# On the Convergence of a Family of Chlodowsky Type Bernstein-Stancu-Schurer Operators

Authors: Lian-Ta Shu; Guorong Zhou; Qing-Bo Cai
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We construct a new family of univariate Chlodowsky type Bernstein-Stancu-Schurer operators and bivariate tensor product form. We obtain the estimates of moments and central moments of these operators, obtain weighted approximation theorem, establish local approximation theorems by the usual and the second order modulus of continuity, estimate the rate of convergence, give a convergence theorem for the Lipschitz continuous functions, and also obtain a Voronovskaja-type asymptotic formula. For the bivariate case, we give the rate of convergence by using the weighted modulus of continuity. We also give some graphs and numerical examples to illustrate the convergent properties of these operators to certain functions and show that the new ones have a better approximation to functionsf for one dimension.


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## Body

## 1. Introduction

In 1912, Bernstein [1] proposed the famous polynomials called nowadays Bernstein polynomials to prove the Weierstrass approximation theorem. Later it was found that Bernstein polynomials possess many remarkable properties, which made them an area of intensive research. Even recently, there are also many papers mentioned about Bernstein type operators, such as [2–5]. A generalization of Bernstein polynomials based on q-integers was proposed by Lupas in 1987 in [6]. However, the Lupas q-Bernstein operators are rational functions rather than polynomials. The Phillips q-Bernstein polynomials were introduced by Phillips in 1997 in [7]. In 2015, Mursaleen et al. [8] first introduced the (p, q)-analogue of Bernstein operators, in the case p=1 these operators coincide with the Phillips q-Bernstein operators. In the same year, they also proposed the (p, q)-analogue of Bernstein-Stancu operators in [9]. And then, in 2017, Khan et al. introduced the Lupas (p, q)-analogue of Bernstein operators. There are some recent papers relevant to Bernstein operators based on (p, q)-integers, such as [10–12]. Also some other positive operators related to (p, q)-integers; we listed some of them as [13–18].In 1932, Chlodowsky introduced the classical Bernstein-Chlodowsky operators as(1)Bn~f;x=∑k=0nfknbnnkxbnk1-xbnn-k,where 0≤x≤bn and {bn} is a sequence of positive numbers such that limn→∞bn=∞, limn→∞bn/n=0. These operators have been studied extensively, including one- and two-dimensional cases, which may be found in [19–25]. In 2017, Mishra et al. [26] introduced the Chlodowsky variant of (p, q) Bernstein-Stancu-Schurer operators as(2)Cn,mα,βf;x,p,q=1pn+mn+m-1/2∑k=0n+mn+mkp,qpkk-1/2xbnk∏s=0n+m-k-1ps-qsxbn×fpn+m-kkp,q+αnp,q+βbn,where n∈N, m,α,β∈N0, with α/β≈1, 0≤x≤bn, 0<q<p≤1, and bn is an increasing sequence of positive terms with the properties bn→∞ and bn/[n]p,q→0 as n→∞. They discussed Korovkin-type approximation properties and rate of convergence of operators (2).Due to the fact that these operators (2) reproduce only constant functions and it seems that there have been no two-dimensional case of their defined operators (2) at present, the first aim of this paper is to give a new type of these operators such that the new ones preserve not only constant functions but also linear functions; the second aim is to introduce the two-dimensional case based on these operators (3), which will be defined in (69). We also discuss weighted approximation properties of these new operators (3) and (69) and compare with the ones (2) by graphics and the absolute error bound of numerical analysis; we will show that the new ones (3) are better than (2) when approximating to functions f.We introduce new Chlodowsky type (p, q)-Bernstein-Stancu-Schurer operators Cn,p,qα,β,l~(f;x) as(3)Cn,p,qα,β,l~f;x=∑k=0n+lcn,p,qk,luxfpn+l-kkp,q+αnp,q+βbn,and the basis function cn,p,qk,l(x) is defined as(4)cn,p,qk,lx=1pn+ln+l-1/2n+lkp,qpkk-1/2xbnk1-xbnp,qn+l-k,where 0<q<p≤1, u(x)=[n]p,q+βx-αbn/[n+l]p,qαbn/[n]p,q+β≤x≤[n+l]p,q+α/[n]p,q+βbn, 0≤α≤β, and bn is an increasing sequence of positive terms with the properties bn→∞, bn/[n]p,q→0 as n→∞.We mention some definitions based on (p, q)-integers, and details can be found in [27–31]. For any fixed real number p>0 and q>0, the (p, q)-integers [k]p,q are defined by (5)kp,q=pk-1+pk-2q+pk-3q2+⋯+pqk-2+qk-1=pk-qkp-q,p≠q≠1;kpk-1,p=q≠1;kq,p=1;k,p=q=1,where [k]q denotes the q-integers and k=0,1,2,…. Also (p, q)-factorial and (p, q)-binomial coefficients are defined as follows:(6)kp,q!=kp,qk-1p,q⋯1p,q,k=1,2,…,1,k=0,nkp,q=np,q!kp,q!n-kp,q!,n≥k≥0.The (p, q)-Binomial expansion is defined by(7)x+yp,qn=1,n=0,x+ypx+qy⋯pn-1x+qn-1y,n=1,2,….Whenp=1, all the definitions of (p, q)-calculus above are reduced to q-calculus.The paper is organized as follows: In Section2, we give some basic definitions regarding (p, q)-integers. In Section 3, we estimate the moments and central moments of these operators (3). In Section 4, we obtain weighted approximation theorem, establish local approximation theorems, estimate the rate of convergence, give a convergence theorem for the Lipschitz continuous functions, and also obtain a Voronovskaja-type asymptotic formula. In Section 5, we give some graphs and numerical examples to illustrate the convergent properties for one variable functions and compare with the ones in (2). In Section 6, we propose the bivariate case, give the rate of convergence by using the weighted modulus of continuity, and give some graphs and numerical analysis for two variables functions.


## 2. Auxiliary Results

Lemma 1 (see [26], Lemma 2).
Forx∈0,bn, we have(8)Cn,mα,β1;x,p,q=1,Cn,mα,βt;x,p,q=n+mp,qx+αbnnp,q+β,Cn,mα,βt2;x,p,q=qn+mp,qn+m-1p,qnp,q+β2x2+n+mp,q2α+pn+m-1bnnp,q+β2x+α2bn2np,q+β2,Cn,mα,βt-x;x,p,q=n+mp,qnp,q+β-1x+αbnnp,q+β,Cn,mα,βt-x2;x,p,q=1-2n+mp,qnp,q+β+qn+mp,qn+m-1p,qnp,q+β2x2+2α+pn+m-1n+mp,qnp,q+β-2αbnxnp,q+β+α2bn2np,q+β2.Lemma 2.
Forx∈αbn/[n]p,q+β,[n+l]p,q+α/[n]p,q+βbn, the following equalities hold:(9)Cn,p,qα,β,l~1;x=1,(10)Cn,p,qα,β,l~t;x=x,(11)Cn,p,qα,β,l~t2;x=x2-pn+l-1x2n+lp,q+pn+l-1bn2α+n+lp,qxnp,q+βn+lp,q-pn+l-1αbn2α+n+lp,qnp,q+β2n+lp,q,(12)Cn,p,qα,β,l~t-x;x=0,(13)Cn,p,qα,β,l~t-x2;x=pn+l-1bn2α+n+lp,qxnp,q+βn+lp,q-pn+l-1αbn2α+n+lp,qnp,q+β2n+lp,q-pn+l-1x2n+lp,q.Proof.
From Lemma1 and (3), we obtain (9), (10), and (11) by some simple computations, and we also get (12) and (13) by (9)–(11); here we omit it.Remark 3.
From (9) and (10) of Lemma 2, we know the operators Cn,p,qα,β,l~(f;x) preserve not only constant functions but also linear functions. That is to say, Cn,p,qα,β,l~(at+b;x)=ax+b, where fixed a,b∈R.Lemma 4 (see Theorem 2.1 of [32]).
For0<qn<pn≤1, set qn:=1-αn, pn:=1-βn such that 0≤βn<αn<1, αn→0, βn→0 as n→∞. The following statements are true:(A)
Iflimn→∞⁡en(βn-αn)=1 and enβn/n→0, then [n]pn,qn→∞.(B)
Iflim¯n→∞en(βn-αn)<1 and enβn(αn-βn)→0, then [n]pn,qn→∞.(C)
Iflim_n→∞en(βn-αn)<1, lim¯n→∞en(βn-αn)=1 and max{enβn/n,enβn(αn-βn)}→0, then [n]pn,qn→∞.Lemma 5.
Consider the sequencesp={pn} and q={qn}, for 0<qn<pn≤1 satisfying the conditions (A), (B), or (C) of Lemma 4 and limn→∞pnn=a, a∈(0,1]; then for fixed x∈αbn/[n]p,q+β,[n+l]p,q+α/[n]p,q+βbn, the following equalities hold:(14)Cn,p,qα,β,l~t3;x=x3+Obnnp,qφ1x,(15)Cn,p,qα,β,l~t4;x=x4+Obnnp,qφ2x,(16)limn→∞⁡npn,qnbnCn,pn,qnα,β,l~t-x2;x=ax,(17)limn→∞npn,qn2bn2Cn,pn,qnα,β,l~t-x4;x=3a2x2,where φ1(x) and φ2(x) are functions depending on x.Proof.
From (3), we have(18)Cn,p,qα,β,l~t3;x=∑k=0n+lcn,p,qk,lxpn+l-kkp,q+αnp,q+βbn3=∑k=0n+lcn,p,qk,lxp3n+3l-3kkp,q3bn3np,q+β3+3p2n+2l-2kkp,q2αbn3np,q+β3+3pn+l-kkp,qα2bn3np,q+β3+α3bn3np,q+β3,≔∆1+∆2+∆3+∆4,and by some computations, we have(19)kp,q2=pk-1kp,q+qkp,qk-1p,q,(20)kp,q3=q3kp,qk-1p,qk-2p,q+pk-2q2p+qkp,qk-1p,q+p2k-2kp,q;thus, using (4), (19), (20), and ∑k=0n+l‍cn,p,qk,l(x)=1, we have(21)∆1=∑k=0n+lcn,p,qk,lxp3n+3l-3kkp,q3bn3np,q+β3=q3n+lp,qn+l-1p,qn+l-2p,qnp,q+β3ux3+pn+l-2q2p+qbnn+lp,qn+l-1p,qnp,q+β3ux2+p2n+2l-2bn2n+lp,qnp,q+β3ux,and, similarly, we have(22)∆2=∑k=0n+lcn,p,qk,lx3p2n+2l-2kkp,q2αbn3np,q+β3=3qαbnn+lp,qn+l-1p,qnp,q+β3ux2+3pn+l-1αbn2n+lp,qnp,q+β3ux,(23)∆3=∑k=0n+lcn,p,qk,lx3pn+l-kkp,qα2bn3np,q+β3=3α2bn2n+lp,qnp,q+β3ux,(24)∆4=∑k=0n+lcn,p,qk,lxα3bn3np,q+β3=α3bn3np,q+β3.Since u(x)=[n]p,q+βx-αbn/[n+l]p,q, from (21), we get(25)q3n+lp,qn+l-1p,qn+l-2p,qnp,q+β3ux3=q3n+l-1p,qn+l-2p,qn+lp,q2x3-3αbnq3n+l-1p,qn+l-2p,qnp,q+βn+lp,q2x2+3α2bn2q3n+l-1p,qn+l-2p,qnp,q+β2n+lp,q2x-α3bn3q3n+l-1p,qn+l-2p,qnp,q+β3n+lp,q2,due to(26)q3n+l-1p,qn+l-2p,qn+lp,q2x3=n+lp,q-pn+l-1n+lp,q-2p,qpn+l-2n+lp,q2x3,and, combining (18), (21)-(24), u(x)=[n]p,q+βx-αbn/[n+l]p,q, (25), and (26), we obtain(27)Cn,p,qα,β,l~t3;x=x3+Obnnp,qφ1x.Next, since (28)kp,q4=q6kp,qk-1p,qk-2p,qk-3p,q+pk-3q32p,q2+2p2kp,qk-1p,qk-2p,q+p2k-4q2p2p,q+3p,qkp,qk-1p,q+p3k-3kp,q,using the same methods and by some computations, we get(29)
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## 3. Approximation Properties

In the sequel, letI=αbn/[n]p,q+β,[n+l]p,q+α/[n]p,q+βbn, where 0≤α≤β and bn is an increasing sequence of positive terms with limn→∞bn=∞ and limn→∞bn/[n]p,q=0. In order to obtain the weighted approximation Theorem 6, let p={pn}, q={qn} be sequences satisfying the conditions (A), (B), or (C) of Lemma 4 and limn→∞pnn=a, a∈(0,1].LetBx2(I) be the set of all functions f defined on I satisfying the condition |f(x)|≤Mf(1+x2), where Mf is the constant depending only on f. We denote the subspace of all continuous functions belonging to Bx2(I) by Cx2(I). Let Cx2∗(I) be the subspace of all functions f∈Cx2(I), for which limx→∞f(x)/1+x2 is finite. The norm on Cx2∗(I) is(32)fx2=supx∈I⁡fx1+x2.Firstly, we discuss the weighted approximation theorem.Theorem 6.
Forf∈Cx2∗(I), we have(33)limn→∞⁡Cn,p,qα,β,l~f-fx2=0.Proof.
By using the Korovkin theorem, we see that it is sufficient to verify the following three conditions:(34)limn→∞⁡Cn,pn,qnα,β,l~ti;x-xix2=0,i=0,1,2.Since Cn,pn,qnα,β,l~(1;x)=1 and Cn,pn,qnα,β,l~(t;x)=x, equality (34) holds true for i=0 and i=1. Finally, for i=2, from Lemma 2, we have(35)Cn,pn,qnα,β,l~t2;x-x2x2=supx∈I⁡Cn,pn,qnα,β,l~t2;x-x21+x2≤pnn+l-1bn2α+n+lpn,qnnpn,qn+βn+lpn,qnsupx∈I⁡x1+x2+pnn+l-1αbn2α+n+lpn,qnnpn,qn+β2n+lpn,qnsupx∈I⁡11+x2+pnn+l-1n+lpn,qnsupx∈I⁡x21+x2≤pnn+l-1bn2α+n+lpn,qnnpn,qn+βn+lpn,qn+pnn+l-1αbn2α+n+lpn,qnnpn,qn+β2n+lpn,qn+pnn+l-1n+lpn,qn.We can obtain limn→∞Cn,pn,qnα,β,l~(t2;x)-x2x2=0 by using Lemma 4. Theorem 6 is proved.We give the following definitions: The space of all real valued continuous bounded functionsf defined on the interval I is denoted by CB(I). The norm on CB(I) is defined by f=sup{|f(x)|:x∈I}. The Peetre’s K-functional is given by(36)K2f;δ=infg∈W2f-g+δg′′,where δ>0 and W2={g∈CB[0,∞):g′,g′′∈CB[0,∞)}. For f∈CB[0,∞), the usual modulus of continuity and the second order modulus of smoothness are defined as follows:(37)ωf;δ=sup0<h≤δ⁡supx∈0,∞⁡fx+h-fx,(38)ω2f;δ=sup0<h≤δ⁡supx∈0,∞⁡fx+2h-2fx+h+fx.By [33], there exists a constant C>0, such that(39)K2f;δ≤Cω2f;δ.Now, we establish local approximation theorems as follows.Theorem 7.
Forf∈CB(I), we have(40)Cn,p,qα,β,l~f;x-fx≤Cω2f;pn+l-1bn2α+n+lp,qx2np,q+βn+lp,q,where C is a positive constant.Proof.
Letg∈W2; by Taylor’s expansion, we have(41)gt=gx+g′xt-x+∫xtt-ug′′udu,and, applying Cn,p,qα,β,l~ to (41), using (9) and (12), we get(42)Cn,p,qα,β,l~g;x-gx=Cn,p,qα,β,l~∫xtt-ug′′udu;x.Thus, from (13), we have(43)Cn,p,qα,β,l~g;x-gx=Cn,p,qα,β,l~∫xtt-ug′′udu;x≤Cn,p,qα,β,l~∫xtt-ug′′udu;x≤Cn,p,qα,β,l~t-x2;xg′′≤pn+l-1bn2α+n+lp,qxnp,q+βn+lp,qg′′.On the other hand, by (3) and (9), we have(44)Cn,p,qα,β,l~f;x≤∑k=0n+lcn,p,qk,luxfpn+l-kkp,q+αnp,q+βbn≤f.Now (43) and (44) imply(45)Cn,p,qα,β,l~f;x-fx≤Cn,p,qα,β,l~f-g;x-f-gx+Cn,p,qα,β,l~g;x-gx≤2f-g+pn+l-1bn2α+n+lp,qxnp,q+βn+lp,qg′′,and, from (36), taking infimum on the right hand side over all g∈W2, we obtain(46)Cn,p,qα,β,l~f;x-fx≤2K2f;pn+l-1bn2α+n+lp,qx2np,q+βn+lp,q.Finally, using (39), we get(47)Cn,p,qα,β,l~f;x-fx≤Cω2f;pn+l-1bn2α+n+lp,qx2np,q+βn+lp,q.Theorem 7 is proved.Theorem 8.
Forf∈CB(I) and δ>0, we have(48)Cn,p,qα,β,l~f;x-fx≤1+1δpn+l-1bn2α+n+lp,qxnp,q+βn+lp,qωf;δ.Proof.
Since(49)ft-fx≤ωf;t-x≤1+t-xδωf;δ,applying Cn,p,qα,β,l~(f;x) to (49), we obtain(50)Cn,p,qα,β,l~f;x-fx≤Cn,p,qα,β,l~ft-fx;x≤1+1δCn,p,qα,β,l~t-x;xωf;δ,using Cauchy-Schwartz inequality, we have(51)Cn,p,qα,β,l~f;x-fx≤1+1δCn,p,qα,β,l~t-x2;xωf;δ≤1+1δpn+l-1bn2α+n+lp,qxnp,q+βn+lp,qωf;δ.Theorem 8 is proved.Corollary 9.
From Theorem8, applied to δ=pn+l-1bn2α+[n+l]p,qx/[n]p,q+β[n+l]p,q, we have(52)Cn,p,qα,β,l~f;x-fx≤2ωf;pn+l-1bn2α+n+lp,qxnp,q+βn+lp,q.Remark 10.
For any fixedx∈I, we have limn→∞pnn+l-1bn(2α+[n+l]pn,qn)x/(npn,qn+β)[n+l]pn,qn=0, and this gives us a rate of pointwise convergence of the operators Cn,p,qα,β,l~(f;x) to f(x).Theorem 11.
Iff is differentiable on I and f′∈CB(I), then for δ>0, we have(53)Cn,p,qα,β,l~f;x-fx≤pn+l-1bn2α+n+lp,qxnp,q+βn+lp,q1+1δpn+l-1bn2α+n+lp,qxnp,q+βn+lp,qωf′;δ.Proof.
Sincef(t)=f(x)+f′(x)(t-x)+f(t)-f(x)-f′(x)(t-x) and we have (12), we can write(54)Cn,p,qα,β,l~f;x-fx≤Cn,p,qα,β,l~ft-fx-f′xt-x;x,and, by the mean value theorem, we obtain (55)ft-fx-f′xt-x≤t-x1+t-xδωf′;δ,so, by the Cauchy-Schwartz inequality, we get(56)Cn,p,qα,β,l~ft-fx-f′xt-x;x≤Cn,p,qα,β,l~t-x;x+1δCn,p,qα,β,l~t-x2;xωf′;δ≤Cn,p,qα,β,l~t-x2;x+1δCn,p,qα,β,l~t-x2;xωf′;δ≤Cn,p,qα,β,l~t-x2;x1+1δCn,p,qα,β,l~t-x2;xωf′;δ,and then we have the desired result by (13).Corollary 12.
From Theorem11, let f∈CB(I), applied to δ=pn+l-1bn2α+[n+l]p,qx/[n]p,q+β[n+l]p,q; then we have(57)Cn,p,qα,β,l~f;x-fx≤2pn+l-1bn2α+n+lp,qxnp,q+βn+lp,qωf′;pn+l-1bn2α+n+lp,qxnp,q+βn+lp,q.Next, we study the rate of convergence of the operatorsCn,p,qα,β,l~(f;x) with the help of functions of Lipschitz class LipM(μ), where M>0 and 0<μ≤1. A function f belongs to LipM(μ) if (58)fy-fx≤My-xμx,y∈R.We have the following theorem.Theorem 13.
Letf∈LipM(μ), 0<μ≤1; we have(59)Cn,p,qα,β,l~f;x-fx≤Mpn+l-1bn2α+n+lp,qxnp,q+βn+lp,qμ/2.Proof.
Obviously,Cn,p,qα,β,l~(f;x) are linear positive operators; since f∈LipM(μ)  (0<μ≤1), we have (60)Cn,p,qα,β,l~f;x-fx≤Cn,p,qα,β,l~ft-fx;x=∑k=0n+lcn,p,qk,luxfpn+l-kkp,q+αnp,q+βbn-fx≤M∑k=0n+lcn,p,qk,luxpn+l-kkp,q+αnp,q+βbn-xμ≤M∑k=0n+lcn,p,qk,luxpn+l-kkp,q+αnp,q+βbn-x2μ/2cn,p,qk,lux2-μ/2.Applying Hölder’s inequality for sums, we obtain(61)Cn,p,qα,β,l~f;x-fx≤M∑k=0n+lcn,p,qk,luxpn+l-kkp,q+αnp,q+βbn-x2μ/2∑k=0n+lcn,p,qk,lux2-μ/2=MCn,p,qα,β,l~t-x2;xμ/2≤Mpn+l-1bn2α+n+lp,qxnp,q+βn+lp,qμ/2.Theorem 13 is proved.Now, we give a Voronovskaja-type asymptotic formula forCn,p,qα,β,l~(f;x).Theorem 14.
Forx∈I, we have the following conclusion:(62)limn→∞⁡npn,qnbnCn,pn,qnα,β,l~f;x-fx=f′′x2ax.Proof.
Letx∈I be fixed. By the Taylor formula, we may write(63)ft=fx+f′xt-x+12f′′xt-x2+rt;xt-x2,where r(t;x) is the Peano form of the remainder, r(t;x)∈C(I); using L’Hospital’s rule, we have(64)limt→x⁡rt;x=limt→x⁡ft-fx-f′xt-x-1/2f′′xt-x2t-x2=limt→x⁡f′t-f′x-f′′xt-x2t-x=limt→x⁡f′′t-f′′x2=0.Since we have (12), applying Cn,p,qα,β,l~(f;x) to (63), we obtain(65)np,qbnCn,p,qα,β,l~f;x-fx=f′′x2np,qbnCn,p,qα,β,l~t-x2;x+np,qbnCn,p,qα,β,l~rt;xt-x2;x.By the Cauchy-Schwarz inequality, we have(66)Cn,p,qα,β,l~rt;xt-x2;x≤Cn,p,qα,β,l~r2t;x;xCn,p,qα,β,l~t-x4;x.From r2(x;x)=0 and (17), we get limn→∞[n]pn,qn/bnCn,pn,qnα,β,l~r(t;x)(t-x)2;x=0. Hence, from (16), we have(67)limn→∞⁡npn,qnbnCn,pn,qnα,β,l~f;x-fx=limn→∞⁡f′′x2npn,qnbnCn,pn,qnα,β,l~t-x2;x=f′′x2ax.Theorem 14 is proved.Remark 15.
For the functionf(t)=t2, from Theorem 14, we have the limit equality(68)limn→∞⁡npn,qnbnCn,pn,qnα,β,l~t2;x-x2=x2′′2ax=ax,which is the corresponding result in (16).


## 4. Graphical and Numerical Analysis I

In this section, we give several graphs and numerical examples to show the convergence ofCn,p,qα,β,l~(f;x) to f(x) with different values of parameters which satisfy the conclusions of Lemmas 4 and 5.Example 16.
Letf(x)=x2; the graphs of Cn,p,qα,β,l~(f;x) with n=100, p=0.99999, α=1, β=2, l=5, bn=5 and different values of q are shown in Figure 1. The graphs of Cn,p,qα,β,l~(f;x) with p=0.99999, q=0.99, α=1, β=2, l=5, bn=10 and different values of n are shown in Figure 2. The graphs of Cn,p,qα,β,l~(f;x) with n=100, α=1, β=2, l=5, bn=10 and different values of p and q are shown in Figure 3. Moreover, we give a comparison on the approximation of Cn,p,qα,β,l~(f;x) (the red one) and Cn,m(α,β)(f;x,p,q) (the yellow one) in Figure 4. In Tables 1 and 2, we show the absolute error bound of the approximation of Cn,p,qα,β,l~(f;x) with α=1, β=2, l=5, bn=10 and different values of n, p, and q to f(x).Table 1
The absolute error bound of the approximation ofCn,p,qα,β,l~(f;x) with α=1,β=2,l=5,bn=10,p=0.999999 and different values of n and q.
q f x - C n , p , q α , β , l ~ f ; x ∞ n = 30 n = 50 n = 80 n = 100 0.99 0.215524 0.169329 0.132302 0.118517 0.999 0.202854 0.146061 0.101294 0.084213 0.9999 0.201487 0.143719 0.098331 0.081025 0.99999 0.201349 0.143485 0.098037 0.080710Table 2
The absolute error bound of the approximation ofCn,p,qα,β,l~(f;x) with α=1,β=2,l=5,bn=10 and different values of n, p, and q, where p=1-1/10m,q=1-1/10m-1.
m ∥ f ( x ) - C n , p , q α , β , l ~ ( f ; x ) ∥ ∞ n=30 n=50 n=80 n=100 6 0.20134927 0.14348509 0.09803662 0.08070974 7 0.20134028 0.14346631 0.09801166 0.08068261 8 0.20133938 0.14346444 0.09800917 0.08067989 9 0.20133930 0.14346425 0.09800892 0.08067962 10 0.20133929 0.14346423 0.09800889 0.08067960Figure 1
The figures ofCn,p,qα,β,l~(f;x) with n=100, p=0.99999, α=1, β=2, l=5, bn=5 and different values of q.Figure 2
The figures ofCn,p,qα,β,l~(f;x) with p=0.99999, q=0.99, α=1, β=2, l=5, bn=10 and different values of n.Figure 3
The figures ofCn,p,qα,β,l~(f;x) with n=100,α=1,β=2,l=5,bn=10 and different values of p and q.Figure 4
The figures ofCn,p,qα,β,l~(f;x) (the red one) and Cn,m(α,β)(f;x,p,q) (the yellow one) with n=100,bn=10,α=β=1,m=l=5,p=0.9999,q=0.999.


## 5. Construction of Bivariate Operators and Weighted Approximation Properties

We introduce the bivariate tensor product (p, q)-analogue of Chlodowsky type Bernstein-Stancu-Schurer operators as follows:(69)Cpn1,pn2,qn1,qn2n1,n2,α,β,l~f;x,y=∑k1=0n1+l∑k2=0n2+lcn1,pn1,qn1k,luxcn2,pn2,qn2k,lvy×fpn1n1+l-k1k1pn1,qn1+αn1pn1,qn1+βbn1,pn2n2+l-k2k2pn2,qn2+αn2pn2,qn2+βbn2,where(70)ux=n1pn1,qn1+βx-αbn1n1+lpn1,qn1,αbn1n1pn1,qn1+β≤x≤n1+lpn1,qn1+αn1pn1,qn1+βbn1,(71)vy=n2pn2,qn2+βy-αbn2n2+lpn2,qn2,αbn2n2pn2,qn2+β≤y≤n2+lpn2,qn2+αn2pn2,qn2+βbn2,(72)cn,p,qk,lx=1pn+ln+l-1/2n+lkp,qpkk-1/2xbnk1-xbnp,qn+l-k,0<qn1,qn2<pn1,pn2≤1, 0≤α≤β, and {bn1}, {bn2} are increasing sequences of positive terms with bn1,bn2→∞, bn1/[n1]pn1,qn1,bn2/[n2]pn2,qn2→0 as n1,n2→∞.Lemma 17.
Letei,j(x,y)=xiyj,i,j∈N, (x,y)∈I×I be the two-dimensional test functions; using Lemma 1, we easily obtain the following equalities:(73)Cpn1,pn2,qn1,qn2n1,n2,α,β,l~e0,0;x,y=1,(74)Cpn1,pn2,qn1,qn2n1,n2,α,β,l~e1,0;x,y=x,(75)Cpn1,pn2,qn1,qn2n1,n2,α,β,l~e0,1;x,y=y,(76)Cpn1,pn2,qn1,qn2n1,n2,α,β,l~e1,1;x,y=xy,(77)Cpn1,pn2,qn1,qn2n1,n2,α,β,l~e2,0;x,y=x2-pn1n1+l-1x2n1+lpn1,qn1+pn1n1+l-1bn12α+n1+lpn1,qn1xn1pn1,qn1+βn1+lpn1,qn1-pn1n1+l-1αbn12α+n1+lpn1,qn1n1pn1,qn1+β2n1+lpn1,qn1,(78)Cpn1,pn2,qn1,qn2n1,n2,α,β,l~e0,2;x,y=y2-pn2n2+l-1y2n2+lpn2,qn2+pn2n2+l-1bn22α+n2+lpn2,qn2yn2pn2,qn2+βn2+lpn2,qn2-pn2n2+l-1αbn22α+n2+lpn2,qn2n2pn2,qn2+β2n2+lpn2,qn2.Lemma 18.
Using Lemma17 and (16), (17) of Lemma 5, we have the following statements: (79)Cpn1,pn2,qn1,qn2n1,n2,α,β,l~t-x;x,y=0,Cpn1,pn2,qn1,qn2n1,n2,α,β,l~s-y;x,y=0,Cpn1,pn2,qn1,qn2n1,n2,α,β,l~t-x2;x,y=Obn1n1pn1,qn1x,Cpn1,pn2,qn1,qn2n1,n2,α,β,l~s-y2;x,y=Obn2n2pn2,qn2y,Cpn1,pn2,qn1,qn2n1,n2,α,β,l~t-x4;x,y=Obn12n1pn1,qn12x2,Cpn1,pn2,qn1,qn2n1,n2,α,β,l~s-y4;x,y=Obn22n2pn2,qn22y2.LetBρ be the space of all functions f defined on I×I satisfying the condition |f(x)|≤Mfρ(x,y), where Mf is a positive constant depending only on f and ρ(x,y)=1+x2+y2 is a weighted function. We denote the subspace of all continuous functions belonging to Bρ by Cρ. Let Cρ∗ be the subspace of all functions f∈Cρ, for which limx2+y2→∞f(x,y)/ρ(x,y) is finite. The norm on Cρ∗ is fρ=supx,y∈I|f(x,y)|/ρ(x,y). For the infinite interval I, f∈Cρ∗, and δ1,δ2>0, Ilarslan and Acar [34] introduced the weighted modulus of continuity as(80)Ωρf;δ1,δ2=supx,y∈I⁡sup0≤k1≤δ1,0≤k2≤δ2⁡fx+k1,y+k2-fx,yρx,yρk1,k2,which satisfies the following inequality:(81)Ωρf;d1δ1,d2δ2≤41+d11+d21+δ121+δ22Ωρf;δ1,δ2,d1,d2>0.From the definition of Ωρ, we have(82)ft,s-fx,y≤ρx,yρt-x,s-yΩρf;t-x,s-y≤1+x2+y21+t-x21+s-y2Ωρf;t-x,s-y.Now, we establish the degree approximation of operators Cpn1,pn2,qn1,qn2n1,n2~ in the weighted space Cρ∗ by the weighted modulus of continuity Ωρ.Theorem 19.
Forf∈Cρ∗, we have the following inequality:(83)supx,y∈I⁡Cpn1,pn2,qn1,qn2n1,n2,α,β,l~f;x,y-fx,yρx,y3≤CΩρf;bn1n1pn1,qn1,bn2n2pn2,qn2,where C is a positive constant.Proof.
From (81) and (82), for δn1,δn2>0, we get(84)ft,s-fx,y=41+x2+y21+t-x21+s-y21+t-xδn11+s-yδn21+δn12×1+δn22Ωρf;δn1,δn2=41+x2+y21+δn121+δn221+t-xδn1+t-x2+t-xδn1t-x2×1+s-yδn2+s-y2+s-yδn2s-y2Ωρf;δn1,δn2,and, applying the operators Cpn1,pn2,qn1,qn2n1,n2~ on the above inequality, we have(85)
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n
2
.Let δn1=bn1/[n1]pn1,qn1 and δn2=bn2/[n2]pn2,qn2; we have(89)Cpn1,pn2,qn1,qn2n1,n2,α,β,l~f;x,y-fx,y≤41+x2+y21+bn1n1pn1,qn11+bn2n2pn2,qn2C1+x2+y22×Ωρf;bn1n1pn1,qn1,bn2n2pn2,qn2,where C,C1,C2,C3,C4 are positive constants. Theorem 19 is proved.


## 6. Graphical and Numerical Analysis II

In this section, we give several graphs and numerical examples to show the convergence ofCpn1,pn2,qn1,qn2n1,n2,α,β,l~(f;x,y) to f(x,y) with different values of parameters which satisfy the conclusions of Lemmas 4 and 5.Example 20.
Letf(x,y)=x2y2; the graphs of Cpn1,pn2,qn1,qn2n1,n2,α,β,l~(f;x,y) with n1=n2=50,pn1=pn2=0.999,qn1=qn2=0.95,α=1,β=2,l=5,bn1=bn2=5, and f(x,y) are shown in Figure 5. The graphs of Cpn1,pn2,qn1,qn2n1,n2,α,β,l~(f;x,y) with n1=n2=100,pn1=pn2=0.99999,qn1=qn2=0.999,α=1,β=2,l=5,bn1=bn2=10, and f(x,y) are shown in Figure 6. In Table 3, we show the absolute error bound of the approximation of Cpn1,pn2,qn1,qn2n1,n2,α,β,l~(f;x,y) with α=1,β=2,l=5,bn1=bn2=10 and different values of n1=n2=n,pn1=pn2=p,qn1=qn2=q, where p=1-1/10m and q=1-1/10m-1(m=6,7,8,9,10), to f(x,y).Table 3
The absolute error bound of the approximation ofCpn1,pn2,qn1,qn2n1,n2,α,β,l~(f;x,y) with α=1,β=2,l=5,bn1=bn2=10 and different values of n1=n2=n,pn1=pn2=p,qn1=qn2=q, where p=1-1/10m,q=1-1/10m-1.
m f x , y - C p n 1 , p n 2 , q n 1 , q n 2 n 1 , n 2 , α , β , l ~ f ; x , y ∞ n=30 n=50 n=80 n=120 6 0.44324007 0.30755815 0.20568442 0.14177371 7 0.44321848 0.30751521 0.20562961 0.14171254 8 0.44321632 0.30751092 0.20562413 0.14170642 9 0.44321610 0.30751049 0.20562358 0.14170581 10 0.44321608 0.30751045 0.20562352 0.14170575Figure 5
The figures of (the upper one)Cpn1,pn2,qn1,qn2n1,n2,α,β,l~(f;x,y) with n1=n2=50,pn1=pn2=0.999,qn1=qn2=0.95,α=1,β=2,l=5,bn1=bn2=5 and f(x,y) (the below one).Figure 6
The figures of (the upper one)Cpn1,pn2,qn1,qn2n1,n2,α,β,l~(f;x,y) with n1=n2=100,pn1=pn2=0.99999,qn1=qn2=0.999,α=1,β=2,l=5,bn1=bn2=10 and f(x,y) (the below one).



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