Differences of Positive Linear Operators on Simplices

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 and 
 
 K
 
 -functionals. Several applications and examples illustrate the general results.


Introduction
Differences of positive linear operators were intensively investigated in the last years; see [1][2][3][4][5][6][7][8][9][10][11][12][13][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. Let ðU n Þ and ðV n Þ be certain positive linear operators. If we know that jU n ð f Þ − V n ð f Þj is small, we can choose ðU n Þ or ðV n Þ 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 in ℝ 2 , where the notation is simpler, but the results can be easily translated to an arbitrary simplex in ℝ n .
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: U ρ n (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][19][20]); we introduce it and use it in composition with the Bernstein operator to get a useful representation of U ρ n . We get estimates of differences of the abovementioned operators, in terms of suitable moduli of smoothness and K -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][22][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 Lemma 1. Section 2 contains the main theoretical results, while Section 3 is devoted to applications and examples.
Let S ≔ fðx, yÞ ∈ ℝ 2 jx, y ≥ 0, x + y ≤ 1g be the canonical simplex in ℝ 2 and EðSÞ 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, and pr i : S ⟶ ℝ, i = 1, 2, will denote the ith coordinate functions restricted on S, which are given by Let F : EðSÞ ⟶ ℝ be a positive linear functional such that Then, one has Denote by CðSÞ the space of real-valued continuous functions on S with the norm k f k = max ðx,yÞ∈S jf ðx, yÞj, f ∈ CðSÞ. Let K be a set of nonnegative integers and for k, l ∈ K let p k,l ∈ CðSÞ, p k,l ≥ 0, satisfy ∑ k,l∈K p k,l = 1. Let F k,l : EðSÞ ⟶ ℝ and G k,l : EðSÞ ⟶ ℝ, k, l ∈ K, be positive linear functionals such that F k,l ð1Þ = 1 and G k,l ð1Þ = 1. Moreover, let DðSÞ be the set of all f ∈ EðSÞ for which Now, consider the bivariate positive linear operators V respectively. For future correspondences, we denote where j·j is the l 1 -norm in ℝ 2 .
In the following, we adopt the definitions of K-functional and modulus of smoothness from [25,26]. Let For r ∈ ℕ, rth order differences on the subset SðrhÞ are defined as The rth order modulus of smoothness of f is a function ω r : CðSÞ × ð0,∞Þ ⟶ ½0,∞Þ given by Let C r ðSÞ be the space of all real-valued (continuous) functions, differentiable on int ðSÞ and whose partial derivatives of order ≤r can be continuously extended to S, with the seminorm For f ∈ CðSÞ, we shall use the following K-functional: Then, there exist c 1 , c 2 > 0 such that for any t > 0 (see [25,26]) Here, c 2 depends only on r (for the general definition on the L p , 1 ≤ p ≤ ∞, spaces of functions on bounded domains, see [25] or, on unbounded domains see [27], p.341.
where M f is defined in Lemma 1.
Proof. Let ðx, yÞ ∈ S. From Lemma 1, we get Theorem 3. If f ∈ CðSÞ, then where η 1 , η 2 > 0, and δ = sup k,l∈K fjb where M g 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 g is differentiable at every point of the line segment connecting the points ðb By the mean value theorem (see, e.g., [24], p. 239), there is a point ða 1 , a 2 Þ on this line segment such that From (16), we get

Proposition 4. For bivariate Bernstein operators and their
Durrmeyer variants, the following properties hold: (i) If f ∈ C 2 ðSÞ, then where M f is the same as in Lemma 1 and (ii) If f ∈ CðSÞ, then Proof. We need to evaluate the terms in (11). So, we get the following results: 0 ≤ k + l ≤ n. Therefore, we easily obtain that μ F n,k,l 2,0 = 0, μ G n,k,l Journal of Function Spaces Using Maple, one obtains σ x, y ð Þ= 〠 k,l=0,⋯,n k+l≤n It is easy to verify that σðx, yÞ ≤ 1/ðn + 4Þ. Now, for δ, we obtain The rest of the proof follows from Theorems 2 and 3. The operators A n are given by

Difference of Bivariate Bernstein Operators and the Bivariate Operators
Here, for f ∈ CðSÞ and ðx, yÞ ∈ S, we introduce the bivariate form of the operators A n as follows Denoting for k, l = 0, ⋯, n, k + l ≤ n, we get b F n,k,l 1 Proposition 5. For bivariate Bernstein operators and bivariate operators A n , the following properties hold: (i) If f ∈ C 2 ðSÞ, then (ii) If f ∈ CðSÞ, then

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 where n ∈ ℕ,f ∈ C½0, 1, and 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 U n,1 on a simplex. In [34], Sauer deeply studied the multivariate genuine Bernstein-Durrmeyer operators. Here, for f ∈ L 1 ðSÞ, we 5 Journal of Function Spaces consider the bivariate form given by with the bivariate Bernstein's fundamental functions given by (28) (see [33], Formula 1.7). These operators satisfy U n ð f Þð x, yÞ = f ðx, yÞ at the vertices of S.
Here, for f ∈ L 1 ðSÞ, we consider the bivariate form of these operators, given by where F ρ n,0,l f ð Þ ≔ It can be easily seen that, for ρ = 1, we obtain the genuine Bernstein-Durrmeyer operators U n . On the other hand, these operators have the following limiting behavior. Proof. Let f = pr j 1 , j = 0, 1, ⋯. Then, F ρ n,k,l pr Since F ρ n,0,l pr we get Similar results can be obtained for pr j 2 , j = 0, 1, ⋯. Using Korovkin's theorem (see [36], p. 534, C.4.3.3), it follows lim ρ⟶∞ F ρ n,μ,ν ðf Þ = f ðμ/n, ν/nÞ. Therefore, Proposition 8. For the bivariate operators U ρ n , the following properties hold: where M f is the same as in Lemma 1 and (ii) If f ∈ CðSÞ, then Proof.
where p n,k,l ðx, yÞ is given by (28). It can be easily seen that, for a = 1, we obtain Kantorovich operators K n introduced in [38]. If we denote (i) If f ∈ C 2 ðSÞ, then where M f is the same as in Lemma Then σ x, y ð Þ= 〠 k,l=0,⋯,n k+l≤n a 2 9 n + a ð Þ 2 p n,k,l x, y ð Þ= a 2 9 n + a ð Þ 2 : Then, the proof follows from Theorems 2 and 3.