z _Œn^Œnqn_C1qn

0

.t x/²p_n;k;qn.qnt /dqnt 1=u

DM

ı²_n;qn.x/

x

r=2

:

Hence, the proof is completed.

4. THE CONSTRUCTION OF OPERATORS FOR THE BIVARIATE CASE

In this section, we introduce the bivariate case of the generalized Durrmeyer type operators (1.2). LetIjDŒ0;_Œn^Œnjqnj

jC1_qnj  jD1; 2. In what follows, let.qnj/; j D1; 2;

be sequences in .0; 1/ such that lim

n_j!1qnj D1:For I DI1I2, let C.I / denote the space of all real valued continuous functions on I with the norm jjfjjC .I /D sup_.x;y/2Ijf .x; y/j:Forf 2C.I /, the bivariate case of the operators given by (1.2) is defined as:

.Ln1;n2;q_n1;q_n2f /.x; y/D (4.1)

Œn1C1²_q

n1

Œn1q_n1

Œn2C1²_q

n2

Œn2q_n2 n1

X

k1D0 n2

X

k2D0

q_n^k1

1 q_n^k2

2 p_n1;k1;qn1.x/ p_n2;k2;qn2.y/

Z ^Œn1qn1

Œn1C1qn1

tD0

Z ^Œn2qn2

Œn2C1qn2

sD0

pn₁;k₁;q_n1.qn1t / pn₂;k₂;q_n2.qn2s/f .t; s/ dq_n1t dq_n2s:

Lemma 5. Leteij.x; y/Dxⁱy^j; .i; j /2N[ f0g N[ f0gwithiCj 2. For the operators given by (4.1), there hold the following equalities:

i) .L_{n1;n2;qn1;qn2}e₀₀/.x; y/D1I

ii) .Ln1;n2;q_n1;q_n2e10/.x; y/D Œn1q_n1

Œn1C1q_n1Œn1C2q_n1 Cqn1Œn1q_n1x

Œn1C2q_n1 I iii) .Ln₁;n₂;q_n1;q_n2e01/.x; y/D Œn₂_qn2

Œn2C1q_n2Œn2C2q_n2 Cq_n2Œn₂_qn2y Œn2C2q_n2 I iv) .Ln1;n2;q_n1;q_n2e20/.x; y/D Œn1q_n1

Œn1C2q_n1Œn1C3q_n1

Œn1 1q_n1q_n⁴₁x² Cq_n³₁xŒn1q_n1Cqn1.1C2qn1/xŒn1q_n1

Œn1C1q_n1 C.1Cqn1/Œn1q_n1

Œn1C1²_q

n1

I

v) .Ln₁;n₂;q_n1;q_n2e02/.x; y/D Œn₂_qn2 Œn2C2q_n2Œn2C3q_n2

Œn2 1q_n2q_n⁴₂y² Cq_n³

2yŒn2q_n2Cqn₂.1C2qn₂/yŒn2q_n2

Œn2C1q_n2 C.1Cqn₂/Œn2q_n2

Œn2C1²_q

n2

:

Next, we state Korovkin type theorem, given by Volkov [37]. With the help of this theorem we study the convergence of the sequence .Ln1;n2;q_n1;q_n2f /.x; y/to the functionf .x; y/.

Theorem 4 ([37]). Let J1; J2Rbe compact intervals of the real line and let fEm;nfgbe a sequence of linear positive operators applying the spaceC.J1J2/ into itself. Suppose that the following relations

i) Em;n.1Ix; y/D1Cam;n.x; y/;

ii) Em;n.tIx; y/DxCbm;n.x; y/;

iii) Em;n.sIx; y/DyCcm;n.x; y/;

iv) Em;n.t²Cs²Ix; y/Dx²Cy²Cdm;n.x; y/;

hold, for each.x; y/2J1J2:

If the sequencefam;n.x; y/g;fbm;n.x; y/g;fcm;n.x; y/g;fdm;n.x; y/gconverge to zero uniformly on J1J2; then the sequence fEm;nfg converges to f, uniformly on J₁J₂, for eachf 2C.J₁J₂/:

Remark 2. In view of Theorem 4 and Lemma 5, it easily follows that for each f 2C.I /,

n1;nlim2!1.L_{n1;n2;qn1;qn2}f /.x; y/Df .x; y/

uniformly onI.

In the following we give some numerical results which show the rate of conver- gence of the operatorLn1;n2;q_n1;q_n2f to certain functions using Maple algorithms.

Example 1. Let us consider f WR²!R, f .x; y/Dx²y²C2x²y 3y². The convergence of the operatorLn₁;n₂;q_n1;q_n2f to the functionf is illustrated in Fig- ure1and Figure2, respectively forn1Dn2D10; qn1Dqn2D0:5andn1Dn2D 100; qn1 Dqn2 D0:9; respectively. We remark that as the values ofn1 andn2 in- crease, the error in the approximation of the function by the operator becomes smal- ler.

FIGURE1. The convergence of.Ln1;n2;q_n1;q_n2f /.x; y/tof .x; y/, forqn1Dqn2D0:5(redf, blueLn1;n2;q_n1;q_n2)

FIGURE2. The convergence of.Ln1;n2;q_n1;q_n2f /.x; y/tof .x; y/, forqn1Dqn2D0:9(redf, blueLn1;n2;q_n1;q_n2)

IfJ1; J2Rare compact intervals and f 2R^J1J2, the modulus of continuity

!.ı1; ı2/, for anyı1> 0; ı2> 0is defined as

!.ı1; ı2/D sup

.x1;y1/;.x2;y2/2J1J2

jf .x1; y1/ f .x2; y2/j W jx1 x2j ı1;jy1 y2j ı2

:

In what follows, let ın₁;q_n1.x/D

.Ln₁;n₂;q_n1;q_n2.t x/²/.x; y/

1=2

D

.Ln₁;q_n1.t x/²/.x/

1=2

; and

ın2;q_n2.y/D

.Ln1;n2;q_n1;q_n2.s y/²/.x; y/

1=2

D

.Ln2;q_n2.s y/²/.y/

1=2

: Next, we recall the following Shisha-Mond theorem [33].

Theorem 5 ([33]). Let J1; J2 R be compact intervals, B.J1J2/ D ff 2 R^J1J2Wf is bounded onJ1J2gand letEWC.J1J2/!B.J1J2/be a linear positive operator. For eachf 2C.J1J2/; .x; y/2J1J2and anyı1> 0; ı2> 0, the following inequality

jE.fIx; y/ f .x; y/j jf .x; y/jjE.1Ix; y/ 1j C

E.1Ix; y/Cı₁¹p

E.1Ix; y/E..t x/²Ix; y/Cı₂¹p

E.1Ix; y/E..s y/²Ix; y/

Cı₁¹ı₂¹E.1Ix; y/p

E..t x/²Ix; y/E..s y/²Ix; y/

!.ı1; ı2/ holds.

Theorem 6. Letf 2C.I /and.x; y/2I. Then the operatorLn₁;n₂;q_n1;q_n2 sat- isfies the following inequality

j.Ln1;n2;q_n1;q_n2f /.x; y/ f .x; y/j 4!.ın1;q_n1.x/; ın2:q_n2.y//:

Proof. Applying Theorem 5 and Lemma 5, and choosing ı1Dın1;q_n1.x/ and ı2Dın2;q_n2.y/we get

j.Ln1;n2;q_n1;q_n2f /.x; y/ f .x; y/j

1Cı_n1¹_;q

n1.x/q

.Ln1;n2;q_n1;q_n2.t x/²/.x; y/

Cı_n2¹_;q

n2.y/q

.Ln1;n2;q_n1;q_n2.s y/²/.x; y/

Cı_n1¹_;q

n1.x/ı_n2¹_;q

n2.y/q

.Ln1;n2;q_n1;q_n2.t x/²/.x; y/.Ln1;n2;q_n1;q_n2.s y/²/.x; y/

!.ın1;q_n1.x/Iın2;q_n2.y//

Hence, we get the required result.

For f 2C.I / andı > 0;the first order complete modulus of continuity for the bivariate case is defined as:

N

!.fIı/Dsup

jf .t; s/ f .x; y/j W.t; s/; .x; y/2I and q

.t x/²C.s y/²ı

:

Further, the partial moduli of continuity with respect toxandy are given by

!1.fIı/Dsup

jf .x1; y/ f .x2; y/j Wy2I2 and jx1 x2j ı

; and

!2.fIı/Dsup

jf .x; y1/ f .x; y2/j Wx2I1 and jy1 y2j ı

: Theorem 7. Forf 2C.I /;there holds the inequality

j.Ln1;n2;q_n1;q_n2f /.x; y/ f .x; y/j 2.!1.fIın1;q_n1.x//C!2.fIın2;q_n2.y///:

Proof. Using the definition of partial moduli of continuity, Lemma 5 and the Cauchy-Schwarz inequality, we have

j.Ln1;n2;q_n1;q_n2f /.x; y/ f .x; y/j .Ln1;n2;q_n1;q_n2jf .t; s/ f .x; y/j/.x; y/

.Ln1;n2;q_n1;q_n2jf .t; s/ f .t; y/j/.x; y/C.Ln1;n2;q_n1;q_n2jf .t; y/ f .x; y/j/.x; y/

!2.fIın2;q_n2.y//

"

1C 1

ın2;q_n2.y/.Ln1;n2;q_n1;q_n2js yj/.x; y/

#

C!1.fIın1;q_n1.x//

"

1C 1

ın1;q_n1.x/.Ln1;n2;q_n1;q_n2jt xj/.x; y/

#

!2.fIın2;q_n2.y//

"

1C 1

ın2;q_n2.y/

.Ln1;n2;q_n1;q_n2.s y/²/.x; y/

1=2#

C!1.fIın1;q_n1.x//

"

1C 1

ın1;q_n1.x/

.Ln1;n2;q_n1;q_n2.t x/²/.x; y/

1=2#

;

from which the required result is immediate.

4.1. Degree of approximation

In this section, let us assume that lim

n_j!1qnj D1; j D1; 2:We study the degree of approximation for the bivariate operators (4.1) by means of the Lipschitz class.

For 0 < ˛1and0 < ˇ1;we define the Lipschitz classLipM.˛; ˇ/for the bivariate case as follows:

jf .t; s/ f .x; y/j Mjt xj^˛js yj^ˇ; for every.t; s/; .x; y/2I:

Theorem 8. Letf 2Lip_M.˛; ˇ/. Then, we have j.Ln1;n2;q_n1;q_n2f /.x; y/ f .x; y/j M ı_n^˛_1;q

n1.x/ı^ˇ_n2;q

n2.y/:

Proof. By our hypothesis, we may write

j.Ln1;n2;q_n1;q_n2f /.x; y/ f .x; y/j .Ln1;n2;q_n1;q_n2jf .t; s/ f .x; y/j/.x; y/

M.Ln1;n2;q_n1;q_n2jt xj^˛js yj^ˇ.x; y/

DM.Ln1;n2;q_n1;q_n2jt xj^˛/.x/.Ln1;n2;q_n1;q_n2js yj^ˇ/.y/:

Now, using the H¨older’s inequality with u1 D 2

˛; v1 D 2

2 ˛ and u2 D 2 ˇ, v2D 2

2 ˇ;and Lemma5, we have

j.Ln₁;n₂;q_n1;q_n2f /.x; y/ f .x; y/j

Mf.Ln1;n2;q_n1;q_n2.t x/²/.x/g^˛=2.Ln1;n2;q_n1;q_n21/.x/

f.Ln1;n2;q_n1;q_n2.s y/²/.y/g^{ˇ =2}.Ln1;n2;q_n1;q_n21/.y/

DM ı_n^˛

1;q_n1.x/ı^ˇ_n

2;q_n2.y/:

Thus, we get the desired result.

LetC¹.I /be the space of functionsf .x; y/whose first order partial derivatives are continuous onI.

Theorem 9. Letf 2C¹.I /. Then, we have

j.Ln1;n2;q_n1;q_n2f /.x; y/ f .x; y/j jjf_x⁰jjC.I /ın1;q_n1.x/C jjf_y⁰jjC.I /ın2;q_n2.y/:

Proof. For.t; s/2I, we have f .t; s/ f .x; y/D

Z t x

f⁰.u; s/duC Z s

y

f⁰.x; v/dv:

Applying the operator .Ln1;n2;q_n1;q_n2:/.x; y/ on both sides of above equality and using the Cauchy-Schwarz inequality, we get

j.Ln1;n2;q_n1;q_n2f /.x; y/ f .x; y/j

Ln1;n2;q_n1;q_n2

ˇ ˇ ˇ ˇ

Z t x

f_u⁰.u; s/du ˇ ˇ ˇ ˇ

.x; y/

C

Ln1;n2;q_n1;q_n2

ˇ ˇ ˇ ˇ

Z s y

f_v⁰.x; v/dv ˇ ˇ ˇ ˇ

.x; y/

jjf_x⁰jjC.I /.Ln₁;q_n1jt xj/.x/C jjf_y⁰jjC.I /.Ln₂;q_n2js yj/.y/

jjf_x⁰jjC.I /

.Ln1;q_n1.t x/²/.x/

1=2

C jjf_y⁰jjC.I /

.Ln2;q_n2.s y/²/.y/

1=2

:

Hence, we get the required result.

LetC².I /denote the space of functionsf .x; y/whose second order partial de- rivatives are continuous onI;endowed with the norm

jjfjjC².I /D jjfjjC.I /C

2

X

iD1

@ⁱf

@xⁱ _{C.I /}C

@ⁱf

@yⁱ _{C.I /}

! :

The Peetre’sK-functional of the functionf 2C.I /is defined by K.fIı/D inf

g2C².I /fjjf gjjC.I /CıjjgjjC².I /g; ı > 0:

Also, from [12, pp.192] it is known that K.fIı/M

N

!2.fIp

ı/Cmi n.1; ı/jjfjjC.I /

; (4.2)

holds for allı > 0. The constantM in the above inequality is independent ofıand f and!N2.fIp

ı/is the second order modulus of continuity for the bivariate case.

Theorem 10. Letf 2C.I /. Then for alln1; n22Nand each .x; y/2I there exists a constantC> 0such that

j.Ln1;n2;q_n1;q_n2f /.x; y/ f .x; y/j C

N

!2

fI1

2 q

Cn1;n2;q_n1;q_n2.x; y/

Cmin

1;Cn1;n2;q_n1;q_n2.x; y/

4

jjfjjC.I /

C!.f;q

n1;n2;q_n1;q_n2.x; y//;

where

n1;n2;q_n1;q_n2.x; y/D

Œn1q_n1.1Cqn1xŒn1C1q_n1/ Œn1C1q_n1Œn1C2q_n1

x 2

C

Œn2q_n2.1Cqn2yŒn2C1q_n2/ Œn2C1q_n2Œn2C2q_n2

y 2

; and

Cn1;n2;q_n1;q_n2.x; y/Dı²_n1;q

n1.x/Cı_n²_2;q

n2.y/C n1;n2;q_n1;q_n2.x; y/:

Proof. First, we define the auxiliary operator .L_n1;n2;q

n1;q_n2f /.x; y/Df .x; y/C.Ln1;n2;q_n1;q_n2f /.x; y/

f

Œn1q_n1.1Cqn1xŒn1C1q_n1/ Œn1C1q_n1Œn1C2q_n1

;Œn2q_n2.1Cqn2yŒn2C1q_n2/ Œn2C1q_n2Œn2C2q_n2

: Applying Lemma4it is obvious that

.L_n1;n2;q

n1;q_n21/.x; y/D1; .L_n1;n2;q

n1;q_n2t /.x; y/Dx; .L_n1;n2;q

n1;q_n2s/.x; y/Dy;

and hence

.L_n1;n2;q

n1;q_n2.t x//.x; y/D0D.L_n1;n2;q

n1;q_n2.s y//.x; y/:

Further, forf 2C.I /, we have (see also Lemma3) jjL_n1;n2;q

n1;q_n2fjjC.I /3jjfjjC.I /: