# Bivariate Chlodowsky-Stancu Variant of (p,q)-Bernstein-Schurer Operators

Authors: Tuba Vedi-Dilek; Eser Gemikonakli
Journal: Journal of Function Spaces
 (2019)

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## Abstract

In this study, it is proposed to define bivariate Chlodowsky variant of(p,q)-Bernstein-Stancu-Schurer operators. Therefore, Korovkin-type approximation theorems and the error of approximation by using full modulus of continuity are presented. Beside this, we introduce a generalization of the bivariate Chlodowsky variant of (p,q)-Bernstein-Stancu-Schurer operators and investigate its approximation in more general weighted space. Moreover, the numerical results are discussed in order to validate the accuracy of the bivariate Chlodowsky variant of (p,q)-Bernstein-Schurer operators.


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## Body

## 1. Introduction

In this area, many operators aboutq-integer and (p,q)-integer [1–25] were studied. It may not be possible to find the exact solution for the models developed for the engineering fields because of its mathematical intractability. Therefore, these operators can be used effectively to find an algorithm for approximating solutions [26].In 2014, Vedi and Ozarslan introduced the Chlodowsky variant ofq-Bernstein-Schurer-Stancu operators in [27] as (1)Cn,pα,βf;q;x=∑k=0n+pfkq+αnq+βbnn+pkqxbnk∏s=0n+p-k-11-qsxbn,where bn is increasing sequence of real numbers satisfying limn→∞⁡bn, n∈N, p∈N0≔0∪N, 0≤α≤β, 0≤x≤bn and 0<q<1 and investigated its approximation properties.In 2017, Vedi and Ozarslan defined the two dimensional Chlodowsky variant ofq-Bernstein-Schurer-Stancu operators in [28] by(2)Cn,m,pα,βf;qn,qm;x,y≔∑k=0n+p∑j=0m+pfkqn+αnqn+βan,jqm+αmqm+βbmΦk,n,qnxanΦj,m,qmybmwhere an and bm are increasing sequences of real numbers satisfying limn→∞⁡an=limm→∞⁡bm=∞ and n∈N, p∈N0≔0∪N, 0≤α≤β, Φk,n,qn(z)=n+pkqnzk∏s=0n+p-k-11-qnsz and investigated its approximation properties on the rectangular unbounded domain.Moreover, Gemikonakli and Vedi-Dilek [29] introduced the Chlodowsky variant of Bernstein-Schurer operators based on p,q-integers as (3)C-n,sf;p,q;x≔∑k=0n+sfpn-kkp,qnp,qbnpkk-1/2-n+s-1n+s/2n+skp,qxbnk∏j=0n+s-k-1pj-qjxbn,where , s∈N, 0≤x≤bn and 0<q<p≤1.Let us discuss some well-known basic definitions of(p,q)-calculus. For 0<q≤p<1, the p,q-numbers are given as [30] (4)kp,q=pk-qkp-q.For each k∈N0 the p,q-factorial is represented by (5)kp,q!=kk-1…1,k=1,2,3,…,1,k=0and p,q-binomial coefficients are defined as(6)nkp,q=np,q!n-kp,q!kp,q!,where n≥k≥0.Then, Chlodowsky variant of(p,q)-Bernstein-Stancu-Schurer operators was constructed in [31] as(7)Cn,mα,βf;x,p,q≔1pn+mn+m-1/2∑k=0n+mn+pkp,qpkk-1/2xbnk×∏s=0n+m-k-1ps-qsxbnfpn+m-kkp,q+αnp,q+βbn,where n∈N, α, β∈N0 with α/β≈1, 0≤x≤bn, 0<q<p≤1 and bn has the same properties as the operator Cn,pα,βf;q;x.Lemma 1.
LetCn,m(α,β)f;x,p,q be defined in [31].
Then the first few moments of the operators are(8)iCn,mα,β1;x,p,q=1,iiCn,mα,βt;x,p,q=n+mp,qx+αbnnp,q+β,iiiCn,mα,βt2;x,p,q=qn+mp,qn+m-1p,qx2np,q+β2+n+mp,q2α+pn+m-1bnx+α2bn2np,q+β2.The bivariate Chlodowsky variant of Bernstein-Stancu-Schurer operators based on(p,q)-integers is defined and then the first few moments of the operator are provided in Section 2. Next, in Section 3, some Korovkin-type theorems are studied. And following this, the order of convergence of the bivariate Chlodowsky variant of Bernstein-Stancu-Schurer operators based on (p,q)-integers by means of the first modulus of continuity is obtained in Section 4. Moreover, in Section 5, we study the generalization of the bivariate Chlodowsky variant of Bernstein-Stancu-Schurer operators based on (p,q)-integers and seek its approximation properties in more general weighted space. Finally, in Section 6, numerical results for the operators constructed are provided in detail.


## 2. Construction of the Operators

LetDan,bm denote(9)Dan,bm=x,y:0≤x≤an,0≤y≤bmand an and bm are increasing sequences of real numbers satisfying(10)limn→∞⁡an=limm→∞⁡bm=∞,limn→∞⁡annpn,qn=limm→∞⁡bmmpm,qm=0.For x,y∈Dan,bm, we construct the bivariate Chlodowsky variant of Bernstein-Stancu-Schurer operators based on (p,q)-integers as(11)C-n,m,sα,βf;pn,qn,pm,qm;x,y≔1pnn+sn+s-1/21pmm+sm+s-1/2∑k=0n+s∑l=0m+sfpnn+s-kkpn,qn+αnpn,qn+βan,pmm+s-llpm,qm+αmpm,qm+βbm×Φkn,pn,qnxanΦkm,pm,qmybm,where Φkn,pn,qn(z)=n+skpn,qnpnll-1/2z/ank∏j=0n+s-k-1pnj-qnjz/an, 0<qn,qm<pn,pm≤1, 0≤α≤β and n,m∈N, s∈N0≔0∪N.Lemma 2.
LetC-n,m,sα,βf;pn,qn,pm,qm;x,y be given in (11).
Then, the first few moments of the operators are(12)iC-n,m,sα,β1;pn,qn,pm,qm;x,y=1,iiC-n,m,sα,βt1;pn,qn,pm,qm;x,y=n+spn,qnx+αannpn,qn+β,iiiC-n,m,sα,βt2;pn,qn,pm,qm;x,y=m+spm,qmy+αbmmpm,qm+β,ivC-n,m,sα,βt12+t22;pn,qn,pm,qm;x,y=qnn+s-1pn,qnn+spn,qnx2+n+spn,qn2α+pnn+s-1anx+α2an2npn,qn+β2+qmm+s-1pm,qmm+spm,qmy2+m+spm,qm2α+pmm+s-1bmy+α2bm2mpm,qm+β2.


## 3. Korovkin-Type Approximation Theorems

In this section, Korovkin-type approximation theorems are given for the bivariate Chlodowsky variant of Bernstein–Stancu-Schurer operators based on(p,q)-integers. For fixed ν≥0 consider the space Cρν, which consists of all continuous functions f satisfying the condition(13)fx,y≤Mfρνx,y,x,y∈0,∞×0,∞≔R+2,ρx,y=1+x2+y2.Obviously, Cρν is a linear normed space with the following norm:(14)fρν=sup0≤x,y<∞⁡fx,yρνx,y.Theorem 3.
Let the numbersA and B be any fixed positive real numbers.
LetDA,B=x,y:0≤x≤A,0≤y≤B with 0<qn,qm<pn,pm<1,limn→∞⁡qn=limm→∞⁡qm=limn→∞⁡pn=limm→∞⁡pm=1 and an and bm be increasing sequences of positive real numbers that satisfy the following properties: (15)limn→∞⁡an=limm→∞⁡bm=∞,limn→∞⁡annpn,qn=limm→∞⁡bmmpm,qm=0.For all f∈C(DA,B), we obtain (16)limn,m→∞⁡maxx,y∈DA,B⁡C-n,m,sα,βf;pn,qn,pm,qm;x,y-fx,y=0.Proof.
Using Lemma2, we have(17)C-n,m,sα,β1;pn,qn,pm,qm;·,·-1CDA,B=0C-n,m,sα,βt1;pn,qn,pm,qm;·,·-xCDA,B≤An+spn,qnnpn,qn+β-1+αannpn,qn+βC-n,m,sα,βt2;pn,qn,pm,qm;·,·-yCDA,B≤Bm+spm,qmmpm,qm+β-1+αbmmpm,qm+β.And again using Lemma 2 we get(18)C-n,m,sα,βt12+t22;pn,qn,pm,qm;·,·-x2+y2=qnn+s-1pn,qnn+spn,qnnpn,qn+β2x2+n+spn,qn2α+pnn+s-1npn,qn+β2anx+α2an2npn,qn+β2+qmm+s-1pm,qmm+spm,qmmpm,qm+β2y2+m+spm,qm2α+pmm+s-1mpm,qm+β2bmy+α2bm2mpm,qm+β2.Finally, from the above equality, we obtain (19)C-n,m,sα,βt12+t22;pn,qn,pm,qm;·,·-x2+y2CDA,B≤qnn+s-1pn,qnn+spn,qnnpn,qn+β2A2+n+spn,qn2α+pnn+s-1npn,qn+β2anA+α2an2npn,qn+β2+qmm+s-1pm,qmm+spm,qmmpm,qm+β2B2+m+spm,qm2α+pmm+s-1mpm,qm+β2bmB+α2bm2mpm,qm+β2.
Therefore, from the hypothesis of the theorem, we get(20)C-n,m,sα,βt1;pn,qn,pm,qm;·,·-xCDA,B→0C-n,m,sα,βt2;pn,qn,pm,qm;·,·-yCDA,B→0C-n,m,sα,βt12+t22;pn,qn,pm,qm;·,·-x2+y2CDA,B→0when n and m→∞.
Hence, the proof is completed by the two dimensional Korovkin theorem.Theorem 4 (see [13]).
There exists a sequence of positive operatorsTn,m, acting from CρR+2 to CρR+2, satisfying the conditions (21)limn,m→∞⁡Tn,m1;·,·-1ρ=0limn,m→∞⁡Tn,mt1;·,·-xρ=0limn,m→∞⁡Tn,mt2;·,·-yρ=0limn,m→∞⁡Tn,mt12+t22;·,·-x2+y2ρ=0and there exists a function f∗∈Cρ for which (22)limn,m→∞⁡Tn,mf∗-f∗ρ≥14,where ρ=1+x2+y2.Now, consider the following operator:(23)Tn,mf;pn,qn,pm,qm;x,y=C-n,m,sα,βf;pn,qn,pm,qm;x,y,x,y∈Dan,bnfx,y,R+2∖Dan,bnTheorem 5.
Letf∈CρR+2 for any γ>0(24)limn,m→∞⁡Tn,mf;pn,qn,pm,qm;·,·-f·Cρ1+γ=0where an, bm, qn, and qm have the same conditions as in Theorem 3.Proof.
For allε>0, there exist sufficiently large positive real numbers A and B such that(25)1+x2+y2-γ<εwhen x>A and y>B.
Letn, m be sufficiently large so that DA,B⊂Dan,bm(26)Tn,mf;pn,qn,pm,qm;·,·-f·Cρ1+γ≤supx,y∈DA,B⁡C-n,m,sα,βf;pn,qn,pm,qm;x,y-fx,y1+x2+y21+γ+supx,y∈Dan,bn∖DA,B⁡C-n,m,sα,βf;pn,qn,pm,qm;x,y-fx,y1+x2+y21+γ=yn,m′+yn,m′′.By Theorem 3, limn,m→∞⁡yn,m′=0 and for the proof of the second term we have(27)yn,m′′≤1+x2+y2-γC-n,m,sα,βf;pn,qn,pm,qm;x,y1+x2+y2+fx,y1+x2+y2.Finally, since f∈CρR+2, the term f(x,y)/1+x2+y2 is bounded. Furthermore, because of the fact that(28)C-n,m,sα,βf;pn,qn,pm,qm;x,y≤Cn,m,sα,β1+t12+t22;qn,qm;x,y,in case Lemma 2 is used, the term C-n,m,sα,βf;pn,qn,pm,qm;x,y/1+x2+y2 is bounded for sufficiently large n and m. Hence, we get by (25) that (29)yn,m′′≤ε1+MSince ε>0 is arbitrary, then limn,m→∞⁡yn,m′′=0. This completes the proof.Now, consider the subspaceCρ0 of Cρ which is defined by(30)Cρ0≔f∈Cρ:limx,y→0⁡fx,y1+x2+y2=0.Theorem 6.
Let the sequencesqn, qm, an, and bm satisfy the same properties as in Theorem 3. Then for all f∈Cρ0R+2, we obtain (31)limn,m→∞⁡Tn,mC-n,m,sα,βf;pn,qn,pm,qm;·,·-f·Cρ=0.Proof.
For allf∈Cρ0R+2, observe that (32)limx,y→∞⁡fx,y1+x2+y2=0,limn,m→∞⁡fpnn+s-kkpn,qn+α/npn,qn+βan,pmm+s-llpm,qm+α/mpm,qm+βbm1+pnn+s-kkpn,qn+α/npn,qn+βan2+pmm+s-llpm,qm+α/mpm,qm+βbm2=0.
Therefore, for allε>0, we can find sufficiently large numbers A and B such that (33)fx,y<ε1+x2+y2for x>A and y>B and there exist natural numbers n0 and m0 such that (34)fpnn+s-kkpn,qn+αnpn,qn+βan,pmm+s-llpm,qm+αmpm,qm+βbm<ε1+pnn+s-kkpn,qn+αnpn,qn+βan2+pmm+s-llpm,qm+αmpm,qm+βbm2for all n>n0 and m>m0.
Hence, for largen and m, we have(35)Tn,mf;pn,qn,pm,qm;·,·-f·Cρ≤supx,y∈DA,B⁡C-n,m,sα,βf;pn,qn,pm,qm;x,y-fx,y1+x2+y2+supx,y∈Dan,bm∖DA,B⁡C-n,m,sα,βf;pn,qn,pm,qm;x,y-fx,y1+x2+y2=zn,m′+zn,m′′.
By Theorem3, it is sufficient to show that zn,m′′→0 as n→∞.
Using (33) and (34), we get(36)zn,m′′≤ε+supx,y∈Dan,bm∖DA,B⁡C-n,m,sα,βf;pn,qn,pm,qm;x,y)1+x2+y2≤ε+εsupx,y∈Dan,bm∖DA,B⁡tn,mqn,qm;x,y=ε1+supx,y∈Dan,bm∖DA,B⁡tn,mqn,qm;x,y⁡where(37)tn,mqn,qm;x,y≔C-n,m,sα,β1;pn,qn,pm,qm;x,y)+C-n,m,sα,βt12;pn,qn,pm,qm;x,y)+C-n,m,sα,βt22;pn,qn,pm,qm;x,y)1+x2+y2.By Lemma 2, it is clear that there exist K independent of n and m such that (38)supx,y∈Dan,bm/DA,Btn,mpn,qn,pm,qm;x,y≤K.Therefore, for n>n0 and m>m0 we have (39)zn,m′′<1+Kε.
This completes the proof.


## 4. Order of Convergence

In this section, we compute the rate of convergence of the operators in terms of the full modulus of continuity and partial modulus of continuities.Letf∈DA,B and x≥0. Then the definition of the modulus of continuity of f is given by(40)ωf;δ=maxx1-x22+y1-y22≤δx,y∈CDA,Bfx1,y1-fx2,y2.It is known that [32] for any δ>0 we know that (41)fx1,y1-fx2,y2≤ωf,δx1-x22+y1-y22δ+1.Also, for any δ>0 we have (42)fx1,y1-fx2,y2≤ω1f,δx1-x2δ+1,fx1,y1-fx2,y2≤ω2f,δy1-y2δ+1.Theorem 7.
For anyf∈C(DA,B), the following inequalities (43)C-n,m,sα,βf;pn,qn,pm,qm;x,y-fx,y≤2ω1f;δm+ω2f;δn(44)C-n,m,sα,βf;pn,qn,pm,qm;x,y-fx,y≤2ωf;δm2+δn2are satisfied, where (45)δn2≔qnn+s-1pn,qnn+spn,qnnpn,qn+β2A2+n+spn,qn2α+pnn+s-1npn,qn+β2anA+α2an2npn,qn+β2,(46)δm2≔qmm+s-1pm,qmm+spm,qmmpm,qm+β2B2+m+spm,qm2α+pmm+s-1mpm,qm+β2bmB+α2bm2mpm,qm+β2.Proof.
We directly have(47)C-n,m,sα,βf;pn,qn,pm,qm;x,y-fx,y=∑k=0n+s∑l=0m+sfpnn+s-kkpn,qn+αnpn,qn+βan,pmm+s-llpm,qm+αmpm,qm+βbm-fx,y×Φkn,pn,qnxanΦkm,pm,qmybm=∑k=0n+s∑l=0m+sfpnn+s-kkpn,qn+αnpn,qn+βan,pmm+s-llpm,qm+αmpm,qm+βbm-fpnn+s-kkpn,qn+αnpn,qn+βan,y+fpnn+s-kkpn,qn+αnpn,qn+βan,y-fx,yΦkn,pn,qnxanΦkm,pm,qmybm.By linearity and positivity of the operators, we get(48)C-n,m,sα,βf;pn,qn,pm,qm;x,y-fx,y≤∑k=0n+s∑l=0m+sfpnn+s-kkpn,qn+αnpn,qn+βan,pmm+s-llpm,qm+αmpm,qm+βbm-fpnn+s-kkpn,qn+αnpn,qn+βan,y×Φkn,pn,qnxanΦkm,pm,qmybm+∑k=0n+s∑l=0m+sfkpn,qnnpn,qnan,y-fx,yΦkn,pn,qnxanΦkm,pm,qmybm≤∑k=0n+s∑l=0m+sω2f;pmm+s-kkpm,qm+αmpm,qm+βbm-yΦkn,pn,qnxanΦkm,pm,qmybm+∑k=0n+s∑l=0m+sω1f;pnn+s-kkpn,qn+αnpn,qn+βan-xΦkn,pn,qnxanΦkm,pm,qmybm=Ω1x,y+Ω2x,y.Using Lemma 1 and Cauchy-Schwartz inequality, we have(49)Ω1x,y=∑k=0n+s∑l=0m+sω2f;pmm+s-llpm,qm+αmpm,qm+βbm-yΦkn,pn,qnxanΦkm,pm,qmybm=∑l=0m+sω2f;pmm+s-llpm,qm+αmpm,qm+βbm-yΦkm,pm,qmybm≤ω2f;δm1+1δm∑j=0m+ppmm+s-llpm,qm+αmpm,qm+βbm-y2Φkm,pm,qmybm1/2.Then, we get(50)Ω1x,y≤2ω2f;δm,where we choose δm as in (46).
In the same way, we obtain(51)Ω2x,y≤2ω1f;δn,where δn is given in (45). Combining (50) and (51), we obtain (43).
Now, by using linearity and the monotonicity of the operators, and taking into account (40), we have (52)C-n,m,sα,βf;pn,qn,pm,qm;x,y-fx,y≤∑k=0n+s∑l=0m+sωf;pnn+s-kkpn,qn+αnpn,qn+βan-x2+pmm+s-llpm,qm+αmpm,qm+βbm-y2×Φkn,pn,qnxanΦkm,pm,qmybm≤∑k=0n+s∑l=0m+sfpnn+s-kkpn,qn+αnpn,qn+βan,pmm+s-llpm,qm+αmpm,qm+βbm-fx,y×Φkn,pn,qnxanΦkm,pm,qmybm≤1+1δ∑k=0n+s∑l=0m+sωf;pnn+s-kkpn,qn+αnpn,qn+βan-x2+pmm+s-llpm,qm+αmpm,qm+βbm-y2×Φkn,pn,qnxanΦkm,pm,qmybmUsing (41) and the Cauchy-Schwartz inequality, we get (44).


## 5. Generalization of the Bivariate Chlodowsky Variant of Bernstein-Schurer Operators Based on(p,q)-Integers

In this section, we introduce generalization of bivariate Chlodowsky variant of Bernstein-Schurer operators based on(p,q)-integers. The generalized operators help us to approximate continuous functions defined on more general weighted spaces. Note that this kind of generalization was considered earlier for the Chlodowsky-Bernstein polynomials [12].Forx≥0, consider any continuous function ωx,y≥1 and define (53)Gft,s=ft,s1+t2+s2wt,s.Let us consider the generalization of the C-n,m,sα,βf;pn,qn,pm,qm;x,y as follows:(54)Ln,pα,βC-n,m,sα,βf;pn,qn,pm,qm;x,y=wx,y1+x2+y2∑k=0n+s∑l=0m+sGfpnn+s-kkpn,qn+αnpn,qn+βan,pmm+s-llpm,qm+αmpm,qm+βbm×Φkn,pn,qnxanΦkm,pm,qmybm,x,y∈Dan,bnfx,y,R+2∖Dan,bn,where (x,y)∈Danbm and an and bm have the same properties of bivariate Chlodowsky variant of Bernstein-Schurer operators based on (p,q)-integers.Theorem 8.
For all continuous functionsf satisfying f(x,y)≤Mfw(x,y), x,y≥0, and limx,y→∞⁡f(x,y)/w(x,y)=0, we have (55)limn,m→∞⁡Ln,pα,βf;pn,qn,pm,qm;·,·-f·,·w=0where w(x,y)=1+x2+y2.Proof.
Clearly,(56)Ln,pα,βf;pn,qn,pm,qm;x,y-fx,y=wx,y1+x2+y2∑k=0n+s∑l=0m+sGfpnn+s-kkpn,qn+αnpn,qn+βan,pmm+s-llpm,qm+αmpm,qm+βbm×Φkn,pn,qnxanΦkm,pm,qmybm-Gfx,y,thus(57)Ln,pα,βf;pn,qn,pm,qm;·,·-f·,·w=supx,y∈R+2⁡Ln,pα,βf;pn,qn,pm,qm;x,y-fx,ywx,y=supx,y∈R+2⁡Tn,mGf;pn,qn,pm,qm;x,y-Gfx,y1+x2+y2.Since f(x,y)≤Mfw(x,y), then Gfx,y≤Mfw(x,y) for x,y≥0 and Gfx,y is continuous function on R+2. Furthermore, from limx,y→∞⁡f(x,y)/w(x,y)=0, we have (58)limx,y→∞⁡Gfx,yρx,y=0.Thus, from Theorem 6 we get the result.Finally, note that takingw(x,y)=1+x2+y2, then the operators Ln,pα,βf;pn,qn,pm,qm;x,y reduce Tn,mGf;pn,qn,pm,qm;x,y.


## 6. Numerical Results and Discussions

In this section, numerical results are presented in order to demonstrate the effectiveness and validate the accuracy ofC-n,m,sα,βf;pn,qn,pm,qm;x,y to fx,y with different values of parameters. The parameters used in the mathematical calculations are given in Table 1.Table 1
C - n , m , s α , β f ; p n , q n , p m , q m ; x , y parameters used for numerical analysis.
Parameters Values n 20 m 5 an n 2 + 4 / n 2 + 10 n bm m 2 + 4 m 2 + 10 m pn 0.989 qn 0.99 pm 0.989 qm 0.99 s 1 α,β VariableFigures1 and 2 are given for C-n,m,sα,βf;pn,qn,pm,qm;x,y as a function of x and y for different α and β values considering two different multivariable functions fx,y=esin⁡xy and fx,y=x2y2 accordingly.Figure 1
Convergence ofC-n,m,sα,βf;pn,qn,pm,qm;x,y to multivariable function fx,y=esin⁡xy.Figure 2
Convergence ofC-n,m,sα,βf;pn,qn,pm,qm;x,y to multivariable function fx,y=x2y2.Figure1 demonstrates the convergence of C-n,m,sα,βf;pn,qn,pm,qm;x,y to the multivariable function fx,y=esin⁡xy. For different x and y values, various α and β values are taken into account. According to the results obtained, the C-n,m,sα,βf;pn,qn,pm,qm;x,y behaves like the function fx,y=esin⁡xy for smaller α and β values. Therefore, the error of approximation of C-n,m,sα,βf;pn,qn,pm,qm;x,y to the function fx,y=esin⁡xy is minimised for α=0.02 and β=0.03, where s=1, n=20, and m=5 for the sequences an and bm as provided in Table 1.Figure2 shows the convergence C-n,m,sα,βf;pn,qn,pm,qm;x,y to the function fx,y=x2y2 but this time for different α and β values, while keeping all the other parameters constant. Unlike in Figure 1, increasing α and β values upto 0.3 and 0.5 respectively provided better approximate results. It is evident that, similar to Figure 1, α and β values have a significant effect on the convergence of C-n,m,sα,βf;pn,qn,pm,qm;x,y to the multivariable function fx,y=x2y2, which provide better approximate results.Comparative results are given in Tables2 and 3, for the errors of the approximation of  C-n,m,sα,βf;pn,qn,pm,qm;x,y to the functions fx,y=esin⁡xy and fx,y=x2y2.Table 2
Errors of approximationC-n,m,sα,βf;pn,qn,pm,qm;x,y to f(x,y)=esin⁡xy, (fx,y-C-n,m,s0.02,0.03f;pn,qn,pm,qm;x,y).
x y 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.0015 0.0039 0.0075 0.0112 0.0149 0.0186 0.0222 0.0258 0.0294 0.0329 0.0364 0.1 0.0044 0.0015 0.0022 0.0032 0.0041 0.0049 0.0056 0.0063 0.0069 0.0074 0.0078 0.2 0.0086 0.0023 0.0034 0.0049 0.0063 0.0075 0.0086 0.0096 0.0104 0.011 0.0116 0.3 0.0128 0.0033 0.0046 0.0066 0.0083 0.0099 0.0114 0.0128 0.0141 0.0153 0.0164 0.4 0.0169 0.0038 0.0051 0.0072 0.0092 0.0109 0.0127 0.0145 0.0163 0.0183 0.0206 0.5 0.0208 0.004 0.0048 0.0066 0.0082 0.0099 0.0117 0.0138 0.0164 0.0198 0.0243 0.6 0.0245 0.0035 0.0033 0.0042 0.0051 0.0061 0.0076 0.01 0.0137 0.0193 0.0274 0.7 0.028 0.0025 0.0006 0.0002 0.0009 0.0012 0.0004 0.0021 0.0074 0.0164 0.0303 0.8 0.0313 0.0007 0.0038 0.0071 0.0104 0.0128 0.0133 0.0105 0.0031 0.0108 0.0332 0.9 0.0342 0.002 0.0101 0.0171 0.0239 0.0294 0.0317 0.0288 0.0185 0.0023 0.0366 1 0.0368 0.0057 0.0184 0.0303 0.0421 0.0517 0.0566 0.0537 0.0393 0.0093 0.0407Table 3
Errors of approximationC-n,m,sα,βf;pn,qn,pm,qm;x,y to f(x,y)=x2y2, fx,y-C-n,m,s0.3,0.5f;pn,qn,pm,qm;x,y.
x y 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0 0 0.0001 0.0002 0.0004 0.0005 0.0007 0.001 0.0013 0.0017 0.0021 0.1 0.0001 0.0002 0.0004 0.0007 0.0012 0.0016 0.0022 0.0029 0.0037 0.0048 0.0066 0.2 0.0003 0.0005 0.001 0.0017 0.0025 0.0033 0.0043 0.0055 0.0069 0.0093 0.0134 0.3 0.0006 0.001 0.0019 0.0029 0.004 0.0052 0.0065 0.0079 0.0099 0.0136 0.0209 0.4 0.001 0.0017 0.003 0.0044 0.0058 0.0072 0.0086 0.0102 0.0127 0.0177 0.0289 0.5 0.0015 0.0024 0.0043 0.0061 0.0078 0.0093 0.0108 0.0124 0.0151 0.0217 0.0374 0.6 0.0021 0.0034 0.0058 0.0081 0.01 0.0116 0.0129 0.0143 0.0173 0.0256 0.0466 0.7 0.0028 0.0045 0.0075 0.0103 0.0124 0.014 0.0151 0.0162 0.0192 0.0292 0.0563 0.8 0.0036 0.0058 0.0094 0.0127 0.0151 0.0166 0.0173 0.0179 0.0209 0.0327 0.0665 0.9 0.0046 0.0072 0.0116 0.0154 0.018 0.0193 0.0195 0.0194 0.0222 0.0361 0.0773 1 0.0056 0.0087 0.014 0.0183 0.0211 0.0221 0.0217 0.0207 0.0233 0.0392 0.0887



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