# On the Approximation Properties ofq−Analogue Bivariate λ-Bernstein Type Operators

Authors: Edmond Aliaga; Behar Baxhaku
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this article, we establish an extension of the bivariate generalization of theq-Bernstein type operators involving parameter λ and extension of GBS (Generalized Boolean Sum) operators of bivariate q-Bernstein type. For the first operators, we state the Volkov-type theorem and we obtain a Voronovskaja type and investigate the degree of approximation by means of the Lipschitz type space. For the GBS type operators, we establish their degree of approximation in terms of the mixed modulus of smoothness. The comparison of convergence of the bivariate q-Bernstein type operators based on parameters and its GBS type operators is shown by illustrative graphics using MATLAB software.


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## Body

## 1. Introduction

Leth∈CS with S=0,1,λ∈−1,1, and m∈ℕ. In 2018, Chen et al. [1] proposed a new generalization of Bernstein operators based on a fixed real parameter λ∈−1,1 as
(1)Bmλh;y=∑k=0mΩ¯m,kλyhkm,x∈S,where the basis functions Ω¯m,kλy are defined as
(2)Ω¯m,0λy=Ωm,0y−λm+1Ωm+1,1y,Ω¯m,kλy=Ωm,ky+λm−2j+1m2−1Ωm+1,ky−m−2j−1m2−1Ωm+1,k+1y,1≤k≤m−1,Ω¯m,mλy=Ωm,my−λm+1Ωm+1,my,The authors studied the established of some Korovkin type approximation properties and the degree of approximation by means of the modulus of continuity, Voronovskaja-type results, and shape-preserving properties for these operators.This work took the attention of researchers from approximation theory for a short time. Since that time, lots of researchers have put forth many relevant studies on this issue, and numerous articles can be given interrelated with their work [2–5].In [6] were introduced the bivariate extension of the operators (1) and studied the degree of approximation in terms of the second order Ditzian-Totik modulus of continuity for two variables. A Kantorovich variant of the λ-Bernstein operators (1) was introduced and studied in [7]. Many authors also considered the univariate and bivariate positive linear operators and studied their approximation behavior; we refer the reader to articles (cf. [8–17]) and references therein. Now, we give some basic definitions based on the q-calculus [18], which are used in this paper. Let 0<q<1 and b,x be any real numbers.Theq-number bq is defined as bq=1−qb/1−q,q≠1 and for b=n∈ℕ,
(3)nq=1+q+⋯+qn−1.Theq-number 1−xqb is defined as 1−xqb=∏j=0∞1−xqj/1−xqj+b and for b=n∈ℕ(4)1−xqn=1−x1−qx⋯1−qn−1x.For the integersn,j such that 0≤j≤n, the q-binomial is defined as
(5)mjq=nq!jq!n−jq!.For an integern, the q-factorial is defined as
(6)nq!=nqn−1q!ifn≥1,and nq!=1ifn=0.Cai et al. [19] considered the generalized Bernstein type operators based on parameters q−analogue and for fixed real parameter λ∈−1,1 as
(7)Bmλh;q,y=∑j=0mΩ¯m,jλ,qyhjqmq,y∈S,where the basis functions Ω¯m,kλ,qy are defined as
(8)Ω¯m,0λ,qy=Ωm,0qy−λmq+1Ωm+1,1qy,(9)Ω¯m,jλ,qy=Ωm,jqy+λmq−2jq+1mq2−1Ωm+1,jy−mq−2qjq−1mq2−1Ωm+1,j+1qy,1≤j≤m−1,(10)Ω¯m,mλ,qy=Ωm,mqy−λm+1Ωm+1,mqy,and Ωm,jqy=mkqyj∏s=0m1−qsy. Note that for q=1, these operators reduce to λ-Bernstein operators (1) and for q=λ=1 (7) educes to Bernstein operators defined in [20].Therefore, linear operators, in particular the limitq-Bernstein operator, are of significant interest for applications.The purpose of this article is to present an extension of the bivariateλ,q-Bernstein type operators involving parameters and obtain the degree of approximation by means of the Lipschitz type space for two variables. Moreover, we consider the associated Generalized Boolean Sum (GBS) operators and study their degree of approximation in terms of the mixed modulus of smoothness for bivariate functions.


## 2. Construction of the Bivariateq,λ-Bernstein Type Operators

ForS2=0,1×0,1, let CS2 be the space of all continuous functions on S2.Forh∈CS2 and λ1,λ2∈−1,1, the bivariate extension of the operator (7) is defined by
(11)Bm1,m2,qm1λ1,λ2,qm2h;y1,y2=∑j1=0m1∑j2=0m2Ωm1,m2,j1,j2λ1,λ2y1,y2hj1q1m1q1,j2q2m2q2,where qmimi∈N is a sequence in 0,1 satisfying
(12)qmi→1andqmimi→ci;fori=1,2.Further,Ωm1,m2,j1,j2λ1,λ2y1,y2=Ωm1,j1λ1y1Ωm2,j2λ2y2, where Ωm1,j1λ1y1 and Ωm2,j2λ2y2 are defined similarly as Ωm,jλy in (8).It is easy to see thatBm1,m2,qm1λ1,λ2,qm2h;y1,y2 is bounded. We denote
(13)Bm1,m2,qm1λ1,λ2,qm2u−y1sv−y2t;x,y=Bm1λ1u−y1s;qm1,y1Bm2λ1v−y2t;qm2,y2=μm1,λ1,sqm1,y1μm2,λ2,tqm2,y2s,t∈N∪0.Lemma 1 (see [19]).
For the operatorsBm1λ1h,q1,y1, we have
(i)
Bm1λ11;q1,y1=1(ii)
Bm1λ1t;q1,y1=y1+m1+1q1y11−y1m1λ1/m1q1m1q1−2m+1q1y1λ1/mq12−11−y1m1/m1q1+q1y11−y1n1−1+λ1/q1m1q1m1q1+11−∏i=0m11−qiy1−y1m1+1−m1+1q1y11−y1m1−1+λ1/m1q12−12m1+1q1y121−y1m1−1−2m1+1q1y1/q1m1q11−y1m1+2/q1m1q11−∏i=0m11−q1iy1−y1m1+1(iii)
Bm1λ1t2;q1,y1=y12+y11−y1/m1q1+y1λ1m1+1q1/m1q1m1q1−1q1y11−y1m1−1+1−y1m1/m1q1−2m1+1q1λ1/m1q1m1q12−1y11−y1m1/m1q1+q12+q1y121−y1m1−1+q13m1−1q1y131−y1m1−2−λ1/q1m1q1m1q1+1m1+1q1y121−y1m1−1−m1+1q1y11−y1m1/q1m1q1+1−∏i=0m11−q1iy1−y1m1+1/q1m1q1+2λ1/m1q1m1q12−1q1m1−1q1m1+1q1y131−y1m1−2−1−q1m1+1q1y121−y1m1−1/q1+m1+1q1y11−y1m1/q12m1q1−1−∏i=0m11−q1iy1−y1m1+1/q12m1q1Letersy1,y2=t1rt2s,r,s∈ℕ0×ℕ0, with r+s≤2. In order to obtain the main results, we need the following lemmas:Lemma 2.
For the operatorsBm1,m2,qm1λ1,λ2,qm2, we have
(i)
Bm1,m2,qm1λ1,λ2,qm21;y1,y2=1(ii)
Bm1,m2,qm1λ1,λ2,qm2e10;y1,y2=y1+m1+1q1y11−y1m1λ1/m1q1m1q1−1−2m1+1q1y1λ1/mq12−11−y1m1/m1q1+q1y11−y1m1−1+λ1/q1m1q1m1q1+11−∏i=0m11−q1iy1−y1m1+1−m1+1q1y11−y1m1+λ1/m1q12−12m1+1q1y121−y1m1−1−2m1+1q1y1/q1m1q11−y1m1+2/q1m1q11−∏i=0m11−q1iy1−y1m1+1(iii)
Bm1,m2,qm1λ1,λ2,qm2e20;y1,y2=y12+y11−y1/m1q1+y1λ1m1+1q1/m1q1q1y11−y1m1−1+1−y1m1/m1q1−2m1+1q1λ1/m1q1m1q12−1y11−y1m1/m1q1+q12+q1y121−y1m1−1+q13m1−1q1y131−y1m1−1−λ1/q1m1q1m1q1+1m1+1q1y121−y1m1−1−m1+1q1y11−y1m1/q1m1q1+1−∏i=0m11−q1iy1−y1m1+1/q1m1q1+2λ1/m1q1m1q12−1q1m1−1q1m1+1q1y131−y1m1−2−1−q1m1+1q1y121−y1m1−1/q1+m1+1q1y11−y1m1/q12m1q1−1−∏i=0m11−q1iy1−y1m1+1/q12m1q1+2/q1m1q11−∏i=0m11−q1iy1−y1m1+1(iv)
Bm1,m2,qm1λ1,λ2,qm2e01;x,y=y2+m2+1q2y21−y2m2λ2/m2q2m2q2−1−2m2+1q2y2λ2/mq22−11−y2m2/m2q2+q2y21−y2m2−1+λ2/q2m2q2m2q2+11−∏i=0m21−q2iy2−y2m2+1−m2+1q2y21−y2m2+λ2/m2q22−12m2+1q2y221−y2m2−1−2m2+1q2y2/q2m2q21−y2m2+2/q2m2q21−∏i=0m21−q2iy2−y2m2+1(v)
Bm1,m2,qm1λ1,λ2,qm2e02;y1,y2=y22+y21−y2/m2q2+y2λ2m2+1q2/m2q2m2q2−1q2y21−y2m2−1+1−y2m2/m2q2−2m2+1q2λ2/m2q2m2q22−1y21−y2m2/m2q2+q22+q2y221−y2m2−1+q23m2−1q2y231−y2m2−1−λ2/q2m2q2m2q2+1m2+1q2y221−y2m2−1−m2+1q2y21−y2m2/q2m2q2+1−∏i=0m21−q2iy2−y2m2+1/q2m2q2+2λ2/m2q2m2q22−1q2m2−1q2m2+1q2y231−y2m2−2−1−q2m2+1q2y221−y2m2−1/q2+m2+1q2y21−y2m2/q22m2q2−1−∏i=0m21−q2iy2−y2m2+1/q22m2q2Corollary 3.
Applying Lemma2, we have
(i)
Bm1,m2,qm1λ1,λ2,qm2u−y1;y1,y2=m1+1q1y11−y1m1λ1/m1q1m1q1−1−2m+1q1y1λ1/mq12−11−y1m1/m1q1+q1y11−y1m1−1+λ1/q1m1q1m1q1+11−∏i=0m11−q1iy1−y1m1+1−m1+1q1y11−y1m1+λ1/m1q12−12m1+1q1y121−y1m1−1−2m1+1q1y1/q1m1q11−y1m1+2/q2m2q21−∏i=0m21−q2iy2−y2m2+1(ii)
Bm1,m2,qm1λ1,λ2,qm2v−y2;y1,y2=m2+1q2y21−y2m2λ2/m2q2m2q2−1−2m2+1q2y2λ2/mq22−11−y2m2/m2q2+q2y21−y2m2−1+λ2/q2m2q2m2q2+11−∏i=0m21−q2iy2−y2m2+1−m2+1q2y21−y2m2+λ2/m2q22−12m2+1q2y221−y2m2−1−2m2+1q2y2/q2m2q21−y2m2+2/q2m2q21−∏i=0m21−q2iy2−y2m2+1(iii)
Bm1,m2,qm1λ1,λ2,qm2u−y12;y1,y2=y11−y1/m1q1+m1+1q1y1λ1/m1q1m1q1−1q1y11−y1m1−1+1−y1m1/m1q1−2y11−y1m1−2m1+1q1λ1/m1q1m1q12−1y11−y1m1/m1q1+q12+q1y121−y1m1−1+q13m1−1q1y131−y1m1−1+4y1m1+1q1y1λ1/m1q12−11−y1m1/m1q1+q1y11−y1m1−1−λ1/q1m1q1m1q1+1m1+1q1y121−y1m1−1−m1+1q1y11−y1m1/q1m1q1+1−∏i=0m11−q1iy1−y1m1+1/q1m1q1−2y11−∏i=0m11−q1iy1−y1m1+1−m1+1q1y11−y1m1+2λ1/m1q1m1q12−1q1m1−1q1m1+1q1y131−y1m1−2−1−q1m1+1q1y121−y1m1−1/q1+m1+1q1y11−ym1/q12m1q1−1−∏i=0m11−q1iy1−y1m1+1/q12m1q1−2y1λ1/m1q12−12m1+1q1y121−y1m1−1−2m1+1q1y1/q1m1q11−y1m1+2/q1m1q11−∏i=0m11−q1iy1−y1m1+1(iv)
Bm1,m2,qm1λ1,λ2,qm2u−y22;y1,y2=y21−y2/m2q2+m2+1q2y2λ2/m2q2m2q2−1q2y21−y2m2−1+1−y2m2/m2q2−2y21−y2m2−2m2+1q2λ2/m2q2m2q22−1y21−y2m2/m2q2+q22+q2y221−y2m2−1+q23m2−1q2y231−y2m2−1+4y2m2+1q2y2λ2/m2q22−11−y2m2/m2q2+q2y21−y2m2−1−λ2/q2m2q2m2q2+1m2+1q2y221−y2m2−1−m2+1q2y21−y2m2/q2m2q2+1−∏i=0m21−q2iy2−y2m2+1/q2m2q2−2y21−∏i=0m21−q2iy2−y2m2+1−m2+1q2y21−y2m2+2λ2/m2q2m2q22−1q2m2−1q2m2+1q2y231−y2m2−2−1−q2m2+1q2y221−y2m2−1/q2+m2+1q2y21−y2m2/q22m2q2−1−∏i=0m21−q2iy2−y2m2+1/q22m2q2−2y2λ2/m2q22−12m2+1q2y221−y2m2−1−2m2+1q2y2/q2m2q21−y2m2+2/q2m2q21−∏i=0m21−q2iy2−y2m2+1(v)
limm1→∞m1qm1Bm1,m2,qm1λ1,λ2,qm2t1−y1;y1,y2=0(vi)
limm2→∞m2qm2Bm1,m2,qm1λ1,λ2,qm2t2−y2;y1,y2=0(vii)
limm1→∞m1qm1Bm1,m2,qm1λ1,λ2,qm2t1−y12;y1,y2=y11−y1,y1∈0,1(viii)
limm2→∞m2qm2Bm1,m2,qm1λ1,λ2,qm2t2−y22;y1,y2=y21−y2,y2∈0,1
In what follows, the Volkov-type approximation theorem is proved forBm1,m2,qm1λ1,λ2,qm2.Theorem 4.
Leth∈CS2. Then, we have
(14)limm1,m2→∞Bm1,m2,qm1λ1,λ2,qm2h−h=0.Proof.
Using Lemma 2, it is obvious(15)limm1,m2→∞Bm1,m2,qm1λ1,λ2,qm2ers;y1,y2=ers,r,s∈0,0,0,1,0,1,and
(16)limm1,m2→∞Bm1,m2,qm1λ1,λ2,qm2e20+e02;y1,y2=e20+e02,uniformly on S2. The result follows using [[21], Thm 2.1].In order to discuss the next results, let us recall the definitions of modulus of continuity and partial modulus of continuityDefinition 5 (see [22]).
Forh∈S2 and δ>0, the full modulus of continuity in the bi-variate case is defined as
(17)ωch;δ=max∑i=12ti−yi2≤δht1,t2−hy1,y2:t1,t2,y1,y2∈S2.
For each fixedi=1,2, the partial modulus of continuity of h with respect to yi is defined
(18)ω1h;δ=suphx1,y−hx2,y:y∈0,1x1−x2≤δ,and
(19)ω2h;δ=suphx,y1−hx2,y:x∈0,1y1−y2≤δ,respectively.Theorem 6.
Forh∈CS2, we have
(20)Bm1,m2,qm1λ1,λ2,qm2h−h≤2ωch;δ,where δ2=μm1,λ1,2qm1,y1+μm2,λ2,2qm2,y2.Proof.
Using the facts that(21)Bm1,m2,qm1λ1,λ2,qm2ht1,t2;y1,y2−hy1,y2≤∑j1=0m1∑j2=0m2Ωm1,m2,j1,j2λ1,λ2y1,y2hj1q1m1q1,j2q2m2q2−hy1,y2.Then, we use the following property of the complete modulus of continuity:
(22)ht1,t2−hy1,y2≤ωcf;δ1+t1−y12+t2−y22δ,we get
(23)Bm1,m2,qm1λ1,λ2,qm2ht1,t2;y1,y2−hy1,y2≤1+Bm1,m2,qm1λ1,λ2,qm2t1−y12+t2−y22;y1,y2δωch;δ.
Applying Cauchy Schwarz inequality, we have(24)Bm1,m2,qm1λ1,λ2,qm2ht1,t2;y1,y2−hy1,y2≤1+1δBm1,m2,qm1λ1,λ2,qm2t−x2;y1,y2+Bm1,m2,qm1λ1,λ2,qm2s−y2;y1,y21/2ωch;δ≤1+1δμm1,λ1,2qm1,y1+μm2,λ2,2qm2,y21/2ωch;δ;∀y1,y2∈S2.
By choosing(25)δ=μm1,λ1,2qm1,y1+μm2,λ2,2qm2,y21/2,we obtain the desired result.Theorem 7.
Leth∈CS2, then the following inequality holds:
(26)Bm1,m2,qm1λ1,λ2,qm2ht1,t2;y1,y2−hy1,y2≤2ω1h;μm1,λ1,2qm1,y1+ω2h;μm2,λ2,2qm2,y2.Proof.
Similarly to previous theorems, using relations (19) we obtain
(27)Bm1,m2,qm1λ1,λ2,qm2ht1,t2;y1,y2−hy1,y2≤Bm1,m2,qm1λ1,λ2,qm2ω2h;∣t2−y2∣;y1,y2+Bm1,m2,qm1λ1,λ2,qm2ω1h;∣t1−y1∣;y1,y2.Applying Cauchy Schwarz inequality withδ1=μm1,λ1,2qm1,y11/2 and δ2=μm2,λ2,2qm2,y21/2, we obtain the desired result.Now, we want to give the quantitative result in terms of the Lipschitz class functionals. For any functionsh:S2→ℝ and 0<ù≤1, the function h is said to be in Lipschitz class LipMùS2 if ∃ a M>0 such that
(28)LipMù≔h:∣hs1,s2−hy1,y2∣≤M∥s−t∥ù,∀s=s1,s2,t=y1,y2∈S2, where ∥s−t∥=s1−y12+s2−y221/2 is the Euclidean norm and M is a positive real constant.The following theorem yields us an estimate of error for functions inLipMù, by the operators Bm1,m2,qm1λ1,λ2,qm2.Theorem 8.
Leth∈LipMù. Then for sufficiently large m and n and for all y1,y2∈S2, there holds the inequality
(29)Bm1,m2,qm1λ1,λ2,qm2h−hCI2≤Mmqm−1+nqn−1u′2,where M>0 is a constant.Proof.
From hypothesis, we have(30)Bm1,m2,qm1λ1,λ2,qm2h;y1,y2−hy1,y2≤Bm1,m2,qm1λ1,λ2,qm2hs1,s2−hy1,y2;y1,y2≤MBm1,m2,qm1λ1,λ2,qm2∥s−t∥ù;y1,y2,where s=s1,s2,t=y1,y2∈S2. Applying Hölder’s inequality and Corollary 3, we obtain
(31)Bm1,m2,qm1λ1,λ2,qm2h;y1,y2−hy1,y2≤MBm1,m2,qm1λ1,λ2,qm2∥s−t∥2;y1,y2u′/2≤Mμm1,λ1,2qm2,y1+μm2,λ2,2qm2,y2ù/2,which leads to the required result on applying Corollary 3.LetC′S2 denote the space of continuous functions hy1,y2 on S2 whose first-order partial derivatives g′y1 and g′y2 are also continuous on S2.Our next result yields us the rate of approximation for continuously differentiable functions onS2 by the operators Bm1,m2,qm1λ1,λ2,qm2.Theorem 9.
Leth∈C′S2. Then for sufficiently large m and n, we have
(32)Bm1,m2,qm1λ1,λ2,qm2h−hCS2≤A∥h′y1∥CS2m1qm1−1/2+∥h′y2∥CS2m2qm2−1/2,where A is some positive constant.Proof.
Fory1,y2∈S2 be arbitrary, we may write
(33)ht1,t2−hy1,y2=∫y1t1h′ηη,t2dη+∫y2t2h′ϕy1,ϕdϕ,fort1,t2∈S2.
Hence, applying the operatorBm1,m2,qm1λ1,λ2,qm2 on both sides of the above equation, we obtain
(34)Bm1,m2,qm1λ1,λ2,qm2h;y1,y2−hy1,y2=Bm1,m2,qm1λ1,λ2,qm2∫y1t1h′ηη,t2dη;y1,y2+Bm1,m2,qm1λ1,λ2,qm2∫y2t2h′ϕy1,ϕdϕ;y1,y2.By using sup-norm onS2(35)∫y1t1h′ηη,t2dη≤h′y1CS2t1−y1,∫y2t2h′ϕy1,ϕdϕ≤h′y2CS2t2−y2.we get
(36)Bm1,m2,qm1λ1,λ2,qm2h;y1,y2−hy1,y2≤h′y1CS2Bm1,m2,qm1λ1,λ2,qm2∣t1−y1∣;y1,y2+h′y2CS2Bm1,m2,qm1λ1,λ2,qm2∣t2−y2∣;y1,y2.Hence, applying the Cauchy-Schwarz inequality and Corollary3, we obtain
(37)Bm1,m2,qm1λ1,λ2,qm2h;y1,y2−hy1,y2≤h′y1CS2Bm1,m2,qm1λ1,λ2,qm2t−y2;y1,y21/2+∥h′y2∥CS2Bm1,m2,qm1λ1,λ2,qm2t2−y22;y1,y21/2,from which the desired result is immediate.The following result yields the degree of approximation ofh by Bm1,m2,qm1λ1,λ2,qm2 in terms of the partial modul of continuity of the partial derivatives of h.Theorem 10.
Leth∈C′S2. Then for sufficiently large m and n, we have
(38)Bm1,m2,qm1λ1,λ2,qm2h−hCS2≤A∑i=12miqmi−121+2ωihyi′;miqmi−12,where ωih′yi;. are the partial modul of continuity of h′yi for i=1,2 and A is some positive constant.Proof.
If we use the mean value theorem in the following form, we have(39)ht1,t2−hy1,y2=t1−y1h′y1η,y2+t2−y2h′y2y1,ù=t1−y1h′y1y1,y2+t1−y1h′y1η,y2−h′y1y1,y2+t2−y2h′y2y1,y2+t2−y2h′y2y1,ù−h′y2y1,y2,where y1<η<t1 and y2<ù<t2. Applying the operator Bm1,m2,qm1λ1,λ2,qm2 to both sides, we deduce that
(40)Bm1,m2,qm1λ1,λ2,qm2h;y1,y2−hy1,y2=h′y1y1,y2Bm1,m2,qm1λ1,λ2,qm2t1−y1;y1,y2+Bm1,m2,qm1λ1,λ2,qm2t1−y1h′y1η,y2−h′y1y1,y2;y1,y2+h′y2y1,y2Bm1,m2,qm1λ1,λ2,qm2t2−y2;y1,y2+Bm1,m2,qm1λ1,λ2,qm2t2−y2h′y2y1,ù−h′y2y1,y2;y1,y2.Sinceh′y1 and h′y2 are continuous in S2, there exist positive constants A1 and A2 such that h′y1≤A1 and h′y1≤A2, for all y1,y2∈S2. Hence, applying the Cauchy-Schwarz inequality, we obtain
(41)Bm1,m2,qm1λ1,λ2,qm2h;y1,y2−hy1,y2≤∑i=12hyi′Bm1,m2,qm1λ1,λ2,qm2ti−yi;y1,y2+∑i=12Bm1,m2,qm1λ1,λ2,qm2ti−yi1+ti−yiδi;y1,y2ωihyi′;δi≤∑i=12MiBm1,m2,qm1λ1,λ2,qm2ti−yi2;y1,y212+ωihyi′;δiBm1,m2,qm1λ1,λ2,qm2ti−yi2;y1,y212+1δiBm1,m2,qm1λ1,λ2,qm2ti−yi2;y1,y2.Choosingδi=miqmi−1/2,i=1,2 we get the required result.


## 3. Construction of GBS Operator of Generalized Bernstein Type

In the last two decades, the study of generalized Boolean sum (GBS) operators of certain linear positive operators has attracted very much attention in the approximation theory. In early 1937 with Bögel [23], a great number of studies are performed related to these operators. To make an analysis in multidimensional spaces, Bögel [23] introduced the concepts of continuity and differentiability in a Bögel space. There are still many authors working on this subject. Agrawal et al. [24] studied the degree of approximation for bivariate Lupas-Durrmeyer type operators based on Pólya distribution with associated GBS operators. Further, Gupta et al. [25] introduced some GBS operators of discrete and integral types and examined the properties of approximations in a Bögel space. Recently, Kajla and Miclacus [16] studied the rate of approximation of Bögel continuous and Bögel differentiable functions by the GBS operators of Bernstein-Durrmeyer type operators. In the last year, Kumar and Shivam [26] constructed the bivariate Kantorovich-type sampling operator, involved with GBS operators, as well as estimation of the rate of convergence of the sequences of these operators.A functionh:S2→ℝ is called B-continuous at point x0,y0∈S2 if
(42)limx,y→y01,y02Δy1,y2hy01,y02;y1,y2=0,for any y1,y2∈S2, (see [23]). The function h:S2→ℝ is B-bounded on S2 if there exists M>0 such that Δx,yht1,t2;y1,y2≤M for every y1,y2,t1,t2∈S2.Throughout this article,BbS2 denotes all B-bounded functions on S2. The space of all B-continuous functions is denoted by CbS2.Motivated by the above authors, we construct the GBS operator ofBm1,m2λ1,λ2, who is defined as follows:
(43)GBSm1,m2λ1,λ2ht1,t2;x,y≔Bm1,m2λ1,λ2ht1,y2+hy1,t2−ht1,t2;y1,y2,for all y1,y2∈S2. More precisely, the q-analogue λ-Bernstein type GBS operator is defined as follows:
(44)GBSm1,m2λ1,λ2ht1,t2;y1,y2=∑j1=0m1∑j2=0m2Ωm1,m2,j1,j2λ1,λ2y1,y2×hj1q1m1q1,y2+hy1,j2q2m2q2−hj1q1m1q1,j2q2m2q2,where the operator GBSm1,m2λ1,λ2 is well-defined on the space CbS2 into CS2 and h∈CbS2.


## 4. Degree of Approximation byGBSm1,m2,qm1,qm2λ1,λ2

Fory1,y2,t1,t2∈S2, the mixed modulus of smoothness of h∈CbS2 is defined by
(45)ωBh;δ1,δ2≔supΔy1,y2ht1,t2;y1,y2:y1−t1<δ1,y2−t2<δ2,and for any δ1,δ2∈0,∞×0,∞. Using (45), we have
(46)ωBh;P1δ1,P2δ2≤1+P11+P2ωBf;δ1,δ2;P1,P2>0,The basic results ofωB were studied by Badea et al. [27, 28] and are similar to the properties of the usual modulus of continuity for bivariate functions. We shall obtain the rate of approximation of the operators (44) to h∈CbS2 in terms of the mixed modulus of continuity for two variables. For this, we apply the Shisha-Mond theorem for B-continuous functions defined by Gonska [29] and Badea and Cottin [28].Theorem 11.
For everyh∈CbS2, at each point y1,y2∈S2 and sufficiently large m and n, the operator (44) satisfy the following results
(47)GBSm1,m2,qm1λ1,λ2,qm2h−hB≤Cλ1λ2ωBh;m1qm1−1/2,m2qm2−1/2,where Cλ1λ2 is a positive constant depending on parameters λ1 and λ2.Proof.
Using (45) and applying the inequality (46), we have
(48)Δy1,y2ht1,t2;y1,y2≤ωBh;t1−y1,t2−y2≤∏i=121+∣ti−yi∣δiωBg;δ1,δ2≤1+∣t1−y1∣δ1−1+∣t2−y2∣δ2−1+δ1δ2−1t1−y1t2−y2ωBg;δ1,δ2,for every y1,y2,t1,t2∈S2 and for any δ1,δ2>0. Taking the definition of Δy1,y2ht1,t2;y1,y2, we may write
(49)hy1,t2+ht1,y2−ht1,t2=hy1,y2−Δy1,y2ht1,t2;y1,y2.Applying the operatorBm1,m2,qm1λ1,λ2,qm2 on both sides of the above inequality
(50)GBSm1,m2,qm1λ1,λ2,qm2h;y1,y2=hy1,y2Bm1,m2,qm1λ1,λ2,qm21;y1,y2−Bm1,m2,qm1λ1,λ2,qm2Δy1,y2ht1,t2;y1,y2;y1,y2.Now, from Lemma2, with the help of Cauchy-Schwarz inequality and Remark 1 (in that order), we obtain
(51)GBSm1,m2,qm1,qm2λ1,λ2h;y1,y2−hy1,y2≤Bm1,m2,qm1λ1,λ2,qm2Δy1,y2ht1,t2;y1,y2;y1,y2≤ωBg;δ1,δ2Bm1,m2,qm1λ1,λ2,qm21;y1,y2+1δ1Bm1,m2,qm1λ1,λ2,qm2t1−y12;y1,y2+1δ2Bm1,m2,qm1λ1,λ2,qm2t2−y22;y1,y2+1δ1δ2Bm1,m2,qm1λ1,λ2,qm2t1−y12;y1,y2Bm1,m2,qm1λ1,λ2,qm2t2−y22;y1,y2.Now, settingδi=miqmi−1/2,i=1,2, the required result is obtained.For0<ù≤1, the Lipschitz class of Bögel continuous functions is defined as
(52)LipMù=h∈CbS2:Δy1,y2ht1,t2;y1,y2≤L∥t−s∥ù,where t=t1,t2,s=y1,y2∈S2 and ∥t−s∥=t1−y12+t2−y221/2 are the Euclidean norm.In the next result, we obtain the degree of approximation of the operatorsGBSm1,m2,qm1λ1,λ2,qm2 for functions in the Lipschitz-class of Bo¨gel continuous functions.Theorem 12.
Ifh∈LipLu′, then for sufficiently large m1 and m2, we have
(53)GBSm1,m2,qm1λ1,λ2,qm2h;y1,y2−hB≤Lm1qm1−1+m2qm2−1ù/2,where L is a positive constant.Proof.
Using the equation (50), Hölder’s inequality, Lemma 2, and Lemma 1, we get
(54)GBSm1,m2,qm1λ1,λ2,qm2h;y1,y2−hy1,y2≤Bm1,m2,qm1λ1,λ2,qm2Δt,rgy,z;y,z≤LBm1,m2,qm1λ1,λ2,qm2∥t−s∥u′;y1,y2≤LBm1,m2,qm,qnλ1,λ2∥t−s∥2;y1,y2ù/2≤Lμm1,λ1,2qm1,y1+μm2,λ2,2qm2,y2ù/2≤Lm1qm1−1+m2qm2−1ù/2.
Thus, we get the desired result.


## 5. Numerical Results and Discussions

Example 13.
Let us choosem1=m2=10,20and30, qm1=qm2=0.8, and λ1=λ2=0.6. Denote Em1,m2λ1,λ2=Bm1,m2,qm1λ1,λ2,qm2hs1,s2;y1,y2−hy1,y2, the error function of approximation by operators. The convergence of the bivariate Bernstein operators: B10,10,qm1λ1,λ2,qm2h;y1,y2 (yellow), then B20,20,qm1λ1,λ2,qm2h;y1,y2 (red) and B30,30,qm1λ1,λ2,qm2h;y1,y2 (magenta) to hy1,y2=2y13sin2πy12y23 (blue) will be illustrated in Figure 1.Example 14.
Form1=m2=10,20 and 30,λ1=λ2=0.6 and qm1=qm2=0.8, the convergence of GBSm1,m2,qm1λ1,λ2,qm2h;y1,y2tohy1,y2=2y13sin2πy12y23 is illustrated in Figure 2. Denote Em1,m2∗λ1,λ2=∣GBSm1,m2,qm1λ1,λ2,qm2hs1,s2;y1,y2−hy1,y2∣, the error function of approximation by operators. This example explains the convergence of the operators GBSm1,m2,qm1λ1,λ2,qm2hy1,y2 that are going to the function hy1,y2if the values of m1,m2 are increasing.Figure 1
The convergence ofBm1,m2,qm1λ1,λ2,qm2h;y1,y2tohy1,y2.Figure 2
The convergence ofBm1,m2,qm1λ1,λ2,qm2h;y1,y2tohy1,y2.Comparative results are given in Figure3, Tables 1 and 2, for the errors of the approximation of GBSm1,m2,qm1λ1,λ2,qm2h;y1,y2 and Bm1,m2,qm1λ1,λ2,qm2h;y1,y2 to the functions hy1,y2=2y13sin2πy12y23 for m1=m2=30, λ1=λ2=0.6, and q1=q2=0.8. Note that (see Tables 1 and 2 and Figure 3) the GBS-Bernstein operator approximation outperforms others.Figure 3
The convergence ofGBS30,30,0.80.6,0.6,0.8h;y1,y2 and B30,30,0.80.6,0.6,0.8h;y1,y2 to hy1,y2.Table 1
Error of approximationEm1,m2λ1,λ2 for m1=m2=10,20 and 30; qm1=qm2=0.8 and λ1=λ2=0.6.
y1,y2E10,100.6,0.6E20,200.6,0.6E30,300.6,0.6(0.1,0.1)0.0001600.0001200.000117(0.1,0.4)0.0018620.0015180.001486(0.2,0.6)0.0127280.0120020.011916(0.3,0.2)0.0021300.0021000.002094(0.6,0.5)0.0050070.0087710.009194(0.75,0.7)0.0940960.1045750.105493(0.9,0.7)0.2467660.2266620.225008(0.9,0.9)0.5948990.5531660.549529Table 2
Error of approximationEm1,m2∗λ1,λ2 for m1=m2=10,20 and 30 and q1=q2=0.8 and λ1=λ2=0.6.
y1,y2E10,10∗0.6,0.6E20,20∗0.6,0.6E30,30∗0.6,0.6(0.1,0.1)0.0001440.0001070.000103(0.1,0.4)0.0009120.0007010.000682(0.2,0.6)0.0018310.0014280.001385(0.3,0.2)0.0005880.0006000.000598(0.6,0.5)0.0404270.0358370.035374(0.75,0.7)0.0132320.0147820.014782(0.9,0.7)0.0718230.0622880.061427(0.9,0.9)0.0358400.0324380.032209



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