# The Approximation of Bivariate Blending Variant Szász Operators Based Brenke Type Polynomials

Authors: Behar Baxhaku; Ramadan Zejnullahu; Artan Berisha
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

We have constructed a new sequence of positive linear operators with two variables by using Szasz-Kantorovich-Chlodowsky operators and Brenke polynomials. We give some inequalities for the operators by means of partial and full modulus of continuity and obtain a Lipschitz type theorem. Furthermore, we study the convergence of Szasz-Kantorovich-Chlodowsky-Brenke operators in weighted space of function with two variables and estimate the rate of approximation in terms of the weighted modulus of continuity.


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## Body

## 1. Introduction

The well-known Mirakjan-Favard-Szasz type operators of one variable are defined as(1)Snf;x=∑i=0∞ωnifin,n∈N,x∈0,∞,where(2)ωnix=e-nxnxii!and f:[0,∞)→R is such that the above exist series. For the convergence of Sn(f;x) to f(x), usually f is supposed to be the exponential growth, that is, fx≤αeβx, for all x∈[0,∞), with α,β>0 (see [1]). Later, in 1969, Jakimovski and Leviatan [2] investigated approximation properties of the generalization of Szász operators by means of the Appell polynomials pk(x)=∑i=0kai(xk-i/(k-i)!) which satisfy the identity (3)gtetx=∑k=0∞pkxtkwhere g(z)=∑k=0∞akzk,z<R,(R>1) and g(1)≠0. Varma and Tasdelen [3] constructed positive linear operators based on orthogonal polynomials, e.g., Brenke polynomials. Suppose that (4)Kt=∑k=0∞akzk,K0≠0,Lt=∑k=0∞bkzk,L0≠0is analytic functions in the disk z<R,(R>1), where ak and bk are real. The generating function for these polynomials is given by (5)KtLtx=∑k=0∞pkxtkfrom which the explicit form of pk(x) is as follows: (6)pkx=∑i=0kal-kblxl,k=0,1,2,….We suppose that(1)
K(1)≠0, ak-lbl/K(1)≥1,k=0,1,2,…,(2)
L:[0,∞)→(0,∞),(3)
(5) and (6) converge for t<R,(R>1),(4)
limu→∞⁡(L(k)(u)/L(u))=1, for k∈1,2,3,4.Atakut and Buyukyazici in [4] introduced the Kantorovich-Szász variant based on Brenke type polynomials defined as(7)Smf;x=cmK1Lbmy∑j=0∞pjbmy∫j/cmj+1/cmftdt,where (bm),(cm) are strictly increasing sequences of positive numbers such that limn→∞⁡(1/cm)=0,bm/cm=1+O(1/cm). The classical Bernstein-Chlodowsky polynomials are defined by(8)Bnf;x=∑k=0npn,kxanfknanwhere pn,kx/an=nkx/ank1-x/ann-k,0≤x≤an and (an) is a sequence of positive numbers with limn→∞⁡an=∞ and limn→∞⁡(an/n)=0. In the last few decades the convergence estimation for linear positive operators is an active area of research amongst researchers. Several new operators have been introduced and their convergence behavior has been discussed (see [5–8]). In [9, 10] authors introduced a bivariate blending variant of the Szász type operators and studied local approximation properties for these operators. Also, they estimated the approximation order in terms of Peetre’s K-functional and partial moduli of continuity.In the present paper, we define new bivariate operators associated with a combination of Szasz-Kantorovich-Chlodowsky operators based on Brenke polynomials as follows:(9)Tn,m,anbm,cmf;x,y=ncmanK1Lbmy∑k=0n∑j=0∞pn,kxanpjbmy∫j/cmj+1/cm∫k/nank+1/nanft,sdtds,where the sequences (an),(bm),(cm) are defined as above and satisfy the following conditions: (10)limn→∞⁡ann=0,limm→∞⁡1cm=0,bmcm=1+O1cm.For operators defined in (36) we have(11)Tn,m,anbm,cmf;x,y=xCn∗Smbm,cmy∗f;x,y=y∗Smbm,cmCxn∗f;x,ywhere (12)Cxn∗f;x,y=nan∑k=0npn,kxan∫k/nank+1/nanft,ydtand(13)Smbm,cmy∗f;x,y=bmK1Lbmy∑j=0∞pjbmy∫j/cmj+1/cmfx,sds.In this study, we give some basic convergence properties for the operators defined by (9) and study local approximation properties for these operators. Furthermore, we study the linear positive operators in a weighted space of function with two variables and estimate the rate of approximation of the operators Tn,m,anbm,cm in the terms of the weighted modulus of continuity.


## 2. Notations and Auxiliary Results

We will subsequently need the following basic results to prove the main results.In what follows, leteij(x,y)=xiyj,(i,j)∈N0×N0, where i+j≤4 is the two dimensional test functions.By simple calculations we get the following lemma.Lemma 1.
LetTn,m,anbm,cm be the bivariate of Szasz-Kantorovich-Chlodowsky-Brenke operators defined by (9). For all m,n∈N, Tn,m,anbm,cm satisfy the following results:
(i)(14)Tn,m,anbm,cme00;x,y=1;
(ii)(15)Tn,m,anbm,cme10;x,y=x+an2n;
(iii)(16)Tn,m,anbm,cme01;x,y=bmyL′bmycmLbmy+2K′1+12K1cm;
(iv)(17)Tn,m,anbm,cme20;x,y=1-1nx2+2annx+an23n2;
(v)(18)Tn,m,anbm,cme02;x,y=bm2L′′bmyy2cm2Lbmy+2bmK′1+K1L′bmyycm2K1Lbmy+3K′′1+6K′1+K13cm2K1;
(vi)(19)Tn,m,anbm,cme03;x,y=bm3L′′′bmyy3cm3Lbmy+3bm22K′1+3K1L′′bmyy2cm3K1Lbmy+bm6K′′1+18K′1+7K1L′bmyy2cm3K1Lbmy+4K′′′1+16K′1+12K′1+7K14cm3K1;
(vii)(20)Tn,m,anbm,cme04;x,y=bm4L4bmyy4cm4+4bm3K′1+2K1L′′′bmyy3cm4K1Lbmy+3bm22K′′1+8K′1+5K1L′′bmyy22cm4K1Lbmy+bm4K′′′1+24K′′1+30K′1+6K1L′bmyycm4K1Lbmy+5K41+40K′′′1+75K′′1+30K′1+K1cm4K1.Proof.
In view of definition of operators defined by (9) we have(21)Tn,m,anbm,cme00;x,y=xCn∗e0;x,yy∗Smbm,cme0;x,yTn,m,anbm,cme10;x,y=xCn∗e1;x,yy∗Smbm,cme0;x,yTn,m,anbm,cme01;x,y=xCn∗e0;x,yy∗Smbm,cme1;x,yTn,m,anbm,cme20;x,y=xCn∗e2;x,yy∗Smbm,cme0;x,yTn,m,anbm,cme02;x,y=xCn∗e0;x,yy∗Smbm,cme2;x,yTn,m,anbm,cme03;x,y=xCn∗e0;x,yy∗Smbm,cme3;x,yTn,m,anbm,cme04;x,y=xCn∗e0;x,yy∗Smbm,cme4;x,ywith the help of these equalities, we can easily prove required results.Lemma 2.
It follows from Lemma1 that(22)Tn,m,anbm,cme10-x;x,y=an2n;Tn,m,anbm,cme10-x;x,y=bm2L′′bmycm2Lbmy-1y+2K′1+12cmK1;Tn,m,anbm,cme10-x2;x,y=xan-xn+an23n2;Tn,m,anbm,cme01-y2;x,y=bm2L′′bmycm2Lbmy-2bmL′bmycmLbmy+1y2+2bmK′1+K1L′bmycm2K1Lbmy-2K′1+1cmK1y+3K′′1+6K′1+K13cm2K1.Proof.
The results follow from linearity of the operatorsTn,m,anbm,cm and Lemma 1.
For sufficiently largen,m, for all (x,y)∈Ian, by taking into consideration Lemma 1, and condition (10), we have the following equalities: (23)Δ1=Tn,m,anbm,cme10-x2;x,y=xCn∗e10-x2;x,y=Oann∑i=12xi;(24)Δ2=Tn,m,anbm,cme10-x4;x,y=xCn∗e10-x4;x,y=Oann∑i=14xi;and(25)Δ1∗=Tn,m,anbm,cme10-y2;x,y=y∗Smbm,cme10-y2;x,y=O1cm∑i=12yi;(26)Δ2∗=Tn,m,anbm,cme10-y4;x,y=y∗Smbm,cme10-y4;x,y=O1cm∑i=04yi.Further, let δnx=Δ1, δmy=Δ1∗, and δmx,y=Δ1+Δ1∗1/2.


## 3. Main Results

To study the convergence of the sequenceTn,m,anbm,cm we shall use the following Korovkin type theorem, established by Volkov [11]. Next, the degree of approximation of the operator {Tn,m,anbm,cm} given by (36) will be established in the space of continuous function on compact set Iab=[0,a]×[0,b]⊂Ian. For Iab=[0,a]×[0,b], let C(Iab), denote the space of all real valued continuous functions on Iab, endowed with the norm fC(Iab)=sup(x,y)∈Iab⁡fx,y.Theorem 3.
Let{Tn,m,anbm,cm} be the sequences of linear positive operators defined by (36). Then for each f∈C(Iab), we have
lim n , m → ∞ ⁡ T n , m , a n b m , c m = f ( x , y ) uniformly on the compact set Iab.Proof.
From Lemma1, we have (27)limn,m→∞Tn,m,anbm,cmeij;x,y=eij=0,i,j∈0,0,1,1,2,2and(28)limn,m→∞Tn,m,anbm,cme20+e02;x,y=e20+e02,uniformly on Iab. The result follows from the well-known Volkov theorem.Example 4.
Let us consider the functionf(x,y)=x2y-xy2. For n=m=100,1000,K(t)=e2t; L(t)=et and an=n; bm=m; cm=m+1/m the convergence of Tn,m,anbm,cmtof(x,y) is illustrated in Figures 1(a) and 1(b), respectively. Further, in Table 1 we compute error estimation for operator (9) to the function f.Table 1
Error estimation for operator (9) to the function f(x,y)=x2y-xy2 for n=m=100.
x (0.01,0.5) (0.1,0.5) (0.4,0.5) (0.5,0.5) (0.6,0.5) (0.8,0.5) (0.9,0.5)  n=m= 1000 0.0037 0.0011 0.0076 0.0105 0.0135 0.0196 0.0226  n = m=100 0.0118 0.0052 0.0187 0.0273 0.0361 0.0548 0.0645Figure 1 (a) (b)Example 5.
ForKt=e2t, L(t)=et the convergence of operators Tn,m,anbm,cm to function f(x,y) is illustrated in Figures 2(a) and 2(b), respectively, where (x,y)=xy+xy2, n=m=100,1000, and an=n; bm=m; cm=m+1/m. In Table 2 there are are compute error estimations for operator (9) to the function f.Table 2
Error estimation for operator (9) to the function f(x,y)=xy+xy2 for n=m=1000.
x (0.01,0.5) (0.1,0.5) (0.3,0.5) (0.4,0.5) (0.6,0.5) (0.8,0.5) (0.9,0.5)  n =m= 1000 0.0116 0.0121 0.0131 0.0136 0.0151 0.0156 0.0161  n = m=100 0.0390 0.0435 0.0535 0.0584 0.0734 0.0784 0.0834Figure 2 (a) (b)An estimation of the rate of convergence can be obtained using the modulus of continuity for two dimensional real valued functions. Letf∈C(Iab) and δ>0. In what follows, we shall use the following modulus of continuity for bivariate real functions: (29)ωf;δn,δm=sup⁡ft,s-fx,y:t,s,x,y∈Iab,t-x≤δn,s-y≤δm.Alternately, the complete modulus of continuity of f which we denote by ω(f;δ) is defined as (30)ωf;δ=supt-x2+s-y2≤δft,s-fx,y:t,s,x,y∈Iab,.Theorem 6.
For anyf∈C(Iab), then we have estimated (31)Tn,m,anbm,cmf;x,y-fx,y≤2ωf;δn,mwhere δn,m=δn,m(x,y).Proof.
From (9) and by definition of ω(f;δ), we can write (32)Tn,m,anbm,cmf;x,y-fx,y≤ncmanK1Lbmy∑k=0n∑j=0∞pn,kxanpjbmy∫j/cmj+1/cm∫k/nank+1/nanωf;t-x2+s-y2;x,ydtds≤ωf;δn,m1+1δm,nncmanK1Lbmy∑k=0n∑j=0∞pn,kxanpjbmy∫j/cmj+1/cm∫k/nank+1/nant-x2+s-y2dtdsUsing the Cauchy-Schwarz inequality, we obtain(33)Tn,m,anbm,cmf;x,y-fx,y≤ωf;δn,m1+1δn,mTn,m,anbm,cm(t-x2+s-y2;x,y1/2≤ωf;δn,m1+1δn,mOann∑i=12xi+O1cm∑i=12yi1/2.Taking δn,m=δn,m(x,y), we obtain the desired result.
The partial modulus of continuity with respect tox and y is given by (34)ω1f;δ=sup0≤y≤b⁡supx1-x2≤δ⁡fx1,y-fx2,y,ω2f;δ=sup0≤x≤a⁡supy1-y2≤δ⁡fx,y1-fx,y2,Theorem 7.
For anyf∈C(Iab), then the inequalities satisfy (35)Tn,m,anbm,cmf;x,y-fx,y≤2ω1f;δn+ω2f;δmwhere δn=δn(x), δm=δm(y).Proof.
Using the definition of partial modulus of continuityω(i)(f;δ),i=1,2, we may write (36)Tn,m,anbm,cmf;x,y-fx,y≤ncmanK1Lbmy∑k=0n∑j=0∞pn,kxanpjbmy∫j/cmj+1/cm∫k/nank+1/nanft,s-ft,ydtds+ncmanK1Lbmy∑k=0n∑j=0∞pn,kxanpjbmy∫j/cmj+1/cm∫k/nank+1/nanft,y-fx,y≤ncmanK1Lbmy∑k=0n∑j=0∞pn,kxanpjbmy∫j/cmj+1/cm∫k/nank+1/nanω2f;s-ydtds+ncmanK1Lbmy∑k=0n∑j=0∞pn,kxanpjbmy∫j/cmj+1/cm∫k/nank+1/nanω1f;t-xdtds=η1x,y+η2x,y.Consider η(x,y). Using Lemma 1 and the well-known properties of the modulus of continuity, we have (37)η1x,y≤ω2f;δm1+ncmδmanK1Lbmy∑k=0n∑j=0∞pn,kxanpjbmy∫j/cmj+1/cm∫k/nank+1/nans-ydtdsBy using Cauchy-Schwarz inequality, we get (38)η1x,y≤ω2f;δm1+1δmncmanK1Lbmy∑k=0n∑j=0∞pn,kxanpjbmy×∫j/cmj+1/cm∫k/nank+1/nans-y2dtds1/2.So, by using (25), we obtain(39)η1x,y≤ω2f;δm1+O1/cm∑i=12yiδm.In the same way we gain (40)η2x,y≤ω1f;δn1+Oan/n∑i=12xiδn.Hence from (39), (40), and (32), we arrive at(41)Tn,m,anbm,cmf;x,y-fx,y≤ω2f;δm1+O1/cm∑i=12yiδm+ω1f;δn1+Oan/n∑i=12xiδn.Finally, choosing δn=Δ1=δn(x) and δm=Δ1∗=δm(y), for all (x,y)∈Iab, we reach the desired result.
For0<γ≤1, we define the Lipschitz class LipL(γ) for bivariate case as follows: (42)LipLγ≔f:ft1,t2-fx,y≤Lr-sγr+x+yγ/2,where r=(t1,t2), s=(x,y) in Iab, and r-s=t1-x2+t2-y21/2 is the Euclidean norm.Theorem 8.
Suppose thatf∈LipL(γ). Then, for every (x,y)∈Iab, we have (43)Tn,m,anbm,cmf;x,y-fx,y≤Lδm1/2x+y1/2,where δm=δm(x,y).Proof.
First, we prove theorem for caseγ=1. Then, for f∈LipL(γ) and for each x,y∈Iab, using the monotonicity and linearity of operators, we may write (44)Tn,m,anbm,cmf;x,y-fx,y≤Tn,m,anbm,cmft1,t2-fx,y;x,y≤LTn,m,anbm,cmr-sr+x+y1/2;x,y≤Lx+y1/2Tn,m,anbm,cmr-s;x,y,where r=(t1,t2) and s=(x,y).
Using the Cauchy- Shwarz inequality and Lemma2, the above inequality implies that (45)Tn,m,anbm,cmf;x,y-fx,y≤Lx+y1/2Tn,m,anbm,cmr-s2,x,y1/2≤Lx+y1/2·Cxn∗t1-x2,x,y+y∗Smbm,cmt2-y2,x,y1/2≤Lδm1/2x,yx+y1/2.Thus, the result holds for γ=1. Secondly, let 0<γ<1. Then, for f∈LipL(γ) and for each x,y∈Iab, we get (46)Tn,m,anbm,cmf;x,y-fx,y≤Tn,m,anbm,cmft1,t2-fx,y;x,y≤LTn,m,anbm,cmr-sγr+x+yγ/2;x,y≤Lx+yγ/2Tn,m,anbm,cmr-sγ;x,y.    ,Now, applying Holder’s inequality with u1=2/γ, u2=2/(2-γ), and Lemma 2, we get (47)Tn,m,anbm,cmf;x,y-fx,y≤Lx+yγ/2Tn,m,anbm,cmr-s2,x,yγ/2≤Lx+yγ/2Cxn∗t1-x2,x,y+y∗Smbm,cmt2-y2,x,yγ/2≤Lx+y1/2Oann∑i=12xi+O1cm∑i=12yi)γ/2.which leads us to the required result.Theorem 9.
Iff(x,y) has continuous partial derivatives fx′ and fy′ and ω(1)(fx′;δ) and ω(2)(fy′;δ) denote the partial moduli of continuity of fx′ and fy′ respectively. Then we have estimate (48)Tn,m,anbm,cmf;x,y-fx,y≤Λan2n+Υbm2L′′bmycm2Lbmy-1y+2K′1+12cmK1+ω1fx′;δn1+δn+ω2fx′;δm1+δm,where Λ,Υ are the positive constants such that ∂f/∂x≤Λ,∂f/∂y≤Υ,(x,y∈Iab).Proof.
From the mean value theorem we have(49)ft1,t2-fx,y=ft1,y-fx,y+ft1,t2-ft1,y=t1-x∂fη,y∂x+t2-y∂fx,ζ∂y=t1-x∂fx,y∂x+t1-x∂fη,y∂x-∂fx,y∂x+t2-y∂fx,y∂y+t2-y∂fx,ζ∂y-∂fx,y∂y,where x<η<t1 and y<ζ<t2. By using the above identity, we get (50)Tn,m,anbm,cmf;x,y-fx,y=Tn,m,anbm,cmt1-x∂fx,y∂x;x,y+Tn,m,anbm,cmt1-x∂fη,y∂x-∂fx,y∂x;x,y+Tn,m,anbm,cmt2-y∂fx,y∂y;x,y+Tn,m,anbm,cmt2-y∂fx,ζ∂y-∂fx,y∂y;x,y.Hence,(51)Tn,m,anbm,cmf;x,y-fx,y≤∂fx,y∂xTn,m,anbm,cmt1-x;x,y+Tn,m,anbm,cmt1-x∂fη,y∂x-∂fx,y∂x;x,y+∂fx,y∂yTn,m,anbm,cmt2-y;x,y++Tn,m,anbm,cmt2-y∂fx,ζ∂y-∂fx,y∂y;x,y≤ΛTn,m,anbm,cmt1-x;x,y+ΥTn,m,anbm,cmt2-y;x,y+Tn,m,anbm,cmt1-xω1fx′;δnt1-xδn+1;x,y+Tn,m,anbm,cmt2-yω1fy′;δmt2-yδm+1;x,y,Since η-x<t1-x and ζ-y<t2-y.
Using last inequalities, we have(52)Tn,m,anbm,cmf;x,y-fx,y≤ΛTn,m,anbm,cmt1-x;x,y+ΥTn,m,anbm,cmt2-y;x,y++ω1fx′;δn1Tn,m,anbm,cmt1-x;x,y+ω1fx′;δn1δn1Tn,m,anbm,cmt1-x2;x,y+ω2fy′;δmTn,m,anbm,cmt2-y;x,y+ω2fy′;δn2δmTn,m,anbm,cmt2-y2;x,y.Now, applying the Cauchy-Schwarz inequality (53)Tn,m,anbm,cmf;x,y-fx,y≤ΛTn,m,anbm,cmt1-x;x,y+ΥTn,m,anbm,cmt2-y;x,y++ω1fx′;δnCxn∗t1-x2;x,y1/2+ω1fx′;δnδnCxn∗t1-x2;x,y+ω2fy′;δmSmbm,cmy∗t2-y2;x,y1/2+ω2fy′;δmδmSmbm,cmy∗t2-y2;x,y.Now choosing δn=δn(x) and δm=δm(y), we have (54)Tn,m,anbm,cmf;x,y-fx,y≤Λan2n+Υbm2L′′bmycm2Lbmy-1y+2K′1+12cmK1+ω1fx′;δn1+δn+ω2fy′;δm1+δm.This completes the proof.


## 4. Weighted Approximation Properties

The weighted Korovkin-type theorems are used for the purpose of this study, which are previously proved by Gadjiev [12, 13]. Therefore we need to introduce the notations of [13]. Let R+2=x,y:x≥0,y≥0 and Bρ(R+2) be the space of all functions having the property fx,y≤Mfρ(x,y), where (x,y)∈R+2 and Mf is a constant depending on function only. By Cρ(R+2) we denote the subspace of all continuous functions belonging to Bρ(R+2). It is clear that Cρ(R+2) is a linear normed space with the norm fρ=sup(x,y)∈R+2⁡fx,y/ρ(x,y). Also, let Cρ∗(R+2) be the subspace of all functions f∈Cρ(R+2), for which limx,y→∞⁡(f(x,y)/(1+x,y2))=kf<∞, where x,y=x2+y2.Theorem 10.
Letf belong to Cρ∗(R+2) and fx,y≤Mfρ(x,y). Then(55)limn,m→∞⁡Tn,m,anbm,cmf;x,y-fCρ∗R+2=0if and only if(i)
Tn,m,anbm,cm1;x,y-1Cρ∗(R+2)=0;(ii)
Tn,m,anbm,cme10;x,y-xCρ∗(R+2)=0;(iii)
Tn,m,anbm,cme01;x,y-yCρ∗(R+2)=0;(iv)
Tn,m,anbm,cme20;x,y-x2Cρ∗(R+2)=0;(v)
Tn,m,anbm,cme02;x,y-y2Cρ∗(R+2)=0;
asn,m→∞ for (x,y)∈Iab.Proof.
The necessity part is trivial; then we need only to prove sufficiency. Let(x,y),(s,t)∈Iab, and f∈Cρ∗(R+2). Since for each f∈Cρ∗(R+2) is uniformly on Iab, for each ϵ>0 there exists some δ>0, such that for each (s,t)∈Iab with s-x2+t-y2<δ implies |f(s,t)-f(x,y)|<ϵ. Now let (x,y)∈Iab and (s,t)∈R+2 and let (x1,y1) be an arbitrary boundary point of Iab such that 0≤x1≤a,0≤y1≤b. Since f is continuous on the boundary points also, then for each ϵ>0 there exists δ>0 such that s-x2+t-y2<δ implies (56)fs,t-fx,y≤fs,t-fx1,y1+fx1,y1-fx,y<ϵ.On the other hand, if s-x2+t-y2≥δ, we have (57)fs,t-fx,y≤Cs-x2+t-y2δ2,where C>0 is constant and (x,y)∈Iab and (s,t)∈R+2. So, we get the following inequality: (58)fs,t-fx,y≤ϵ+Cs-x2+t-y2δ2,for (x,y)∈Iab,(s,t)∈R+2. Now applying the operators Tn,m,anbm,cm(f;x,y) in the last inequality and taking relations (i)-(v) of Theorem 10, sufficiency is obtained easily.
Now we estimate the rate of approximation of the operatorsTn,m,anbm,cm in the terms of the weighted modulus of continuity Ω(f;δn,δm) (see [14]) defined by (59)Ωf;δn,δm=supx,y∈R+2⁡suph1≤δ1,h2≤δ2⁡fx+h1,y+h2-fx,yρx,yρh1,h2,f∈Cρ∗R+.and it satisfied the following properties: (60)ft,s-fx,y≤8Ωf;δn,δm1+x2+y2gt,xgt,ywhere g(t,x)=1+t-x/δn1+t-x2 and g(s,y)=1+s-y/δm1+s-y2.Theorem 11.
For eachf∈Cρ∗(R+), there exists a positive constant B independent of an,bm, and cm such that the inequality(61)Tn,m,anbm,cmf;x,y-fρ3≤BΩf;δn,δm,is satisfied for a sufficiently large n, m, where δn=an/n and δm=1/cm.Proof.
By the linearity and monotonicity ofTn,m,anbm,cm applied to inequality (60) we obtain(62)Tn,m,anbm,cmf;x,yf;x,y-fx,y≤8ncmanK1LbmyΩf;δn,δm1+x2+y2×∑k=0npn,kxan∫k/nank+1/nangx,tdt×∑j=0∞pjbmy∫j/cmj+1/cmgs,yds.(63)≤8Ωf;δn,δm1+x2+y21+nδnan∑k=0npn,kxan∫k/nank+1/nant-xdt+nδnan∑k=0npn,kxan∫k/nank+1/nant-x2dt+nδnan∑k=0npn,kxan∫k/nank+1/nant-xt-x2dt×1+cmδmK1Lbmy∑j=0∞pjbmy∫j/cmj+1/cms-yds+cmδmK1Lbmy∑j=0∞pjbmy∫j/cmj+1/cms-ys-y2ds+cmK1Lbmy∑j=0∞pjbmy∫j/cmj+1/cms-y2dswhere g(x,t)=1+t-x/δn1+t-x2 and g(s,y)=1+s-y/δm1+s-y2.
Applying Cauchy-Schwarz inequality, we have(64)Tn,m,anbm,cmf;x,y-fx,y≤8Ωf;δn,δm1+x2+y2×1+Δ1δn+δ1+Δ1Δ2δn×1+Δ1∗δn+δ1+Δ1∗Δ2∗δn.where Δ1,Δ1∗Δ2, and Δ2∗ are defined by Lemma 2.
Combining (64) and all identities in Lemma 2, we obtain(65)Tn,m,anbm,cmf;x,y-fx,y≤8Ωf;δn,δm1+x2+y2×1+1δnOann∑i=12xi+Oann∑i=12xi1δnOann∑i=12xiOann∑i=14xi×1+1δmO1cm∑i=12yi+O1cm∑i=12yi+1δmO1cm∑i=12yiO1cm∑i=14yiChoosing δn=an/n and δm=1/cm, and for sufficiently large value of n and m, we obtain (66)Tn,m,anbm,cmf;x,y-fρ3≤BΩf;δn,δm,where B is a constant independent of an,bm, and cm.


## 5. Conclusion

We studied a new sequence generalization of the Szasz-Kantorovich-Chlodowsky type operators defined by means of the Brenke type polynomials defined by (9). This type of modification enables better error estimation for a certain function in comparison to the Szász-Kantorovich-Chlodowsky operators and Szasz-Chlodowsky-type operators. We find the rate of convergence using weighted Korovkin-type theorem. We give some inequalities for these operators means of partial and full modulus of continuity and also obtain a Lipschitz type theorem. At the end, we mentioned results on the weighted modulus of continuity due to Ispir for the operators Tn,m,anbm,cm.



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*Source: 3658389-2018-12-02.xml*