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Kyungpook Mathematical Journal 2022; 62(3): 467-484

Published online September 30, 2022

Copyright © Kyungpook Mathematical Journal.

Some Approximation Results by Bivariate Bernstein-Kantorovich Type Operators on a Triangular Domain

Reşat Aslan∗ and Aydin Izgi

Department of Mathematics, Faculty of Sciences and Arts, Harran University, 63100 Şanlıurfa, Turkey
e-mail : resat63@hotmail.com and a_izgi@harran.edu.tr

Received: February 3, 2021; Revised: January 17, 2022; Accepted: January 24, 2022

Abstract

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In this work, we define bivariate Bernstein-Kantorovich type operators on a triangular domain and obtain some approximation results for these operators. We start off by computing some moment estimates and prove a Korovkin type convergence theorem. Then, we estimate the rate of convergence using the partial and complete modulus of continuity, and derive a Voronovskaya-type asymptotic theorem. Further, we calculate the order of approximation with regard to the Peetre's K-functional and a Lipschitz type class. In addition, we construct the associated GBS type operators and compute the rate of approximation using the mixed modulus of continuity and class of the Lipschitz of Bögel continuous functions for these operators. Finally, we use the two operators to approximate example functions in order to compare their convergence.

Keywords: Bernstein-Kantorovich operators, Modulus of continuity, Voronovskaya-type asymptotic theorem, Peetre’s K-functional, GBS type operators

1. Introduction

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In [28], Weierstrass showed in his famous approximation theorem that any continuous function f on a compact set can be approximated uniformly by a polynomial sequence pn. Since the proof of Weierstrass's approximation theorem is very long and complex, many authors have subsequently worked on the proof of this theorem, but the most elegant one was presented by Bernstein [6]. In 1930, Kantorovich [18] introduced approximations for Lebesgue integrable functions. In [19], Kingsley introduced and studied the Bernstein polynomials in the bivariate case of the class C(k). Stancu [26] obtained a new method for dealing with Bernstein operators for two variables. Very recently, Pop and Fărc{a}ş [22] investigated several approximation properties of bivariate Kantorovich type operators. Kajla and Goyal [17] obtained direct results for the modified Bernstein–Kantorovich operators and considered the bivariate case of these operators. Moreover, Deshwal et al. [11] proposed and studied bivariate operators of Bernstein-Kantorovich type on a triangle. In [16], Kajla constructed generalized Bernstein-Kantorovich–type operators on a triangle and presented Voronovskaja-type and Grüss Voronovskaja-type asymptotic theorems; he estimated of the rate of approximation using Peetre's K-functional. In 2017, Goyal et al. [15] considered a bivariate extension of the Bernstein-Durrmeyer type operators on a triangle domain. Başcanbaz-Tunca et al. [6] considered bivariate Cheney-Sharma operators which preserve the Lipschitz condition. Agrawal et al. [1] discussed the deferred weighted A-statistical approximation and investigated the convergence estimates for the functions in a Bögel space by Bernstein-Kantorovich type operators on a triangle. Local and global approximation results in terms of modulus of continuity, Peetre’s K-functional, second-order modulus of smoothness and statistical convergence for certain Bernstein-Kantorovich, bivariate Bernstein-Kantorovich and Bernstein-Stancu operators are studied in very recent papers [2, 20, 25].

Now, let ∇:=(x,y):−1≤x,y≤1,x+y≤0 be a triangular domain and let C(∇) be the set of all real functions h that are continuous on ∇ and bounded on ℝ×ℝ. The norm on C(∇) is

h=sup(x,y)∈∇h(x,y).

Inspired by the works mentioned above, we construct bivariate Bernstein-Kantorovich type operators for (x,y)∈∇ and h:∇→ℝ as follows:

 Rn(h;x,y)=∑ k=0 n ∑l=0n−k ν n,k,l (x,y)∫ 2kn+1−1 2k+1n+1−1 ∫ 2l n+1 −1 2 l+1 n+1 −1 h(s,t)dsdt,

where νn,k,l(x,y)=n+122nkn−kl1+x2k1+y2l1−1+x2−1+y2n−k−l.

The structure of this work is planned as follows. In Section 2, we give some auxiliary results, such as computing moment estimates and proving a Korovkin's type approximation using Volkov's theorem [27]. In Section 3, we investigate the rate of approximation with regard to the partial and complete modulus of continuity and derive a Voronovskaya-type asymptotic theorem. In Section 4, we discuss the rate of convergence in terms of the Peetre's K-functional and Lipschitz type class. In Section 5, we construct the GBS type of newly defined operators and estimate the order of convergence in terms of the mixed modulus of smoothness and the Lipschitz class of Bögel-continuous function. Finally, we give a comparison of the convergence of the newly defined operators and their associated GBS operators with example computations.

2. Basic Results

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Lemma 2.1. Let ru,v:∇→ℝ,ru,v(s,t)=sutv. Then, for any (x,y)∈∇ and 0≤u,v≤4, the operators given by (1.1) satisfy the following equalities:

(i) Rn(r0,0;x,y)=1,(ii) Rn(r1,0;x,y)=x1−1n+1,(iii) Rn(r0,1;x,y)=y1−1n+1,(iv) Rn(r1,1;x,y)=xy1−3n+1n+12−x+y+1nn+12,(v) Rn(r2,0;x,y)=x21−3n+1n+12+n+13n+12,(vi) Rn(r0,2;x,y)=y21−3n+1n+12+n+13n+12,(vii) Rn(r3,0;x,y)=x31−6n2+n+1n+13+x3n2−nn+13,(viii) Rn(r0,3;x,y)=y31−6n2+n+1n+13+y3n2−nn+13,(ix) Rn(r4,0;x,y)=x41−10n3−5n2+10n+1n+14+6x2n3+10n2−5nn+14 +39n2−36n+15n+14,(x) Rn(r0,4;x,y)=y41−10n3−5n2+10n+1n+14+6y2n3+10n2−5nn+14 +39n2−36n+15n+14.

Lemma 2.2. Let ku,v:∇→ℝ,ku,v=s−xut−yv. Then, for any (x,y)∈∇ and 0≤u,v≤4, we have the following central moments:

(i) Rn(k0,0;x,y)=1,(ii) Rn(k1,0;x,y)=−xn+1,(iii) Rn(k0,1;x,y)=−yn+1,(iv) Rn(k1,1;x,y)=xy(1−n)n+12−x+y+1nn+12,(v) Rn(k2,0;x,y)=x2(1−n)+n+13n+12,(vi) Rn(k0,2;x,y)=y2(1−n)+n+13n+12,(vii) Rn(k4,0;x,y)=x43n2−20n+1n+14+2x233n2−8n+1n+14+39n2−36n2+15n+14,(viii) Rn(k0,4;x,y)=y43n2−20n+1n+14+2y233n2−8n+1n+14+39n2−36n2+15n+14.

Lemma 2.3. For the operators given by (1.1), we have the following relations:

(i)limn→∞nRn(s−x;x,y)=−x,(ii)limn→∞nRn(t−y;x,y)=−y,(iii)limn→∞nRn(s−x2;x,y)=1−x2,(iv)limn→∞nRn(t−y2;x,y)=1−y2,(v)limn→∞nRn(s−xt−y;x,y)=−xy+x+y+1,(vi)limn→∞n2Rn(s−x4;x,y)=3x4+22x2+13,(vii)limn→∞n2Rn(t−y4;x,y)=3y4+22y2+13.

Theorem 2.4. If h(x,y)∈C(∇), then operators given by (1.1) convergence uniformly to h on ∇ asn→∞.

Proof. As a consequence of [27], we have to show operators given by (1.1) verifies that:

(i)limn→∞Rn(r0,0)−1C(∇)→0,(ii)limn→∞Rn(r1,0)−xC(∇)→0,(iii)limn→∞Rn(r0,1)−yC(∇)→0,(iv)limn→∞Rn(r2,0+r0,2)−(x2+y2)C(∇)→0.

From Lemma 2.1. (i), it is obvious that

(i)limn→∞Rn(r0,0)−1C(∇)→0.

In view of Lemma 2.1. (ii)-(iii), we get

(ii)limn→∞Rn(r1,0)−xC(∇)=limn→∞max−1≤x≤1Rn(r1,0)−x=limn→∞max−1≤x≤1−xn+1≤limn→∞1n+1→0.

Similarly, we obtain

(iii)limn→∞Rn(r0,1)−yC(∇)→0.

Also, from Lemma 2.1. (iv), we have

(iv)limn→∞ Rn(r 2,0+r 0,2)−(x2+y2)C(∇)=limn→∞max−1≤x,y≤1Rn(r2,0+r0,2)−(x2+y2)=limn→∞max−1≤x,y≤1(x2+y2)1−3n+1n+1 2 +2n+ 23n+1 2 −(x2+y2)≤limn→∞8n+1→0,

which gives the proof of the Theorem 2.4.

2. Basic Results

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Lemma 2.1. Let ru,v:∇→ℝ,ru,v(s,t)=sutv. Then, for any (x,y)∈∇ and 0≤u,v≤4, the operators given by (1.1) satisfy the following equalities:

(i) Rn(r0,0;x,y)=1,(ii) Rn(r1,0;x,y)=x1−1n+1,(iii) Rn(r0,1;x,y)=y1−1n+1,(iv) Rn(r1,1;x,y)=xy1−3n+1n+12−x+y+1nn+12,(v) Rn(r2,0;x,y)=x21−3n+1n+12+n+13n+12,(vi) Rn(r0,2;x,y)=y21−3n+1n+12+n+13n+12,(vii) Rn(r3,0;x,y)=x31−6n2+n+1n+13+x3n2−nn+13,(viii) Rn(r0,3;x,y)=y31−6n2+n+1n+13+y3n2−nn+13,(ix) Rn(r4,0;x,y)=x41−10n3−5n2+10n+1n+14+6x2n3+10n2−5nn+14 +39n2−36n+15n+14,(x) Rn(r0,4;x,y)=y41−10n3−5n2+10n+1n+14+6y2n3+10n2−5nn+14 +39n2−36n+15n+14.

Lemma 2.2. Let ku,v:∇→ℝ,ku,v=s−xut−yv. Then, for any (x,y)∈∇ and 0≤u,v≤4, we have the following central moments:

(i) Rn(k0,0;x,y)=1,(ii) Rn(k1,0;x,y)=−xn+1,(iii) Rn(k0,1;x,y)=−yn+1,(iv) Rn(k1,1;x,y)=xy(1−n)n+12−x+y+1nn+12,(v) Rn(k2,0;x,y)=x2(1−n)+n+13n+12,(vi) Rn(k0,2;x,y)=y2(1−n)+n+13n+12,(vii) Rn(k4,0;x,y)=x43n2−20n+1n+14+2x233n2−8n+1n+14+39n2−36n2+15n+14,(viii) Rn(k0,4;x,y)=y43n2−20n+1n+14+2y233n2−8n+1n+14+39n2−36n2+15n+14.

Lemma 2.3. For the operators given by (1.1), we have the following relations:

(i)limn→∞nRn(s−x;x,y)=−x,(ii)limn→∞nRn(t−y;x,y)=−y,(iii)limn→∞nRn(s−x2;x,y)=1−x2,(iv)limn→∞nRn(t−y2;x,y)=1−y2,(v)limn→∞nRn(s−xt−y;x,y)=−xy+x+y+1,(vi)limn→∞n2Rn(s−x4;x,y)=3x4+22x2+13,(vii)limn→∞n2Rn(t−y4;x,y)=3y4+22y2+13.

Theorem 2.4. If h(x,y)∈C(∇), then operators given by (1.1) convergence uniformly to h on ∇ asn→∞.

Proof. As a consequence of [27], we have to show operators given by (1.1) verifies that:

(i)limn→∞Rn(r0,0)−1C(∇)→0,(ii)limn→∞Rn(r1,0)−xC(∇)→0,(iii)limn→∞Rn(r0,1)−yC(∇)→0,(iv)limn→∞Rn(r2,0+r0,2)−(x2+y2)C(∇)→0.

From Lemma 2.1. (i), it is obvious that

(i)limn→∞Rn(r0,0)−1C(∇)→0.

In view of Lemma 2.1. (ii)-(iii), we get

(ii)limn→∞Rn(r1,0)−xC(∇)=limn→∞max−1≤x≤1Rn(r1,0)−x=limn→∞max−1≤x≤1−xn+1≤limn→∞1n+1→0.

Similarly, we obtain

(iii)limn→∞Rn(r0,1)−yC(∇)→0.

Also, from Lemma 2.1. (iv), we have

(iv)limn→∞ Rn(r 2,0+r 0,2)−(x2+y2)C(∇)=limn→∞max−1≤x,y≤1Rn(r2,0+r0,2)−(x2+y2)=limn→∞max−1≤x,y≤1(x2+y2)1−3n+1n+1 2 +2n+ 23n+1 2 −(x2+y2)≤limn→∞8n+1→0,

which gives the proof of the Theorem 2.4.

4. Local Approximation

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In this section, we will estimate the rate of convergence in terms of Peetre's K-functional and a Lipschitz-type function. The norm on C2(∇) and Peetre's K-functional are given, respectively as follows:

hC2(∇)=hC(∇)+∑2j=1 ∂jh∂xj C(∇)+ ∂jh∂yj C(∇),
K2(h,ζ)=infh−gC(∇)+ζgC2(∇):g∈C2(∇)  (ζ>0).

Also, for an absolute constant D>0 such that

K2(h,ζ)≤Dω2∗(h,ζ),

where ω2∗(h,ζ) denote the second order of modulus of continuity. (See: [3, 29]).

Further, for h∈C(∇), (x,y),(t,s)∈∇ and β,γ∈0,1, the Lipschitz-type class for bivariate case is assigned as

Lipα(β,γ)=h∈C(∇):h(t,s)−h(x,y);x,y≤α t−xβs−yγ.

Theorem 4.1. Let h∈Lipα(β,γ). Then, for each (x,y)∈∇ the following inequality verifies that

 Rn(h;x,y)−h(x,y)≤α Πn1(x)β2Πn2(y)γ2

where Πn1(x)=Rn(k2,0;x,y) and Πn2(y)=Rn(k0,2;x,y).

Proof. In view of (4.2) and using the definition of (1.1), then

 Rn(h;x,y)−h(x,y)≤Rnh(t,s)−h(x,y);x,y≤αRnt−xβs−yγ;x,y=αRn(t−xβ;x,y)Rns−yγ;x,y.

Utilizing the Hölder's inequality with p1,q1=2β,22−β and p2,q2=2γ,22−γ, we

get

 Rn(h;x,y)−h(x,y)≤αRn(t−x)2;x,yβ2Rnr0,0;x,y2−β2×Rn(s−y)2;x,yγ2Rnr0,0;x,y2−γ2.

From Lemma 2.1. (i), which leads to the required result as

 Rn(h;x,y)−h(x,y)≤α Πn1(x)β2Πn2(y)γ2.

Theorem 4.2. Let h∈C1(∇). Then, we obtain the following inequality

 Rn(h;x,y)−h(x,y)≤hxΠn1(x)+hyΠn2(y)

where Πn1(x) and Πn2(y) are same as in Theorem 4.1..

Proof. For a given fixed point (x,y)∈∇, we may write

h(u,v)−h(x,y)=∫xuht(t,v)dt+∫yvhs(x,s)ds.

Operating Rn(.;x,y) to the both sides of above equality, then

 Rn(h;x,y)−h(x,y)≤Rn(∫xuht(t,v)dt;x,y)+Rn(∫yvhs(x,s)ds;x,y).

Since

∫xuht(t,v)dt≤hxu−xand∫yvhs(x,s)ds≤hyv-y, 

hence,

 Rn(h;x,y)−h(x,y)≤hxRn(u−x;x,y)+hyRn(v−y;x,y).

Applying the Cauchy-Schwarz inequality to the above inequality, one has

 Rn(h;x,y)−h(x,y)≤hxRn((u−x)2;x,y)12Rn(r0,0;x,y)12+hyRn((v−y)2;x,y)12Rn(r0,0;x,y)12≤hxΠn1(x)+hyΠn2(y),

which gives the proof of Theorem 4.2.

Theorem 4.3. Let g∈C(∇). Then, for the operators defined by (1.1) we have the following inequality

|Rn(g;x,y)−g(x,y)|≤Nω2∗(g;An (x,y)2+min{1,An}gC(∇)              +ϖ(g;χn(x,y)),

where χn(x,y)=x2+y2n+1,An(x,y)=Πn1 (x)+Πn2 (y)+χn2(x,y) and N>0 is a constant.

Proof. By first, we define the following auxiliary operators

Rn∗(g;x,y)=Rn(g;x,y)−g(xnn+1,ynn+1)+gx,y.

From Lemma 2.1., it is clear that

Rn∗((s−x);x,y)=0, Rn∗((t−y);x,y)=0.

For h∈C2(∇), (s,t)∈∇, applying Taylor's formula, then

h(s,t)−h(x,y)=h(s,y)−h(x,y)+h(s,t)−h(s,y)=∂h(x,y)∂x(s−x)+∫sx(s−u)∂2h(u,y)∂2udu+∂h(x,y)∂y(t−y)+∫ty(t−v)∂2h(x,v)∂2vdv.

Operating Rn(.;x,y) to (4.4), hence

Rn∗(h;x,y)−h(x,y)=Rn∗∫sx(s−u)∂2h(u,y)∂2udu;x,y+Rn∗∫ty(t−v)∂2h(x,v)∂2vdv;x,y=Rn∫sx(s−u)∂2h(u,y)∂2udu;x,y−∫xnn+1xxnn+1−u∂2h(u,y)∂2udu+Rn∫ty(t−v)∂2h(x,v)∂2vdv;x,y−∫ynn+1yynn+1−v∂2h(x,v)∂2vdv.

Moreover,

Rn(h;x,y)−h(x,y)≤Rn∫sxs−u∂2h(u,y)∂2udu;x,y+∫xnn+1xxnn+1−u∂2h(u,y)∂2udu+Rn∫tyt−v∂2h(x,v)∂2vdv;x,y+∫ynn+1yynn+1−v∂2h(x,v)∂2vdv≤Rn((s−x)2;x,y)+(xnn+1−x)2+Rn((t−y)2;x,y)+(ynn+1−y)2h C2(∇).

Choosing χn(x,y)=x2+y2n+1,An(x,y)=Πn1 (x)+Πn2 (y)+χn2(x,y), we get

Rn(h;x,y)−h(x,y)≤An(x,y)hC2(∇).

Taking Lemma 2.2. into account, thus

Rn∗(g;x,y)≤Rn(g;x,y)+g(xnn+1,ynn+1)+g(x,y)≤3gC(∇).

From (4.5) and (4.6), we obtain

Rn(g;x,y)−g(x,y)≤Rn∗(g−h;x,y)+Rn∗(h;x,y)−h(x,y)+h(x,y)−g(x,y+g(xnn+1,ynn+1)−g(x,y)≤4g−h+Rn(h;x,y)−h(x,y)+g(xnn+1,ynn+1)−g(x,y)≤4g−h+An(x,y)hC2(∇)+ϖ(g;χn(x,y)).

Taking the infimum over all h∈C2(∇) on the right hand side of (4.7), we arrive

Rn(g;x,y)−g(x,y)≤4K2(g;An(x,y)4)+ϖ(g;χn(x,y)).

Applying (4.1) to the above inequality, it gives the required result as

Rn(g;x,y)−g(x,y)≤Nω2∗(g;An(x,y)2+min{1,An}gC(∇)              +ϖ(g;χn(x,y)).

5. GBS(Generalized Boolean Sum) Type Operators

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The concept of the B-continuous and B-differentiable functions were firstly used by Bögel [8, 9]. Dobrescu and Matei [12] considered the GBS(Generalized Boolean Sum) type of the bivariate Bernstein operators. Next, using of the B-continuous functions by the GBS operators, which is related to a quantitative variant of the Korovkin's type theorem, was firstly improved by Badea []. Pop and Fărcas [21] obtained some approximation of B-continuous and B-differentiable functions by GBS type of Bernstein bivariate operators. We refer also some papers of various linear positive operators, which are related to the GBS operators, ([23, 24, 5, 13]).

Let a function h:∇→ℝ. For any (x,y),(t0,s0)∈∇, the mixed difference of the function h is given by

ϕ(x,y)ht0,s0;x,y=h(x,y)−h(x,s0)−h(t0,y)+h(t0,s0).

A function h:∇→ℝ is called Bögel-continuous (B-continuous) at (t0,s0)∈ ∇, if

lim(x,y)→(t0,s0)ϕ(x,y)ht0,s0;x,y=0.

A function h:∇→ℝ is called Bögel-differentiable (B-differentiable) at (t0,s0)∈∇, if

lim(x,y)→(t0,s0)ϕ(x,y)ht0,s0;x,y(x−t0)(y−s0)<∞.

Note that, by Cb(∇) and Db(∇), we denote the sets of all B-differentiable and B-continuous functions on ∇, respectively. Also, C(∇)⊂Cb(∇), see details in [10].

The mixed modulus of smoothness for h∈Cb(∇) is given by

ωmixedh;δ1,δ2:=supϕ(x,y)ht0 ,s0 ;x,y:t0−x<δ1,s0−y<δ2

where (x,y),(t0,s0)∈∇ and δ1,δ2∈ℝ+. Also for all λ1,λ2≥0, the following inequality

holds

ωmixedh;λ1δ1,λ2δ2≤1+λ11+λ2ωmixedh;δ1,δ2.

In view of (5.4), one has

ϕ(x,y)ht0,s0;x,y≤ωmixedh;t0−x,s0−y ≤1+t0−xδ11+s0−yδ2ωmixedh;δ1,δ2.

Some details on ωmixed can be found [14].

Let h∈Cb(∇), the Lipschitz class Lipα(β,γ) with α>0, (t0,s0),(x,y)∈∇ and β,γ∈0,1 is defined by

Lipα(β,γ)=h∈Cb(∇):ϕ(x,y)ht0 ,s0 ;x,y≤αt0 -xβs0 -yγ.

Now, for h∈Cb(∇) and (t0,s0),(x,y)∈∇, we define the associated GBS type of operators (1.1) as follows:

 Pn(h;x,y)=∑ k=0 n ∑l=0n−k ν n,k,l (x,y)∫ 2kn+1−1 2k+1n+1−1 ∫ 2l n+1 −1 2 l+1 n+1 −1 h(x,s0 )+h(t0 ,y)−h(t0 ,s0 )ds0dt0,

where νn,k,l(x,y)=n+122nkn−kl1+x2k1+y2l1−1+x2−1+y2n−k−l.

Theorem 5.1. For all h∈Cb(∇) and (x,y)∈∇, the operators given by (5.6) verify that

 Pn(h;x,y)−h(x,y)≤4 ωmixedh;2n+1,2n+1.

Proof. From (5.1), it is clear

h(x,y)−ϕ(x,y)ht0,s0;x,y=h(x,s0)+h(t0,y)−h(t0,s0).

Operating Rn(.;x,y) and using the definition of (5.6)

Pn(h;x,y)−h(x,y)=−Rn(ϕ(x,y)ht0,s0;x,y;x,y).

Utilizing the Cauchy-Schwarz inequality to the above equation and in view of (5.5), thus

|Pn(h;x,y)−h(x,y)|≤Rn(ϕ(x,y)ht0,s0;x,y;x,y)≤Rn(r0,0;x,y)+δ1−1Rn(t0 −x)2;x,y+δ2−1Rn(s0−y)2;x,y+1δ1δ2Rn(t0 −x)2;x,yRn(s0 −y)2;x,yωmixedh;δ1,δ2.

Using Lemma 2.1. (i), Lemma 2.2. (v)-(vi) and (5.3), we get

Pn(h;x,y)−h(x,y)≤1+δ1−12n+1+δ2−12n+1+δ1−1δ2−12n+12n+1ωmixedh;δ1,δ2.

Taking δ1=δ2=2n+1, hence we get

 Pn(h;x,y)−h(x,y)≤4 ωmixedh;2n+1,2n+1.

Theorem 5.2. Let h∈Lipα(β,γ). Then, operators (5.6) satisfy the following inequality:

 Pn(h;x,y)−h(x,y)≤α Πn1(x)β2Πn2(y)γ2.

Proof. In view of (4.2) and by (5.6), we get

 Pn(h;x,y)−h(x,y)≤Rnϕ(x,y)ht0,s0;x,y;x,y≤αRnt0−xβs0−yγ;x,y=αRnt0−xβ;x,yRns0−yγ;x,y.

Applying the Hölder's inequality with p1,q1=2β,22−β, p2,q2=2γ,22−γ to the above inequality, thus

 Pn(h;x,y)−h(x,y)≤αRn(t0−x)2;x,yβ2Rnr0,0;x,y2−β2×Rn(s0−y)2;x,yγ2Rnr0,0;x,y2−γ2.

Considering to Lemma 2.1. (i) and Lemma 2.2. (v)-(vi), then

 Pn(h;x,y)−h(x,y)≤α Πn1(x)β2Πn2(y)γ2,

which completes the proof.

6. Graphics and Error Estimation Tables

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In this section, we present graphs (made with Maple) and error estimation tables comparing of the convergence of the operators Rn(h;x,y) and Pn(h;x,y), from (1.1) and (5.6), to some example functions h(x,y).

Example 6.1. In Figure 1, we show the function h(x,y)=x3sin(x−y) in blue, and the approximations Rn(h;x,y) from (1.1), for n =5 and 20, in yellow. In Table Table 1, we give the error of the same approximation operators for n = 5, 25 and 125. It is clear from Table 1 that, as the value of n increases, the absolute error between the operators and the function h(x,y) decreases.

Table 1 . Error of approximation operators Rn(h;x,y) to the function h(x,y)=x3sin(x−y) for n=5,25,125.

(x,y)n=5n=25n=125
(0.095,-0.095)0.0805650.0100060.001126
(0.05,-0.05)0.0778400.0080230.000554
(0.035,-0.035)0.0772890.0076250.000472
(-0.025,0.025)0.0770270.0074370.000424
(-0.01,-0.01)0.0764140.0071960.000363
(0.015,-0.075)0.0757180.0071790.000389
(-0.03,-0.06)0.0751310.0070400.000364
(-0.04,-0.09)0.0748030.0069330.000361
(-0.02,-0.08)0.0747160.0068940.000333
(-0.01,-0.099)0.0744210.0068300.000325


Figure 1. Approximations Rn(h;x,y) of h(x,y)=x3sin(x−y).

Example 6.2. In Figure 2, we show the function h(x,y)=x2−sin(y−x) in blue; the operator Rn(h;x,y) from (1.1), for n = 5 and 25 is shown in red and its associated GBS operators from (5.6) are shown in green. In addition, in Table 2 we show the error of the operator approximations when n=250. It is evident from the table that GBS type operator gives a better approximation.

Table 2 . Error of approximation operators Rn(h;x,y) and Pn(h;x,y) to the function h(x,y)=x2−sin(y−x) for n=250.

(x,y)|R250(h;x,y)−h(x,y)||P250(h;x,y)−h(x,y)|
(0.01,-0.01)0.003735396720.00007801713
(-0.02,0.02)0.004442183520.00015595725
(0.03,-0.04)0.003138383560.00026983045
(-0.03,-0.05)0.003732199450.00007189246
(0.05,-0.05)0.002764565580.00038854967
(0.06,-0.06)0.002517285060.00046541592
(-0.07,-0.05)0.004142466410.00006892313
(-0.07,-0.07)0.003915069740.00000000012
(0.08,-0.08)0.002023803120.00061769752
(0.09,-0.09)0.001775411940.00069296224


Figure 2. Approximations Rn(h;x,y) and Pn(h;x,y) of h(x,y)=x2−sin(y−x).
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