# Differences of Positive Linear Operators on Simplices

Authors: Ana-Maria Acu; Gülen Başcanbaz-Tunca; Ioan Rasa
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The aim of the paper is twofold: we introduce new positive linear operators acting on continuous functions defined on a simplex and then estimate differences involving them and/or other known operators. The estimates are given in terms of moduli of smoothness andK-functionals. Several applications and examples illustrate the general results.


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## Body

## 1. Introduction

Differences of positive linear operators were intensively investigated in the last years; see [1–14] and the references therein. The operators involved in these studies act usually on continuous functions defined on real intervals, and the differences are estimated in terms of moduli of smoothness and K-functionals. In some papers, operators having equal central moments up to a certain order are considered. Other articles deal with operators constructed with the same fundamental functions and different functionals in front of them.The study of differences of positive linear operators is important from a theoretical point of view, but also from a practical one. LetUn and Vn be certain positive linear operators. If we know that Unf−Vnf is small, we can choose Un or Vn taking into account other qualities of them like shape-preserving properties and smoothness/Lipschitz preserving properties.This paper is concerned with differences of positive linear operators acting on continuous functions defined on simplices. For the sake of simplicity, we consider only the case of the canonical simplex inℝ2, where the notation is simpler, but the results can be easily translated to an arbitrary simplex in ℝn.We consider the bivariate versions of some classical operators like Bernstein, Durrmeyer, Kantorovich, and genuine Bernstein-Durrmeyer operators. These bivariate versions were already studied in literature from other points of view. We introduce the bivariate versions of other operators:Unρ (see [15, 16]) and the operators defined in [17]. All these operators are constructed with the fundamental Bernstein polynomials on the two-dimensional simplex. A different kind of operator is the bivariate version of the univariate Beta operator of Mühlbach and Lupas (see [18–20]); we introduce it and use it in composition with the Bernstein operator to get a useful representation of Unρ.We get estimates of differences of the abovementioned operators, in terms of suitable moduli of smoothness andK-functionals.To resume, the aim of our paper is twofold: we introduce new operators on a simplex and then estimate differences involving them and other known operators.The list of applications and examples can be enlarged. In particular, we will be interested for a future work in studying differences of bivariate versions of operators, which preserve exponential functions (see [21–23]). We also intend to deepen the study of the newly introduced Beta operators on the simplex and to consider the composition of it with other operators, leading to new applications and-why not-new theoretical aspects/problems. Given a Markov operator (i.e., a positive linear operator which preserves the constant functions), the study of its iterate is important not only in Approximation Theory but also in Ergodic Theory and other areas of research. We intend to investigate from this point of view the newly introduced operators, which are in fact Markov operators.We end this Introduction by presenting some notation and a fundamental inequality expressed in Lemma1. Section 2 contains the main theoretical results, while Section 3 is devoted to applications and examples.LetS≔x,y∈ℝ2x,y≥0,x+y≤1 be the canonical simplex in ℝ2 and ES denote a space of real-valued continuous functions of two variables defined on S, containing the polynomials. Throughout the paper, we will denote by 1 the constant function, namely,
(1)1:S⟶ℝ,1x,y=1,x,y∈S,and pri:S⟶ℝ,i=1,2,will denote the ith coordinate functions restricted on S, which are given by
(2)pr1x,y=xandpr2x,y=y,x,y∈S.LetF:ES⟶ℝ be a positive linear functional such that F1=1. Set
(3)b1F≔Fpr1,b2F≔Fpr2,μi,jF≔Fpr1−b1F1ipr2−b2F1j,i,j∈ℕ.Then, one has(4)μ1,0F=0,μ2,0F=Fpr12−b1F2≥0,μ0,1F=0,μ0,2F=Fpr22−b2F2≥0.LetC2S be the space of all real-valued (continuous) functions, differentiable on intS and whose partial derivatives of order ≤2 can be continuously extended to S, having
(5)f=supfx,y:x,y∈S<∞.Lemma 1.
Iff∈C2S, then
(6)Ff−fb1F,b2F≤Mfμ2,0F+μ0,2F,whereMf≔maxfxx,fxy,fyy.Proof.
Consider the line segment connectingb1F,b2F with t1,t2∈S. From Taylor’s formula (see [24], p.245), there is a point c1,c2 on this line segment, different from b1F,b2F and t1,t2, such that
(7)ft1,t2=fb1F,b2F+fxb1F,b2Ft1−b1F+fyb1F,b2Ft2−b2F+12fxxc1,c2t1−b1F2+2fxyc1,c2t1−b1Ft2−b2F+fyyc1,c2t2−b2F2.
Therefore, we can write(8)f−fb1F,b2F1−fxb1F,b2Fpr1−b1F1−fyb1F,b2Fpr2−b2F1≤12fxxpr1−b1F12+2fxypr1−b1F1pr2−b2F1+fyypr2−b2F12≤12fxx+fxypr1−b1F12+fxy+fyypr2−b2F12≤Mfpr1−b1F12+pr2−b2F12,which gives the result.


## 2. Difference of Bivariate Positive Linear Operators

Denote byCS the space of real-valued continuous functions onS with the norm f=maxx,y∈Sfx,y,f∈CS. Let K be a set of nonnegative integers and for k,l∈K let pk,l∈CS,pk,l≥0, satisfy ∑k,l∈Kpk,l=1. Let Fk,l:ES⟶ℝ and Gk,l:ES⟶ℝ,k,l∈K, be positive linear functionals such that Fk,l1=1 and Gk,l1=1. Moreover, let DS be the set of all f∈ES for which
(9)∑k,l∈KFk,lfpk,l∈CS,∑k,l∈KGk,lfpk,l∈CS.Now, consider the bivariate positive linear operatorsV and W acting from DS into CS defined, for f∈DS,by
(10)Vfx,y=∑k,l∈KFk,lfpk,lx,y,Wfx,y=∑k,l∈KGk,lfpk,lx,y,respectively. For future correspondences, we denote
(11)σx,y≔∑k,l∈Kμ2,0Fk,l+μ2,0Gk,l+μ0,2Fk,l+μ0,2Gk,lpk,lx,y,(12)δ≔supk,l∈Kb1Fk,l,b2Fk,l−b1Gk,l,b2Gk,l,where · is the l1-norm in ℝ2.In the following, we adopt the definitions ofK-functional and modulus of smoothness from [25, 26]. Let
(13)Sh=x∈Sx+ht∈Sfor0≤t≤1,h∈ℝ2.Forr∈ℕ, rth order differences on the subset Srh are defined as
(14)Δhrfx=∑k=0r−1r−krkfx+kh.Therth order modulus of smoothness of f is a function ωr:CS×0,∞⟶0,∞ given by
(15)ωrf,α=sup0<h≤αΔhrf,α>0.LetCrS be the space of all real-valued (continuous) functions, differentiable on intS and whose partial derivatives of order ≤r can be continuously extended to S, with the seminorm
(16)gCrS=∑γ1+γ2=r∂rg∂xγ1∂yγ2<∞,γi≥0,i=1,2,γ1+γ2=r.Forf∈CS, we shall use the following K-functional:
(17)Krf,t=inff−g+tgCrS:g∈CrS.Then, there existc1,c2>0 such that for any t>0 (see [25, 26])
(18)c1Krf,tr≤ωrf,t≤c2Krf,tr.Here,c2 depends only on r (for the general definition onthe Lp,1≤p≤∞, spaces of functions on bounded domains, see [25] or, on unbounded domains see [27], p.341.Theorem 2.
Iff∈DS∩C2S, then
(19)V−Wfx,y≤Mfσx,y+ω1f,δ,where Mf is defined in Lemma 1.Proof.
Letx,y∈S. From Lemma 1, we get
(20)V−Wfx,y≤∑k,l∈KFk,lf−Gk,lfpk,lx,y≤∑k,l∈Kpk,lx,yFk,lf−fb1Fk,l,b2Fk,l+Gk,lf−fb1Gk,l,b2Gk,l+fb1Fk,l,b2Fk,l−fb1Gk,l,b2Gk,l≤Mf∑k,l∈Kpk,lx,yμ2,0Fk,l+μ0,2Fk,l+μ2,0Gk,l+μ0,2Gk,l+ω1f;b1Fk,l,b2Fk,l−b1Gk,l,b2Gk,l≤Mfσx,y+ω1f,δ.Theorem 3.
Iff∈CS, then
(21)V−Wfx,y≤η1ω1f,δ+η2ω2f,σx,y,where η1,η2>0, and δ=supk,l∈Kb1Fk,l−b1Gk,l+b2Fk,l−b2Gk,l.Proof.
Letg∈C2S. From Theorem 2, we get
(22)V−Wfx,y≤Vf−gx,y+Wg−fx,y+V−Wgx,y≤2f−g+Mgσx,y+gb1Fk,l,b2Fk,l−gb1Gk,l,b2Gk,l,where Mg is the same notation as in Lemma 1 for g. Since partial derivatives of g exist and are continuous everywhere in S, it follows that gis differentiable at every point of the line segment connecting the points b1Fk,l,b2Fk,l and b1Gk,l,b2Gk,l in S, k,l∈K. By the mean value theorem (see, e.g., [24], p. 239), there is a point a1,a2on this line segment such that
(23)gb1Fk,l,b2Fk,l−gb1Gk,l,b2Gk,l=gxa1,a2b1Fk,l−b1Gk,l+gya1,a2b2Fk,l−b2Gk,l.
From (16), we get
(24)gb1Fk,l,b2Fk,l−gb1Gk,l,b2Gk,l≤gxb1Fk,l−b1Gk,l+gyb2Fk,l−b2Gk,l≤gC1Ssupk,l∈Kb1Fk,l−b1Gk,l+b2Fk,l−b2Gk,l≤δgC1S,k,l∈K.
Moreover, sinceMg≤gC2S, (22) gives that
(25)V−Wfx,y≤2f−g+δgC1S+σx,ygC2S≤K1f,δ+K2f,σx,y.
Finally, from (18), we obtain
(26)V−Wfx,y≤η1ω1f,δ+η2ω2f,σx,y.


## 3. Applications

### 3.1. Difference of Bivariate Bernstein Operators and Their Durrmeyer Variants

For everyn≥1,f∈CS, and x,y∈S, the nth bivariate Bernstein operator Bn:CS⟶CS is defined by
(27)Bnfx,y=∑k,l=0,⋯,nk+l≤nfkn,lnpn,k,lx,y,where
(28)pn,k,lx,y≔n!k!l!n−k−l!xkyl1−x−yn−k−l,with k,l=0,⋯,n,k+l≤n,x,y∈S, (see, e.g., [28], p. 115).Forf∈L1S, the bivariate Durrmeyer operators Mn:L1S⟶CS are defined by
(29)Mnfx,y=∑k,l=0,⋯,nk+l≤nn+1n+2∫01∫01−spn,k,ls,tfs,tdtdspn,k,lx,y,see, e.g., [29].Now, denoting(30)Fn,k,lf≔fkn,ln,0≤k+l≤n,Gn,k,lf≔n+1n+2∫01∫01−spn,k,ls,tfs,tdtds,the bivariate Bernstein operators and bivariate Durrmeyer operators can be written as
(31)Bnfx,y=∑k,l=0,⋯,nk+l≤nFn,k,lfpn,k,lx,y,Mnfx,y=∑k,l=0,⋯,nk+l≤nGn,k,lfpn,k,lx,y,respectively.Proposition 4.
For bivariate Bernstein operators and their Durrmeyer variants, the following properties hold:(i)
Iff∈C2S, then(32)Bn−Mnfx,y≤Mfσx,y+ω1f,3n+3,where Mf is the same as in Lemma 1 and
(33)σx,y=−x2−y2+x+yn2+x2+y2+2n+4n+32n+4≤1n+4.(ii)
Iff∈CS, then(34)Bn−Mnfx,y≤η1ω1f,3n+3+η2ω2f,σx,y.Proof.
We need to evaluate the terms in (11). So, we get the following results:
(35)b1Fn,k,l=kn,b2Fn,k,l=ln,b1Gn,k,l=k+1n+3,b2Gn,k,l=l+1n+3,
0≤k+l≤n. Therefore, we easily obtain that
(36)μ2,0Fn,k,l=0,μ2,0Gn,k,l=k+1n+2−kn+32n+4,μ0,2Fn,k,l=0,μ0,2Gn,k,l=l+1n+2−ln+32n+4.
Using Maple, one obtains(37)σx,y=∑k,l=0,⋯,nk+l≤nk+1n+2−kn+32n+4+l+1n+2−ln+32n+4pn,k,lx,y=−x2−y2+x+yn2+x2+y2+2n+4n+32n+4.
It is easy to verify thatσx,y≤1/n+4.
Now, forδ, we obtain
(38)δ=max0≤k+l≤nb1Fn,k,l−b1Gn,k,l+b2Fn,k,l−b2Gn,k,l=max0≤k+l≤nn−3knn+3+n−3lnn+3=3n+3.
The rest of the proof follows from Theorems2 and 3.


### 3.2. Difference of Bivariate Bernstein Operators and the Bivariate OperatorsAn

LetΠn be the space of polynomials over 0,1 of degree at most n. In [17], Aldaz et al. introduced a Bernstein operator An:C0,1⟶Πn that fixes 1 and x2. The operators An are given by
(39)Anfx=∑k=0nfkk−1nn−11/2nkxk1−xn−k.Here, forf∈CS and x,y∈S, we introduce the bivariate form of the operators An as follows
(40)Anfx,y=∑k,l=0,⋯,nk+l≤nfkk−1nn−1,ll−1nn−1pn,k,lx,y.Denoting(41)Fn,k,lf=fkn,ln,Gn,k,lf=fkk−1nn−1,ll−1nn−1,for k,l=0,⋯,n, k+l≤n, we get
(42)b1Fn,k,l=kn,b2Fn,k,l=ln,b1Gn,k,l=kk−1nn−1,b2Gn,k,l=ll−1nn−1,μ2,0Fn,k,l=μ0,2Fn,k,l=μ2,0Gn,k,l=μ0,2Gn,k,l=0,δ=maxk,l=0,⋯,nk+l≤nb1Fn,k,l−b1Gn,k,l+b2Fn,k,l−b2Gn,k,l=2n.Proposition 5.
For bivariate Bernstein operators and bivariate operatorsAn, the following properties hold:
(i)
Iff∈C2S, then(43)Bn−Anfx,y≤ω1f,2n.(ii)
Iff∈CS, then(44)Bn−Anfx,y≤η1ω1f,2n.


### 3.3. Difference of Bivariate Bernstein Operators and Bivariate Genuine Bernstein-Durrmeyer Operators

In 1987, Chen [30] and Goodman and Sharma [31] constructed the following positive linear operators
(45)Un,1fx=f0pn,0x+f1pn,nx+∑k=1n−1pn,kxn−1∫01pn−2,k−1tftdt,where n∈ℕ,f∈C0,1, and
(46)pn,kx=nkxk1−xn−k,x∈0,1,0≤k≤n.For the historical background of these operators, we refer to [32]. In 1991, Goodman and Sharma [33] constructed and studied the multivariate form of the operators Un,1on a simplex. In [34], Sauer deeply studied the multivariate genuine Bernstein-Durrmeyer operators. Here, for f∈L1S, we consider the bivariate form given by
(47)Unfx,y=f0,01−x−yn+f1,0xn+f0,1yn+∑l=1n−1pn,0,lx,yn−1∫01pn−2,l−1tf0,tdt+∑k=1n−1pn,k,0x,yn−1∫01pn−2,k−1sfs,0ds+∑k=1n−1pn,k,n−kx,yn−1∫01pn−2,k−1tft,1−tdt+∑k+l≤n−1k≥1,l≥1pn,k,lx,yn−1n−2∫01∫01−tpn−3,k−1,l−1s,tfs,tdsdtwith the bivariate Bernstein’s fundamental functions given by (28) (see [33], Formula 1.7). These operators satisfy Unfx,y=fx,yat the vertices of S.Proposition 6.
For bivariate Bernstein operators and bivariate genuine Bernstein-Durrmeyer operators, the following properties hold:(i)
Iff∈C2S, then(48)Bn−Unfx,y≤Mfσx,y,where Mf is the same as in Lemma 1 and
(49)σx,y=x+y−x2−y2n−1nn+1≤12n+1.(ii)
Iff∈CS, then(50)Bn−Unfx,y≤η2ω2f;σx,y.Proof.
If we denote(51)Fn,k,lf≔fkn,ln,0≤k+l≤n,Gn,0,0f=f0,0,k=l=0,Gn,n,0f=f1,0,k=n,l=0,Gn,0,nf=f0,1,k=0,l=n,Gn,k,0f=n−1∫01pn−2,k−1sfs,0ds,1≤k≤n−1,l=0,Gn,0,lf=n−1∫01pn−2,l−1tf0,tdt,k=0,1≤l≤n−1,Gn,k,n−kf=n−1∫01pn−2,k−1tft,1−tdt,1≤k≤n−1,l=n−k,Gn,k,lf=n−1n−2∫01∫01−tpn−3,k−1,l−1s,tfs,tdsdt,1≤k+l≤n−1,then for the bivariate Bernstein operators, we have
(52)Bnfx,y=∑k,l=0,⋯,nk+l≤nFn,k,lfpn,k,lx,y.
The bivariate genuine Bernstein-Durrmeyer operators are given by(53)Unfx,y=∑k,l=0,⋯,nk+l≤nGn,k,lfpn,k,lx,y.
Now, for0≤k+l≤n, we get
(54)b1Fn,k,l=kn,b2Fn,k,l=ln,b1Gn,k,l=kn,b2Gn,k,l=ln.
Hence, we obtain(55)μ2,0Fn,k,l=0,μ2,0Gn,k,l=kn−kn2n+1,μ0,2Fn,k,l=0,μ0,2Gn,k,l=ln−ln2n+1.
Therefore,δ=0 and
(56)σx,y=∑k,l=0,⋯,nk+l≤nkn−kn2n+1+ln−ln2n+1pn,k,lx,y=x+y−x2−y2n−1nn+1,x,y∈S.
The proof is concluded by using Theorems2 and 3.


## 3.1. Difference of Bivariate Bernstein Operators and Their Durrmeyer Variants

For everyn≥1,f∈CS, and x,y∈S, the nth bivariate Bernstein operator Bn:CS⟶CS is defined by
(27)Bnfx,y=∑k,l=0,⋯,nk+l≤nfkn,lnpn,k,lx,y,where
(28)pn,k,lx,y≔n!k!l!n−k−l!xkyl1−x−yn−k−l,with k,l=0,⋯,n,k+l≤n,x,y∈S, (see, e.g., [28], p. 115).Forf∈L1S, the bivariate Durrmeyer operators Mn:L1S⟶CS are defined by
(29)Mnfx,y=∑k,l=0,⋯,nk+l≤nn+1n+2∫01∫01−spn,k,ls,tfs,tdtdspn,k,lx,y,see, e.g., [29].Now, denoting(30)Fn,k,lf≔fkn,ln,0≤k+l≤n,Gn,k,lf≔n+1n+2∫01∫01−spn,k,ls,tfs,tdtds,the bivariate Bernstein operators and bivariate Durrmeyer operators can be written as
(31)Bnfx,y=∑k,l=0,⋯,nk+l≤nFn,k,lfpn,k,lx,y,Mnfx,y=∑k,l=0,⋯,nk+l≤nGn,k,lfpn,k,lx,y,respectively.Proposition 4.
For bivariate Bernstein operators and their Durrmeyer variants, the following properties hold:(i)
Iff∈C2S, then(32)Bn−Mnfx,y≤Mfσx,y+ω1f,3n+3,where Mf is the same as in Lemma 1 and
(33)σx,y=−x2−y2+x+yn2+x2+y2+2n+4n+32n+4≤1n+4.(ii)
Iff∈CS, then(34)Bn−Mnfx,y≤η1ω1f,3n+3+η2ω2f,σx,y.Proof.
We need to evaluate the terms in (11). So, we get the following results:
(35)b1Fn,k,l=kn,b2Fn,k,l=ln,b1Gn,k,l=k+1n+3,b2Gn,k,l=l+1n+3,
0≤k+l≤n. Therefore, we easily obtain that
(36)μ2,0Fn,k,l=0,μ2,0Gn,k,l=k+1n+2−kn+32n+4,μ0,2Fn,k,l=0,μ0,2Gn,k,l=l+1n+2−ln+32n+4.
Using Maple, one obtains(37)σx,y=∑k,l=0,⋯,nk+l≤nk+1n+2−kn+32n+4+l+1n+2−ln+32n+4pn,k,lx,y=−x2−y2+x+yn2+x2+y2+2n+4n+32n+4.
It is easy to verify thatσx,y≤1/n+4.
Now, forδ, we obtain
(38)δ=max0≤k+l≤nb1Fn,k,l−b1Gn,k,l+b2Fn,k,l−b2Gn,k,l=max0≤k+l≤nn−3knn+3+n−3lnn+3=3n+3.
The rest of the proof follows from Theorems2 and 3.


## 3.2. Difference of Bivariate Bernstein Operators and the Bivariate OperatorsAn

LetΠn be the space of polynomials over 0,1 of degree at most n. In [17], Aldaz et al. introduced a Bernstein operator An:C0,1⟶Πn that fixes 1 and x2. The operators An are given by
(39)Anfx=∑k=0nfkk−1nn−11/2nkxk1−xn−k.Here, forf∈CS and x,y∈S, we introduce the bivariate form of the operators An as follows
(40)Anfx,y=∑k,l=0,⋯,nk+l≤nfkk−1nn−1,ll−1nn−1pn,k,lx,y.Denoting(41)Fn,k,lf=fkn,ln,Gn,k,lf=fkk−1nn−1,ll−1nn−1,for k,l=0,⋯,n, k+l≤n, we get
(42)b1Fn,k,l=kn,b2Fn,k,l=ln,b1Gn,k,l=kk−1nn−1,b2Gn,k,l=ll−1nn−1,μ2,0Fn,k,l=μ0,2Fn,k,l=μ2,0Gn,k,l=μ0,2Gn,k,l=0,δ=maxk,l=0,⋯,nk+l≤nb1Fn,k,l−b1Gn,k,l+b2Fn,k,l−b2Gn,k,l=2n.Proposition 5.
For bivariate Bernstein operators and bivariate operatorsAn, the following properties hold:
(i)
Iff∈C2S, then(43)Bn−Anfx,y≤ω1f,2n.(ii)
Iff∈CS, then(44)Bn−Anfx,y≤η1ω1f,2n.


## 3.3. Difference of Bivariate Bernstein Operators and Bivariate Genuine Bernstein-Durrmeyer Operators

In 1987, Chen [30] and Goodman and Sharma [31] constructed the following positive linear operators
(45)Un,1fx=f0pn,0x+f1pn,nx+∑k=1n−1pn,kxn−1∫01pn−2,k−1tftdt,where n∈ℕ,f∈C0,1, and
(46)pn,kx=nkxk1−xn−k,x∈0,1,0≤k≤n.For the historical background of these operators, we refer to [32]. In 1991, Goodman and Sharma [33] constructed and studied the multivariate form of the operators Un,1on a simplex. In [34], Sauer deeply studied the multivariate genuine Bernstein-Durrmeyer operators. Here, for f∈L1S, we consider the bivariate form given by
(47)Unfx,y=f0,01−x−yn+f1,0xn+f0,1yn+∑l=1n−1pn,0,lx,yn−1∫01pn−2,l−1tf0,tdt+∑k=1n−1pn,k,0x,yn−1∫01pn−2,k−1sfs,0ds+∑k=1n−1pn,k,n−kx,yn−1∫01pn−2,k−1tft,1−tdt+∑k+l≤n−1k≥1,l≥1pn,k,lx,yn−1n−2∫01∫01−tpn−3,k−1,l−1s,tfs,tdsdtwith the bivariate Bernstein’s fundamental functions given by (28) (see [33], Formula 1.7). These operators satisfy Unfx,y=fx,yat the vertices of S.Proposition 6.
For bivariate Bernstein operators and bivariate genuine Bernstein-Durrmeyer operators, the following properties hold:(i)
Iff∈C2S, then(48)Bn−Unfx,y≤Mfσx,y,where Mf is the same as in Lemma 1 and
(49)σx,y=x+y−x2−y2n−1nn+1≤12n+1.(ii)
Iff∈CS, then(50)Bn−Unfx,y≤η2ω2f;σx,y.Proof.
If we denote(51)Fn,k,lf≔fkn,ln,0≤k+l≤n,Gn,0,0f=f0,0,k=l=0,Gn,n,0f=f1,0,k=n,l=0,Gn,0,nf=f0,1,k=0,l=n,Gn,k,0f=n−1∫01pn−2,k−1sfs,0ds,1≤k≤n−1,l=0,Gn,0,lf=n−1∫01pn−2,l−1tf0,tdt,k=0,1≤l≤n−1,Gn,k,n−kf=n−1∫01pn−2,k−1tft,1−tdt,1≤k≤n−1,l=n−k,Gn,k,lf=n−1n−2∫01∫01−tpn−3,k−1,l−1s,tfs,tdsdt,1≤k+l≤n−1,then for the bivariate Bernstein operators, we have
(52)Bnfx,y=∑k,l=0,⋯,nk+l≤nFn,k,lfpn,k,lx,y.
The bivariate genuine Bernstein-Durrmeyer operators are given by(53)Unfx,y=∑k,l=0,⋯,nk+l≤nGn,k,lfpn,k,lx,y.
Now, for0≤k+l≤n, we get
(54)b1Fn,k,l=kn,b2Fn,k,l=ln,b1Gn,k,l=kn,b2Gn,k,l=ln.
Hence, we obtain(55)μ2,0Fn,k,l=0,μ2,0Gn,k,l=kn−kn2n+1,μ0,2Fn,k,l=0,μ0,2Gn,k,l=ln−ln2n+1.
Therefore,δ=0 and
(56)σx,y=∑k,l=0,⋯,nk+l≤nkn−kn2n+1+ln−ln2n+1pn,k,lx,y=x+y−x2−y2n−1nn+1,x,y∈S.
The proof is concluded by using Theorems2 and 3.


## 4. The DifferenceUnρ−Unr

Letρ>0 and n∈ℕ. The operators Un,1ρ:C0,1⟶∏n are introduced by Paltanea in [35] (see also [15, 16]). These operators are defined by
(57)Un,1ρf;x≔∑k=1n−1∫01tkρ−11−tn−kρ−1βkρ,n−kρftdtpn,kx+f01−xn+f1xn,where f∈C0,1, x∈0,1, and β·,· are Euler’s Beta function.Here, forf∈L1S, we consider the bivariate form of these operators, given by
(58)Unρfx,y=f0,01−x−yn+f1,0xn+f0,1yn+∑l=1n−1Fn,0,lρfpn,0,lx,y+∑k=1n−1Fn,k,0ρfpn,k,0x,y+∑k=1n−1Fn,k,n−kρfpn,k,n−kx,y+∑k≥1,l≥1k+l≤n−1,Fn,k,lρfpn,k,lx,y,where
(59)Fn,0,lρf≔∫01tlρ−11−tn−lρ−1f0,tdtBlρ,n−lρ,Fn,k,0ρf≔∫01skρ−11−sn−kρ−1fs,0dsBkρ,n−kρ,Fn,k,n−kρf≔∫01tkρ−11−tn−kρ−1ft,1−tdtBkρ,n−kρ,Fn,k,lρf≔∬Sskρ−1tlρ−11−s−tn−k−lρ−1fs,tdsdt∬Sskρ−1tlρ−11−s−tn−k−lρ−1dsdt.It can be easily seen that, forρ=1, we obtain the genuine Bernstein-Durrmeyer operators Un. On the other hand, these operators have the following limiting behavior.Theorem 7.
For anyf∈CS, one has limρ⟶∞Unρf=Bnf,uniformly.Proof.
Letf=pr1j, j=0,1,⋯. Then,
(60)Fn,k,lρpr1j=kρ+j−1⋯kρnρ+j−1⋯nρ,fork,l≥1,limρ⟶∞Fn,k,lρpr1j=knj=pr1jkn,ln.
Since(61)Fn,0,lρpr1j=1,j=0,0,j>0,Fn,k,0ρpr1j=Fn,k,n−kρpr1j=kρ+j−1⋯kρnρ+j−1⋯nρ,we get
(62)limρ⟶∞Fn,0,lρpr1j=pr1j0,ln,limρ⟶∞Fn,k,0ρpr1j=knj=pr1jkn,0,limρ⟶∞Fn,k,n−kρpr1j=knj=pr1jkn,n−kn.
Similar results can be obtained forpr2j,j=0,1,⋯.
Using Korovkin’s theorem (see [36], p. 534, C.4.3.3), it follows limρ⟶∞Fn,μ,νρf=fμ/n,ν/n. Therefore,
(63)limρ⟶∞Unρf=Bnf,f∈CS.Proposition 8.
For the bivariate operatorsUnρ, the following properties hold:
(i)
Iff∈C2S, then(64)Unρ−Unrfx,y≤Mfσx,y,
whereMf is the same as in Lemma 1 and
(65)σx,y=nr+nρ+2x+y−x2−y2n−1nnρ+1nr+1.(ii)
Iff∈CS, then(66)Unρ−Unrfx,y≤η2ω2f;σx,y.Proof.
Sinceb1Fn,k,lρ=k/n, b1Fn,k,lρ=l/n, we get
(67)μ2,0Fn,k,lρ=kn−kn2nρ+1,μ0,2Fn,k,lρ=ln−ln2nρ+1.
Therefore,δ=0 and
(68)σx,y=nr+nρ+2x+y−x2−y2n−1nnρ+1nr+1.


## 5. Difference of Bivariate Bernstein Operators and Their Kantorovich Variants

In 2017, F. Altomare et al. [37] introduced Kantorovich operators on S as follows
(69)ℂnfx,y=∑k,l=0,⋯,nk+l≤npn,k,lx,y2∬Sfk+asn+a,l+atn+adsdt,where pn,k,lx,yis given by (28). It can be easily seen that, for a=1, we obtain Kantorovich operators Kn introduced in [38].If we denote(70)Gk,lf≔2∬Sfk+asn+a,l+atn+adsdt,the bivariate Kantorovich operators can be written as
(71)ℂnfx,y=∑k,l=0,⋯,nk+l≤nGk,lfpn,k,lx,y.Proposition 9.
For bivariate Bernstein operators and bivariate Bernstein-Kantorovich operators, the following properties hold:(i)
Iff∈C2S, then(72)Bn−ℂnfx,y≤Mfσx,y+ω1f,4a3n+a,where Mf is the same as in Lemma 1 and
(73)σx,y=a29n+a2.(ii)
Iff∈CS, then(74)Bn−ℂnfx,y≤η1ω1f,4a3n+a+η2ω2f,σx,y.Proof.
As in the previous examples, taking Bernstein operators as(75)Bnfx,y=∑k,l=0,⋯,nk+l≤nFk,lfpn,k,lx,y,we get
(76)b1Fk,l=kn,b2Fk,l=ln,b1Gk,l=3k+a3n+a,b2Gk,l=3l+a3n+a.
Therefore, we easily obtain that(77)μ2,0Fk,l=0,μ2,0Gk,l=a218n+a2,μ0,2Fk,l=0,μ0,2Gk,l=a218n+a2.
Then(78)σx,y=∑k,l=0,⋯,nk+l≤na29n+a2pn,k,lx,y=a29n+a2.
Moreover, we have(79)δ=max0≤k+l≤nb1Fk,l−b1Gk,l+b2Fk,l−b2Gk,l=max0≤k+l≤na3k−n3nn+a+a3l−n3nn+a≤4a3n+a.
Then, the proof follows from Theorems2 and 3.


## 6. A Beta Operator onCS

Forρ∈0,∞, f∈CS, and x,y∈S, let us define
(80)Bρfx,y=fx,y,x,y∈0,0,1,0,0,1,∫01sρx−11−sρ1−x−1fs,0dsBρx,ρ1−x,x∈0,1,y=0,∫01tρy−11−tρ1−y−1f0,tdtBρy,ρ1−y,x=0,y∈0,1,∫01uρx−11−uρ1−x−1fu,1−uduBρx,ρ1−x,y=1−x,x∈0,1,∬Ssρx−1tρy−11−s−tρ−ρx−ρy−1fs,tdsdt∬Ssρx−1tρy−11−s−tρ−ρx−ρy−1dsdt,x,y∈intS.Forρ=n∈ℕ, this is the bivariate version of the operator B¯n; see [39] and the references therein.Theorem 10.
Bρ is a positive linear operator acting between CS and CS. Moreover,
(81)Bρ1=1,and if φi,jx,y=pr1ipr2jx,y, x,y∈S, i≥0, j≥0, integers, then
(82)Bρφi,jx,y=ρxρx+1⋯ρx+i−1ρρ+1⋯ρ+i−1·ρyρy+1⋯ρy+j−1ρρ+1⋯ρ+j−1.Proof.
It is easy to prove (81) and (82) by direct calculation. It remains to prove that if f∈CS, then, Bρf∈CS. To do this, it suffices to verify that Bρf is continuous at each point of the boundary of S. Let us prove that if 0<a<1 then
(83)limx,y⟶a,0x,y∈intSBρfx,y=Bρfa,0.
LetVa,0:CS⟶ℝ, Va,0g=Bρga,0. For x,y∈intS define Ux,y:CS⟶ℝ, Ux,yg=Bρgx,y, g∈CS. Then, Ux,y and Va,0 are positive linear functionals of norm 1.
Letε>0. Then, there exists a polynomial function p on S such that f−p∞≤ε/4. Using (82), it is easy to verify that
(84)limx,y⟶a,0x,y∈intSUx,yp=Va,0p.
Consequently, there existsδ>0 with
(85)Ux,yp−Va,0p≤ε2,x,y−a,01≤δ.
So, ifx,y−a,01≤δ, we have
(86)Ux,yf−Va,0f≤Ux,yf−Ux,yp+Ux,yp−Va,0p+Va,0p−Va,0f≤f−p∞+ε2+p−f∞≤ε.
This shows that(87)limx,y⟶a,0x,y∈intSUx,yf=Va,0f,f∈CS,and then (83) is proved.
The continuity ofBρf at the other boundary points can be proved similarly.Proposition 11.
For eachf∈CS, one has
(88)limρ⟶∞Bρf=f.Proof.
Using Theorem10, it is easy to verify that (88) is valid for the functions 1,pr1,pr2,pr12+pr22. But these functions form a Korovkin test system (see [36], p. 534, C.4.3.3), so that (88) holds for each f∈CS.In what follows, we formulate aConjecture 12.
Iff∈CS is convex and x,y∈S, then, the function ρ⟶Bρfx,y is decreasing on 0,∞.
It is supported by the following facts.(i)
The unidimensional version of the conjecture is valid: see [14, 40].(ii)
Bρ is a positive linear operator preserving the affine functions; this implies(89)Bρf≥f,f∈CSconvex.
Now, (88) combined with (89) support the conjecture.
(iii)
The conjecture is valid for the functionspr1k,pr2k,1−pr1−pr2k,k∈ℕ
In the sequel, we present two results under the hypothesis that the conjecture is true. To this end, let us introduce some notation.
Letf∈C2S and
(90)m1f≔minfxxx,y−fxyx,y:x,y∈S,m2f≔minfyyx,y−fxyx,y:x,y∈S,M1f≔maxfxxx,y+fxyx,y:x,y∈S,M2f≔maxfyyx,y+fxyx,y:x,y∈S.
Then, the functionsφx,y≔fx,y−1/2m1fx2−1/2m2fy2 and ψx,y≔1/2M1fx2+1/2M2fy2−fx,y are convex on S; indeed, for each of them, the Hessian matrix is positive semidefinite.Theorem 13.
Iff∈C2S and σ>ρ>0, then
(91)σ−ρ2σ+1ρ+1m1fx1−x+m2fy1−y≤Bρfx,y−Bσfx,y≤σ−ρ2σ+1ρ+1M1fx1−x+M2fy1−y,x,y∈S.Proof.
Letσ>ρ>0. If the Conjecture is true, we have Bρφ≥Bσφ and Bρψ≥Bσψ. Thus
(92)0≤Bρφx,y−Bσφx,y=Bρfx,y−Bσfx,y+x2m1fρ−σ1−xσ+1ρ+1+y2m2fρ−σ1−yσ+1ρ+1.
Consequently,(93)Bρfx,y−Bσfx,y≥σ−ρ2σ+1ρ+1m1fx1−x+m2fy1−y.
Moreover,(94)0≤Bρψx,y−Bσψx,y=x2M1fσ−ρ1−xσ+1ρ+1+y2M2fσ−ρ1−yσ+1ρ+1−Bρfx,y+Bσfx,y.
Now,(95)Bρfx,y−Bσfx,y≤σ−ρ2σ+1ρ+1M1fx1−x+M2fy1−y.
So, combining (93) and (95), we have proved the theorem.Theorem 14.
Iff∈C2S and σ>ρ>0, then
(96)n−1σ−ρ2nσ+1nρ+1m1fx1−x+m2fy1−y≤Unρfx,y−Unσfx,y≤n−1σ−ρ2nσ+1nρ+1M1fx1−x+M2fy1−y,x,y∈S.Proof.
It is easy to verify thatUnρ=Bn∘Bnρ and, if u≔pr12, then
(97)Bnρux,y=n−1ρx2+ρ+1xnρ+1.
Using these facts and supposing that the conjecture is true, we have(98)Unρφ≥UnσφandUnρψ≥Unσψ,σ>ρ>0.
Now,(99)0≤Unρφx,y−Unσφx,y=Unρfx,y−Unσfx,y−n−1σ−ρ2nσ+1nρ+1x1−xm1f−n−1σ−ρ2nσ+1nρ+1y1−ym2f.
Thus,(100)Unρfx,y−Unσfx,y≥n−1σ−ρ2nσ+1nρ+1m1fx1−x+m2fy1−y.
FromUnρψ≥Unσψ, we get a similar upper bound for Unρfx,y−Unσfx,y, which concludes the proof.



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