# Approximation by One and Two Variables of the Bernstein-Schurer-Type Operators and Associated GBS Operators on Symmetrical Mobile Interval

Authors: Reşat Aslan; Aydın İzgi
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this article, we purpose to study some approximation properties of the one and two variables of the Bernstein-Schurer-type operators and associated GBS (Generalized Boolean Sum) operators on a symmetrical mobile interval. Firstly, we define the univariate Bernstein-Schurer-type operators and obtain some preliminary results such as moments, central moments, in connection with a modulus of continuity, the degree of convergence, and Korovkin-type approximation theorem. Also, we derive the Voronovskaya-type asymptotic theorem. Further, we construct the bivariate of this newly defined operator, discuss the order of convergence with regard to Peetre’sK-functional, and obtain the Voronovskaya-type asymptotic theorem. In addition, we consider the associated GBS-type operators and estimate the order of approximation with the aid of mixed modulus of smoothness. Finally, with the help of the Maple software, we present the comparisons of the convergence of the bivariate Bernstein-Schurer-type and associated GBS operators to certain functions with some graphical illustrations and error estimation tables.


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## Body

## 1. Introduction

In [1], Bernstein suggested his polynomials that still inspire many studies today as follows:
(1)Brμ;x=∑j=0rrjxj1−xr−jμjr,x∈0,1,for any r∈ℕ and any μ∈C0,1.In 1962, the operatorsSrμ;x:C0,p+1⟶C0,1, which are called Bernstein-Schurer, are proposed by Schurer [2] as follows:
(2)Srμ;x=∑j=0r+pr+pjxj1−xr+p−jμjr,x∈0,1,for any r∈ℕ and μ∈C0,p+1.Very recently, many modifications and generalizations of the Bernstein or Bernstein-Schurer operators for univariate and bivariate cases are discussed by many authors. For instance, Acar et al. [3] established local and global approximation results in terms of modulus of continuity for a new type of the Bernstein-Durrmeyer operators on mobile interval. Izgi [4] presented a new type of the Bernstein polynomials and studied several approximation results of the univariate and bivariate of these operators. For the parameter α∈ℝ, Chen et al. [5] defined a new generalization of the Bernstein operator and derived the order of convergence and Voronovskaya-type asymptotic relation for the α−Bernstein operator. Kajla and Acar [6] constructed a new kind of the α−Bernstein operator and studied a uniform convergence estimate, some direct results involving the asymptotic theorems for these operators. Acar et al. [7] introduced the Kantorovich modifications of the p,q−Bernstein operators for bivariate functions using a new p,q−integral and obtained the uniform convergence and rate of approximation in terms of modulus of continuity for these operators. Further, for λ∈−1,1, Cai [8] introduced the Bézier version of the Kantorovich-type λ−Bernstein polynomials and gained the global and direct approximation theorems. Acar and Kajla [9] introduced an extension of the bivariate generalized Bernstein operators with nonnegative real parameters and studied the degree of approximation with regard to Peetre’s K-functional and Lipschitz-type functions. Bărbosu [10] demonstrated the uniform convergence and estimated the degree of approximation of the bivariate of the Bernstein-Schurer operators. Căbulea [11] considered the generalizations of the Kantorovich and Durrmeyer type of the Bernstein-Schurer operators and evaluated in connection with the modulus of continuity the order of approximation of these operators. Also, for some recent works, we can refer the readers to ([12–23]).By the motivation of the all the above-mentioned works, we define the univariate Bernstein-Schurer-type operators on a symmetrical mobile interval. Let the intervals beDr=−r/r+1−p,r/r+1+p, I=−1,1, and CDr be the set of all continuous and bounded functions on Dr. For a function μ∈CDr and x∈Dr, the univariate Bernstein-Schurer operators Fr:CDr⟶CI are defined as
(3)Frμ;x=r+12rr+p∑j=0r+pφr,jpxμ2j−rr+1,where ∈ℕ,p∈ℕ0=ℕ∪0,φr,jpx=r+pjr/r+1+xjr/r+1−xr+p−j.The goal of the present work is to obtain some approximation features of the operators given by (3). We show the uniform convergence, estimate the degree of convergence with the help of modulus of continuity, and prove the Voronovskaya-type asymptotic theorem for the (3) operators. Next, we define the bivariate of (3) operators, compute the order of convergence by using Peetre’s K-functional, and derive the Voronovskaya-type asymptotic theorem for the bivariate case. Further, we construct the associated GBS type of bivariate operators and estimate their degree of convergence in terms of mixed modulus of smoothness. Finally, by the help of the Maple software, we give comparisons of the convergence of bivariate of (3) operators and related GBS operators to the certain functions with some graphics and error estimation tables.


## 2. Main Results

Lemma 1.
Let the operatorsFrμ;x be defined by (3). Then, for all x∈Dr, the following moments verify
(4)Fr1;x=1,(5)Frt;x=x+pr+1x+rrr+1,(6)Frt2;x=x2+p2+2pr−p−rr2x2+2pr+prr+1x+p2+p+rr+12,(7)Frt3;x=x3+p3+3pr2+3p2r−3p2−3r2−3pr+2p+2rr3x3+3pr+pr+p−1r2r+1x2+r+p3p2+3p+3r−2rr+12x+pp2+3p+3rr+13,(8)Frt4;x=x4+p4+4p3r+6p2r2+4pr3−6p3−6r3−18pr2−18p2rr4+11p2+22pr+11r2−6p−6rr4x4+4p−1r+pr+p−1r+p−2r3r+1x3+2r+pr+p−13p2−3p−3r+8r2r+12x2+4r+pp3−3pr−3r2+7p+9r−6rr+13x+p4+2p3−6p2r−12pr2−4r3+15p2+30pr+15r2−10p−10rr+14.Proof.
From (3), it becomes
(9)Fr1;x=r+12rr+p∑j=0r+pr+pjrr+1+xjrr+1−xr+p−j=r+12rr+p2rr+1r+p=1,Frt;x=r+12rr+p∑j=0r+pr+pjrr+1+xjrr+1−xr+p−j2j−rr+1=r+12rr+p2r+pr+1∑j=0r+p−1r+p−1jrr+1+xj+1rr+1−xr+p−j−1−rr+1r+12rr+p∑j=0r+pr+pjrr+1+xjrr+1−xr+p−j=r+12r2r+pr+1rr+1+x−rr+1=x+pr+1x+rrr+1,Frt2;x=r+12rr+p∑j=0r+pr+pjrr+1+xjrr+1−xr+p−j2j−rr+12=r+12rr+p4r+pr+p−1r+12∑j=0r+p−2r+p−2jrr+1+xj+2rr+1−xr+p−j−2+r+12rr+p4r+pr+12∑j=0r+p−1r+p−1jrr+1+xj+1rr+1−xr+p−j−1−r+12rr+p4rr+pr+12∑j=0r+p−1r+p−1jrr+1+xj+1rr+1−xr+p−j−1+r+12rr+prr+12∑j=0r+pr+pjrr+1+xjrr+1−xr+p−j=r+12r24r+pr+p−1r+12rr+1+x2+r+12r4r+pr+12rr+1+x−r+12r4rr+pr+12rr+1+x+rr+12=x2+p2+2pr−p−rr2x2+2pr+prr+1x+p2+p+rr+12.
The last two identities can be obtained by applying similar methods; hence, we have omitted the details.Lemma 2.
For allx∈Dr, we obtain the following central moments:
(10)Frt−x;x=pr+1x+rrr+1,Frt−x2;x=p2−p−rr2x2+2p2rr+1x+p2+p+rr+12,Frt−x4;x=p4−6p3−6p2r−12pr2+11p2+14pr+3r2−6p−6rr4x4+4p4−4p3−6p2r−3pr2+5p2+8pr+3r2−2p−2rr3r+1x3+23p4−6p3−18p2−18pr2+11p2+26pr+15r2−8p−8rr2r+12x2+4p4−6p2r−9pr2+7p2+16pr+9r2−6p−6rrr+13x+p4+2p3−6p2r−12pr2+15p2+30pr+15r2−10p−10rr+14.Proof.
The proof of this lemma can be directly obtained by using the linearity of (3) operators and as a consequence of Lemma 1.Corollary 3.
For allx∈Dr, the following identities hold:
(11)limr⟶∞rFrt−x;x=px+1,limr⟶∞rFrt−x2;x=1−x2,limr⟶∞r2Frt−x4;x=31−3x2x2+1.
In the next theorem we show the uniform convergence of (3) operators. As it is known, the space CDr denotes the real-valued continuous functions on Dr, and it is equipped with the norm for a function μ as follows:
(12)μCDr=supx∈Drμx.Theorem 4.
Let the operatorsFrμ;x be given by (3). Then, for all x∈Dr, Frμ;x converges to μ uniformly on Dr.Proof.
From (4), it is obvious that
(13)limr⟶∞Fr1;.−1CDr=0.
By (5), we arrive
(14)limr⟶∞Frt;.−xCDr=limr⟶∞maxx∈Dpx2r+1rr+1<limr⟶∞ppr+p+2rrr+1=0.
Similarly, using (6), then
(15)limr⟶∞Frt2;.−x2CDr=limr⟶∞maxx∈Drp2−p−rr2x2+2p2rr+1x+p2+p+rr+12<limr→∞p2−3p+r+4r+12=0.
Hence, according to the Korovkin theorem [24], the (3) operators converge uniformly to μ on Dr.Further, we will obtain the degree of approximation of (3) operators. Let the modulus of continuity for a function μ∈Ca,b be given by
(16)ωμ,δ=supt,y∈a,bt−y≤δμt−μy.Sinceδ>0,ωμ,δ has some useful properties which can be found in [25].Theorem 5.
Letμ∈CDr. Then, for every x∈Dr, the following inequality is verified:
(17)Frμ;x−μx≤2ωμ;γrx,where γrx≔Frt−x2;x.Proof.
Taking into account the following common property of the modulus of continuity:(18)μt−μx≤1+t−xδωμ;δ,by the linearity of the operator (3), then
(19)Frμ;x−μx≤1+1δFrt−x;xωμ;δ.
Utilizing the Cauchy-Schwarz inequality yields(20)Frμ;x−μx≤1+1δFrt−x2;xωμ;δ≤1+γrxδωμ;δ.
If we takeδ=γrx≔Frt−x2;x, thus Theorem 5 is proven.Theorem 6.
Let the operatorsFrμ;x be given by (3). Then, for any μ∈CDr such that μ′,μ″∈CDr, the following identity holds:
(21)limr⟶∞rFrμ;x−μx=px+1μ′x+121−x2μ″x,uniformly on Dr.Proof.
Suppose thatμ,μ′,μ″∈CDr and x∈Dr are fixed. By the Taylor formula, hence
(22)μt=μx+μ′xt−x+12μ″xt−x2+ξt;xt−x2,where ξt;x is a form of Peano of the rest term, and since ξ.;x∈CDr,limt⟶xξt;x=0.
OperatingFr.;x to (22), then
(23)Frμ;x−μx=μ′xFrt−x;x+12μ″xFrt−x2;x+Frξt;xt−x2;x.
From Lemma2, it becomes
(24)limr⟶∞rFrμ;x−μx=px+1μ′x+121−x2μ″x+limr→∞rFrξt;xt−x2;x.
Applying the Cauchy-Schwarz inequality, one has(25)rFrξt;xt−x2;x≤Frξ2t;x;xr2Frt−x4;x.
Owing toξt;x∈CDr,limt⟶xξt;x=0, thus
(26)limr⟶∞Frξ2t;x;x=0,uniformly on Dr with Theorem 4.
Combining (25) and (26) and by Lemma 2, we get
(27)limr⟶∞rFrξt;xt−x2;x=0.
Hence,(28)limr⟶∞rFrμ;x−μx=px+1μ′x+121−x2μ″x,which gives the proof.


## 3. Construction of the Bivariate Bernstein-Schurer-Type Operators

Let the intervals beDr1,r2=−r1/r1+1−p1,r1/r1+1+p1×−r2/r2+1−p2,r2/r2+1+p2,I2=I×I=−1,1×−1,1, and by CDr1,r2, we denote the set of all real-valued continuous functions on Dr1,r2, and it is equipped with the norm μCDr1,r2=supx,y∈Dr1,r2μx,y.We define the bivariateFr1,r2:CDr1,r2⟶CI2 of operators given by (3) as follows:
(29)Fr1,r2μ;x,y=r1+12r1r1+p1r2+12r2r2+p2∑j1=0r1+p1∑j2=0r2+p2φr1,r2,j1,j2p1,p2x,yμ2j1−r1r1+1,2j2−r2r2+1,where φr1,r2,j1,j2p1,p2x,y=r1+p1j1r2+p2j2r1/r1+1+xj1r2/r2+1+yj2r1/r1+1−xr1+p1−j1r2/r2+1−yr2+p2−j2,r1,r2∈ℕ×ℕ, p1,p2∈ℕ0×ℕ0, and x,y∈Dr1,r2.It can be seen that the operators given by (29) are positive and linear.Lemma 7.
Letea1,a2x,y=xa1ya2,x,y∈Dr1,r2,a1,a2∈ℕ0×ℕ0 with a1+a2≤4, be the bivariate test functions. Then, one has
(30)Fr1,r2e0,0;x,y=1,Fr1,r2e1,0;x,y=x+p1r1+1x+r1r1r1+1,Fr1,r2e0,1;x,y=y+p2r2+1y+r2r2r2+1,Fr1,r2e2,0;x,y=x2+p12+2p1r1−p1−r1r12x2+2p1r1+p1r1r1+1x+p12+p1+r1r1+12,Fr1,r2e0,2;x,y=y2+p22+2p2r2−p2−r2r22y2+2p2r2+p2r2r2+1y+p22+p2+r2r2+12,Fr1,r2e3,0;x,y=x3+p13+3p1r12+3p12r1−3p12−3r12−3p1r1+2p1+2r1r13x3+3p1r1+p1r1+p1−1r12r1+1x2+r1+p13p12+3p1+3r1−2r1r1+12x+p1p12+3p1+3r1r1+13,Fr1,r2e0,3;x,y=y3+p23+3p2r22+3p22r2−3p22−3r22−3p2r2+2p2+2r2r23y3+3p2r2+p2r2+p2−1r22r2+1y2+r2+p23p22+3p2+3r2−2r2r2+12y+p2p22+3p2+3r2r2+13,Fr1,r2e4,0;x,y=x4+p14+4p13r1+6p12r12+4p1r13−6p13−6r13−18p1r12−18p12r1r14+11p12+22p1r1+11r12−6p1−6r1r14x4+4p1−1r1+p1r1+p1−1r1+p1−2r13r1+1x3+2r1+p1r1+p1−13p12−3p1−3r1+8r12r1+12x2+4r1+p1p13−3p1r1−3r12+7p1+9r1−6r1r1+13x+p14+2p13−6p12r1−12p1r12+−4r13+15p12+30p1r1+15r12−10p1−10r1r1+14,Fr1,r2e0,4;x,y=y4+p24+4p23r2+6p22r22+4p2r23−6p23−6r23−18p2r22−18p22r2r24+11p22+22p2r2+11r22−6p2−6r2r24y4+4p2−1r2+p2r2+p2−1r2+p2−2r23r2+1y3+2r2+p2r2+p2−13p22−3p2−3r2+8r22r2+12y2+4r2+p2p23−3p2r2−3r22+7p2+9r2−6r2r2+13y+p24+2p23−6p22r2−12p2r22+−4r23+15p22+30p2r2+15r22−10p2−10r2r2+14.Proof.
The proof of the above equalities can be reached easily as a consequence of Lemma1 and by (29); hence, we have omitted the details.Corollary 8.
In view of Lemma7, the following relations hold true:
(31)Fr1,r2t0−x;x,y=p1r1+1x+r1r1r1+1,Fr1,r2s0−y;x,y=p2r2+1y+r2r2r2+1,Fr1,r2t0−x2;x,y=p12−p1−r1r12x2+2p12r1r1+1x+p12+p1+r1r1+12,Fr1,r2s0−y2;x,y=p22−p2−r2r22y2+2p22r2r2+1y+p22+p2+r2r2+12,Fr1,r2t0−x4;x,y=p14−6p13−6p12r1−12p1r12+11p12+14p1r1+3r12−6p1−6r1r14x4+4p14−4p13−6p12r1−3p1r12+5p12+8p1r1+3r12−2p1−2r1r13r1+1x3+23p14−6p13−18p12−18p1r12+11p12+26p1r1+15r12−8p1−8r1r12r1+12x2+4p14−6p12r1−9p1r12+7p12+16p1r1+9r12−6p1−6r1r1r1+13x+p14+2p13−6p12r1−12p1r12+15p12+30p1r1+15r12−10p1−10r1r1+14,Fr1,r2s0−y4;x,y=p24−6p23−6p22r2−12p2r22+11p22+14p2r2+3r22−6p2−6r2r24y4+4p24−4p23−6p22r2−3p2r22+5p22+8p2r2+3r22−2p2−2r2r23r2+1y3+23p24−6p23−18p22−18p2r22+11p22+26p2r2+15r22−8p2−8r2r22r2+12y2+4p24−6p22r2−9p2r22+7p22+16p2r2+9r22−6p2−6r2r2r2+13y+p24+2p23−6p22r2−12p2r22+15p22+30p2r2+15r22−10p2−10r2r2+14.Theorem 9.
Let the operatorsFr1,r2μ;x,y be given by (29). Then for any μ∈CDr1,r2, we arrive at
(32)limr1,r2⟶∞Fr1,r2μ−μ=0.Proof.
It is seen from the following that(33)Fr1,r2e0,0−e0,0⟶0,Fr1,r2e1,0−e1,0⟶0,Fr1,r2e0,1−e0,1⟶0,Fr1,r2e2,0+e0,2−e2,0+e0,2⟶0,as r1,r2⟶∞. Thus, these results complete the proof, as required by the Volkov theorem [26].Moreover, for the operators given by (29), we want to derive the Voronovskaya-type asymptotic theorem and estimate the degree of convergence with the help of Peetre’s K-functional.Suppose thatC2Dr1,r2 represents the space of all functions of μ∈CDr1,r2 such that ∂iμ/∂xi,∂iμ/∂yi∈CDr1,r2fori=1,2. The norm on C2Dr1,r2 and Peetre’s K-functional of μ∈CDr1,r2 are given as follows, respectively.
(34)μC2Dr1,r2=μCDr1,r2+∑i=12∂iμ∂xiCDr1,r2+∂iμ∂yiCDr1,r2,Kμ,ζ=infμ−ϱCDr1,r2+ζϱC2Dr1,r2:ϱ∈C2Dr1,r2,where ζ>0.For a constantC>0, the following inequality
(35)Kμ,ζ≤Cω2∗μ,ζholds, where ω2∗μ,ζ denotes the second order of the modulus of continuity of μ∈CDr1,r2. Also, for μ∈CDr1,r2, the ordinary modulus of continuity is defined as
(36)ωμ,δ=supμu1,u2−μv1,v2:u1,u2,v1,v2∈CDr1,r2,u1−v12+u2−v22≤δ.Theorem 10.
Suppose thatμ∈C2Dr1,r2. Then, the following relation holds
(37)limr1⟶∞r1Fr1,r1μ;x,y−μ;x,y=p1x+1μx′x,y+p2y+1μy′x,y+121−x2μxx″x,y+1−y2μyy″x,y,uniformly on Dr1,r2.Proof.
For the arbitraryx,y∈Dr1,r2, using the Taylor formula, it becomes
(38)μt0,s0=μx,y+μx′x,yt0−x+μy′x,ys0−y+12μxx″x,yt0−x2+μyy″x,ys0−y2+2μxy″x,yt0−xs0−y+χt0,s0;x,yt0−x4+s0−y4,for t0,s0∈Dr1,r2, where χ.,.;x,y∈CDr1,r2 and χt0,s0;x,y⟶0, as t0,s0⟶x,y.
OperatingFr1,r1.;x,y on (38) yields
(39)Fr1,r1μt0,s0;x,y=μx,y+μx′x,yFr1,r1t0−x;x,y+μy′x,yFr1,r1s0−y;x,y+12μxx″x,yFr1,r1t0−x2;x,y+μyy″x,yFr1,r1s0−y2;x,y+2μxy″x,yFr1,r1t0−xs0−y;x,y+Fr1,r1χt0,s0;x,yt0−x4+s0−y4;x,y.
If we use the Cauchy-Schwarz inequality to the last part of (39), one has
(40)Fr1,r1χt0,s0;x,yt0−x4+s0−y4;x,y≤Fr1,r1χ2t0,s0;x,y;x,yFr1,r1t0−x4;x,y+Fr1,r1s0−y4;x,y.
Considering Theorem9 and because of χ.,.;x,y∈CDr1,r2,χt0,s0;x,y⟶0, as t0,s0⟶x,y, then
(41)limr1⟶∞Fr1,r1χ2t0,s0;x,y;x,y=0,uniformly on Dr1,r2. Also, from Corollary 8, it is clear to see
(42)limr1⟶∞r1Fr1,r1t0−xs0−y;x,y=0.
Further, by Corollary3, it follows that
(43)limr1⟶∞r12Fr1,r1t0−x4;x,y=31−3x2x2+1,limr1⟶∞r12Fr1,r1s0−y4;x,y=31−3y2y2+1.
Thus,(44)limr1⟶∞r1Fr1,r1χt0,s0;x,yt0−x4+s0−y4;x,y=0.
Hence, the desired sequel is arrived as follows:(45)limr1⟶∞r1Fr1,r1μ;x,y−μ;x,y=p1x+1μx′x,y+p2y+1μy′x,y+121−x2μxx″x,y+1−y2μyy″x,y,uniformly on Dr1,r2.Theorem 11.
Suppose thatμ∈CDr1,r2. Then, the following inequality is verified:
(46)Fr1,r2μ;x,y−μx,y≤C1ω2∗μ;Sr1,r2x,y+min1,Sr1,r2x,yμC2Dr1,r2+ωμ;ξr1,r2x,y,where C1>0 is a constant and independent of μ and Sr1,r2x,y and γr12x=Fr1,r2t0−x2;x,y,γr22y=Fr1,r2s0−y2;x,y,ξr1,r2x,y=p1r1+1x+r1/r1r1+12+p2r2+1y+r2/r2r2+12, Sr1,r2x,y=γr12x+γr22y+ξr1,r22.Proof.
Let us define the following auxiliary operators:(47)Fr1,r2¯μ;x,y=Fr1,r2μ;x,y+μx,y−μx+p1r1+1x+r1r1r1+1,y+p2r2+1y+r2r2r2+1.
By Lemma7, we obtain
(48)Fr1,r2¯t0−x;x,y=0,Fr1,r2¯s0−y;x,y=0.
Suppose thatμ∈C2Dr1,r2 and t0,s0∈Dr1,r2; by the Taylor formula, we may write
(49)μt0,s0−μx,y=∂μx,y∂xt0−x+∫t0xt0−u∂2μu,y∂2udu+∂μx,y∂ys0−y+∫s0ys0−v∂2μx,v∂2vdv.
Now, operatingFr1,r2¯ on (49), we get
(50)Fr1,r2¯μ;x,y−μx,y=Fr1,r2¯∫xt0t0−u∂2μu,y∂2udu;x,y+Fr1,r2¯∫ys0s0−v∂2μx,v∂2vdv;x,y=Fr1,r2∫xt0t0−u∂2μu,y∂2udu;x,y−∫xx+p1r1+1x+r1/r1r1+1x+p1r1+1x+r1r1r1+1−u∂2μu,y∂2udu+Fr1,r2∫s0ys0−v∂2μx,v∂2vdv;x,y−∫yy+p2r2+1y+r2/r2r2+1y+p2r2+1y+r2r2r2+1−v∂2μx,v∂2vdv.
Hence,(51)Fr1,r2μ;x,y−μx,y≤Fr1,r2∫xt0t0−u∂2μu,y∂2udu;x,y+∫xx+p1r1+1x+r1/r1r1+1x+p1r1+1x+r1r1r1+1−u∂2μu,y∂2udu+Fr1,r2∫ys0s0−v∂2μx,v∂2vdv;x,y+∫yy+p2r2+1y+r2/r2r2+1y+p2r2+1y+r2r2r2+1−v∂2μx,v∂2vdv≤Fr1,r2t0−x2;x,y+x+p1r1+1x+r1r1r1+1−x2+Fr1,r2s0−y2;x,y+y+p2r2+1y+r2r2r2+1−y2μC2Dr1,r2.
If we chooseξr1,r2=x+p1r1+1x+r1/r1r1+12+p2r2+1y+r2/r2r2+12,γr12x=Fr1,r2t0−x2;x,y,γr22y=Fr1,r2s0−y2;x,y,Sr1,r2x,y=γr12x+γr22y+ξr1,r22, then
(52)Fr1,r2μ;x,y−μx,y≤Sr1,r2x,yμC2Dr1,r2.
Also, by Lemma7 and (52), we arrive at
(53)Fr1,r2¯μ;x,y≤Fr1,r2μ;x,y+μx,y+μx+p1r1+1x+r1r1r1+1,y+p2r2+1y+r2r2r2+1≤3μCDr1,r2.
Next, from (53)
(54)Fr1,r2μ;x,y−μx,y≤Fr1,r2¯μ−κ;x,y+Fr1,r2¯κ;x,y−κx,y+μx+p1r1+1x+r1r1r1+1,y+p2r2+1y+r2r2r2+1−μx,y+κx,y−μx,y≤4μ−κCDr1,r2+Sr1,r2x,yκC2Dr1,r2+ωμ;ξr1,r2x,y.
Consequently, on (54), utilizing the infimum on the right-hand side over all μ∈C2Dr1,r2 and by (35), it becomes
(55)Fr1,r2μ;x,y−μx,y≤C1ω2∗μ;Sr1,r2x,y+min1,Sr1,r2x,yμC2Dr1,r2+ωμ;ξr1,r2x,y,which ends the proof.


## 4. Construction of the GBS Type ofFr1,r2μ;x,y

The notion of theB-continuous and B-differentiable functions was firstly used by Bögel [27, 28]. Dobrescu and Matei [29] considered the GBS (Generalized Boolean Sum) kind of the bivariate of the Bernstein polynomials. Next, using the B-continuous functions by the GBS operators, which is related to quantitative version of the Korovkin-type convergence theorem, firstly has been improved by Badea et al. [30, 31]. Pop and Fărcas [32] obtained some approximation of the B-continuous and B-differentiable functions by GBS type of the Bernstein bivariate operators. İspir [33] established quantitative estimates for the GBS of the Chlodowsky-Szász-kind operators. Recently, some authors introduced the GBS operators of various operators (we refer the readers to [34–42]).Let a functionμ:U×V⟶ℝ, where U,V are compact real intervals of ℝ. With x,y,t0,s0∈U×V, the mixed difference of the function μ is given as
(56)ϕx,yμt0,s0;x,y=μx,y−μx,s0−μt0,y+μt0,s0.A functionμ:U×V⟶ℝ is named Bögel-continuous (B-continuous) at t0,s0∈U×V, if
(57)limx,y⟶t0,s0ϕx,yμt0,s0;x,y=0.A functionμ:U×V⟶ℝ is named Bögel-differentiable (B-differentiable) at t0,s0∈U×V, if the below result which denoted by DBfx,y exists and finite
(58)limx,y⟶t0,s0ϕx,yμt0,s0;x,yx−t0y−s0=DBfx,y.Note that, byCbU×V and DbU×V, we represent the sets of all B-differentiable and B-continuous functions on U×V, respectively.A functionμ:Y⊂U×V⟶ℝ is named Bögel-bounded (B−bounded) on Y, if there consists W>0 such that ϕx,yμt0,s0;x,y≤W for every t0,s0,x,y∈Y.Also, ifY is a compact subset of ℝ2, hence all B-continuous functions are B−bounded on Y⟶ℝ.Further, byBbY, we represent the set of all B-bounded functions on Y and equipped with the norm μB=supx,y,t0,s0∈Yϕx,yμt0,s0;x,y. Considering the definition of B-continuous, then CY⊂CbY (see details by [43]).The mixed modulus of smoothness forμ∈CbDr1,r2 is given by
(59)ωmixedμ;δ1,δ2≔supϕx,yμt0,s0;x,y:t0−x<δ1,s0−y<δ2,where x,y,t0,s0∈Dr1,r2, and δ1,δ2∈ℝ+. Also, for all κ1,κ2≥0, the following inequality satisfy
(60)ωmixedμ;κ1δ1,κ2δ2≤1+κ11+κ2ωmixedμ;δ1,δ2.More details about the mixed modulus of smoothness can be found in [30, 31].Now, for allx,y∈Dr1,r2, for any μ∈CbDr1,r2 and r1,r2∈ℕ×ℕ, p1,p2∈ℕ0×ℕ0, we construct the GBS-type operators given by (29) operators as follows:
(61)Gr1,r2μ;x,y=Fr1,r2μx,s0+μt0,y−μt0,s0;x,y.Exactly, for anyμ∈CbDr1,r2 and for all x,y∈Dr1,r2, the GBS type of Fr1,r2 operators is given as
(62)Gr1,r2μ;x,y=r1+12r1r1+p1r2+12r2r2+p2∑j1=0r1+p1∑j2=0r2+p2φr1,r2,j1,j2p1,p2x,y×μx,2j2−r2r2+1+μ2j1−r1r1+1,y−μ2j1−r1r1+1,2j2−r2r2+1,where φr1,r2,j1,j2p1,p2x,y is defined as in (29).It is clear that the operators given by (62) are linear and positive. In the following theorem, with regard to the mixed modulus of smoothness, we estimate the order of approximation of the (62) operators.Theorem 12.
For allμ∈CbDr1,r2 and for each x,y∈Dr1,r2, the operators given by (62) satisfy the following inequality:
(63)Gr1,r2μ;x,y−μx,y≤4ωmixedμ;αr1,p1,αr2,p2,whereαr1,p1=p12−3p1+r1+4/r1+12 and αr2,p2=p22−3p2+r2+4/r2+12.Proof.
In view of (60), it gives
(64)ϕx,yμt0,s0;x,y≤ωmixedμ;t0−x,s0−y≤1+t0−xδ11+s0−yδ2ωmixedμ;δ1,δ2,for all x,y,t0,s0∈Dr1,r2, and for any δ1,δ2∈ℝ+. By (56), we have
(65)μx,s0+μt0,y−μt0,s0=μx,y−ϕx,yμt0,s0;x,y.
OperatingFr1,r2 and by the definition of Gr1,r2, then
(66)Gr1,r2μ;x,y=μx,yFr1,r2e0,0;x,y−Fr1,r2ϕx,yμt0,s0;x,y;x,y.
It follows that(67)Gr1,r2μ;x,y−μx,y≤Fr1,r2ϕx,yμt0,s0;x,y;x,y.
From (64) and by utilizing the Cauchy-Schwarz inequality, one has
(68)Gr1,r2μ;x,y−μx,y≤1+Fr1,r2t0−x2;x,yδ1+Fr1,r2s0−y2;x,yδ2+Fr1,r2t0−x2;x,yFr1,r2s0−y2;x,yδ1δ2ωmixedμ;δ1,δ2.
Taking into account Corollary8, it becomes
(69)Fr1,r2t0−x2;x,y≤p1r1x2+2p12r1r1+1x+p12+p1+r1r1+12≤p12−3p1+r1+4r1+12=αr1,p1,Fr1,r2s0−y2;x,y≤p2r2y2+2p22r2r2+1y+p22+p2+r2r2+12≤p22−3p2+r2+4r2+12=αr2,p2.
Hence,(70)Gr1,r2μ;x,y−μx,y≤1+p12−3p1+r1+4/r1+12δ1+p22−3p2+r2+4/r2+12δ2+p12−3p1+r1+4/r1+12p22−3p2+r2+4/r2+12δ1δ2ωmixedμ;δ1,δ2.
If we chooseδ1=αr1,p1 and δ2=αr2,p2, then the desired result is arrived.


## 5. Graphics and Error Estimation Tables

In this section, with the help of the Maple software, we give some plots and error estimation tables for the comparison of the convergence behavior of (29) and (62) operators to the certain functions.Example 1.
Letμx,y=cosxy/1+y2. In Figure 1, for r1=r2=5 (red), r1=r2=15 (green), r1=r2=75 (blue), and by choosing p1=p2=1, we illustrate the convergence of (29) operators to μx,y=cosxy/1+y2 (yellow). Also, in Table 1, we determine the error of approximation (29) operators to μx,y=cosxy/1+y2 for the certain values of −1≤x,y≤1 and r1=r2=5,50,500, respectively. It is clear from Table 1 that as the values of r1 and r2 increase then the error of approximation (29) operators to μx,y=cosxy/1+y2 decreases.Example 2.
Letμx,y=sin5/2xye−y/3. In Figure 2, for r1=r2=15, p1=p2=2, we compare the convergence of (29) (red) and the associated GBS operators (62) (blue) to μx,y=sin5/2xye−y/3 (yellow). Also, in Table 2, we evaluate the error of approximation (29) and (62) operators to μx,y=sin5/2xye−y/3 for r1=r2=300, p1=p2=2, and the certain points of −1≤x,y≤1. It is obvious that, for r1=r2=300, p1=p2=2, the absolute difference between (29) operators to μx,y is greater than that of (62) operators to μx,y. Namely, the (62) operators has better approximation than (29) operators.Figure 1
The convergence ofFr1,r2μ;x,y operators to μx,y=cosxy/1+y2 (yellow) for r1=r2=5 (red), r1=r2=15 (green), r1=r2=75 (blue), and p1=p2=1.Table 1
Error of approximationFr1,r2μ;x,y operators to μx,y=cosxy/1+y2 for r1=r2=5, r1=r2=50, r1=r2=500, and p1=p2=1.
x,yF5,5μ;x,y−μx,yF50,50μ;x,y−μx,yF500,500μ;x,y−μx,y(0.6, 0.6)0.05896620820.00570599670.0005642954(0.8, −0.1)0.05325583510.00490724920.0004809529(0.5, −0.5)0.02716143760.00244266830.0002455107(0.4, −0.3)0.02035348420.00141315110.0001327585(0.3, −0.7)0.01705429630.00100608000.0000896256(−0.2, 0.8)0.01484699300.00155559070.0001550674(−0.6, 0.4)0.01274179860.00190204170.0001971948(−0.1, 0.1)0.00916945970.00033254680.0000204350(−0.3, −0.7)0.00813015080.00079647890.0000801801(−0.9, 0.9)0.00203628630.00030146920.0000358039Figure 2
The convergence ofFr1,r2μ;x,y (red) and Gr1,r2μ;x,y (blue) operators to μx,y=sin5/2xye−y/3 (yellow) for r1=r2=15 and p1=p2=2.Table 2
Error of approximationFr1,r2μ;x,y and Gr1,r2μ;x,y operators to μx,y=sin5/2xye−y/3 for r1=r2=300 and p1=p2=2.
x,yF300,300μ;x,y−μx,yG300,300μ;x,y−μx,y(-0.9, 0.9)0.01365975030.0000249663(0.5, 0.5)0.01270766380.0000753517(-0.8, -0.1)0.01185238440.0000165295(-0.3, -0.6)0.01174228170.0000287716(-0.6, -0.4)0.01022287130.0000248461(0.4, 0.8)0.01019885600.0001669195(0.9, 0.7)0.00814555650.0005454158(0.1, -0.4)0.00701952070.0000801529(-0.7, 0.3)0.00561122390.0000346506(-0.2, 0.2)0.00034169400.0000879462



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*Source: 9979286-2021-05-04.xml*