Modiﬁed Operators Interpolating at Endpoints

: Some classical operators (e.g., Bernstein) preserve the afﬁne functions and consequently interpolate at the endpoints. Other classical operators (e.g., Bernstein–Durrmeyer) have been modiﬁed in order to preserve the afﬁne functions. We propose a simpler modiﬁcation with the effect that the new operators interpolate at endpoints although they do not preserve the afﬁne functions. We investigate the properties of these modiﬁed operators and obtain results concerning iterates and their limits, Voronovskaja-type results and estimates of several differences.


Introduction
Let X be a compact Hausdorff space and C(X) the space of all real-valued, continuous functions on X, endowed with the usual ordering and the supremum norm. Denote by 1 ∈ C(X) the function of constant value 1.
Let S : C(X) → C(X) be a Markov operator, i.e., a positive linear operator with S1 = 1. Consequently, S is a bounded operator and its norm induced by the supremum norm on C(X) is equal to 1.
In this paper we are concerned with Markov operators of the form where s ≥ 1, A j : C(X) → R are positive linear functionals with A j (1) = 1, and p j ∈ C(X) are linearly independent functions, p j ≥ 0, s ∑ j=0 p j = 1.
As shown in [1], the properties of S are strongly influenced by the value of ν. In particular, there are significant differences between the case ν = 0 and the cases ν > 0.
The Bernstein, Kantorovich, Durrmeyer, genuine Bernstein-Durrmeyer operators are defined, respectively by: The operators which preserve the affine functions (like B n and U n ) interpolate the function at the endpoints and so ν = 2. The aim of this paper is to study a slight modifica-tionL n of a given operator L n : C[0, 1] → C[0, 1] (which does not interpolate at endpoints, like Kantorovich or Durrmeyer operators) such thatL n interpolates the function at the endpoints and so ν = 2, without preserving the affine functions. For example, SinceL n f interpolates f at 0 and 1, it provides a better approximation near these endpoints. We will give estimates forL n f − f andL n f − L n f . Using methods from [2][3][4][5][6] we will investigate differences between classical operators and modified operators. General results concerning the iterates and the invariant measures for operators S of the form (1) are given in [1]. In this paper we extend these results in order to cover the case of the modified operatorsL n . As a Markov operator,L n preserves the constant functions; we will identify the other functions fixed by it. They play an essential role in studying the generalized convexity induced byL n .
All the above considerations can be extended to the multivariate versions of the operators, defined on the canonical simplex of R d .

Limit of Iterates
In this section, we are concerned with the limit of iterates of certain Markov operators. The next theorem completes and gives another proof of the result contained in Ex. 2.3 [1]. Theorem 1. Let ν = 2. Then, there exists an eigenvector k = (k 0 , k 1 , . . . , k s ) t of the matrix M, corresponding to the eigenvalue 1, such that k and (1, 1, . . . , 1) t are linearly independent. Moreover, Proof. Due to the structure of the stochastic matrix M, 1 is an eigenvalue having the linearly independent eigenvectors (1, 1, . . . , 1) t and k. As in (2.2) [1], we have for f ∈ C(X) It is known (see [7][8][9]) that where P is a matrix of the form Since Mk = k, we obtain M m k = k and so Bk = k. It follows that a contradiction with the fact that k and (1, 1, . . . , 1) t are linearly independent. Thus, From (5) and (6), we infer that Using (9), we obtaim the matrix P and then the matrix B. Now, a straightforward calculation leads to (4) and the proof is finished.

Corollary 1.
Let h := k 0 p 0 + · · · + k s p s , where k is the above eigenvector of M. Then, Sh = h.

Remark 1. Let E be the linear subspace generated by 1 and h. Another basis of E is formed by the functions
Proof. (i) is a direct consequence of (1), (2) and (3).
As a positive linear functional on C(X), with A(1) = 1, A is continuous and thus (4) implies Consequently, with Moreover, using (7) and (9), we see that ϕ 0 ≥ 0 and Now, (12) shows that A is a convex combination of A 0 and A 1 .

Modified Durrmeyer Operators
For the sake of simplicity, we fix a number n ≥ 3 and consider the polynomials The corresponding functionals (see Section 1) will be We have which means that Q n (see (3) with s = n and ν = 2) is symmetric with respect to the main diagonal and the second diagonal.

The Linking Operator H n,ρ and Voronovskaja-Type Results
In [10], Pȃltȃnea introduced a class of operators that constitute a link between the genuine Bernstein-Durrmeyer operators and the classical Bernstein operators. In [11], Heilmann and Raşa proposed an explicit representation depending on parameter ρ ≥ 1, which is a link between Bernstein-Durrmeyer operators (for ρ = 1) and Kantorovich operators (for ρ → ∞). These operators are defined as follows: Consider the modification of the operators H n,ρ as follows: The central moments are important in the Shisha and Mond technique used in approximation by positive linear operators. Next, we compare the moments of classical operators with the moments of modified operators.
By elementary calculations, we find Therefore, With similar calculations, Figures 1 and 2 illustrate the difference ∆ for n = 10, ρ = 2 and n = 20, ρ = 5. We see that indeed the second central moments ofH n,ρ are smaller than those of H n,ρ near the endpoints.

Remark 3.
We see once more that the approximation furnished byH n,ρ near the endpoints is better than the approximation furnished by H n,ρ .

Remark 4.
For ρ = 1 in Theorem 2, we obtain the Voronovskaja formula for the modified Bernstein-Durrmeyer operators. The case ρ → ∞, corresponding to the modified Kantorovich operators, was investigated in [13].

Differences in Positive Linear Operators
This section deals with estimates of the differences between classical operators and modified operators. Very recently, estimates of the differences of certain positive linear operators were obtained in [14][15][16][17][18].
Let E(I) be a space of real-valued continuous functions on an interval I ⊂ R containing the polynomials. Denote e i (x) := x i , x ∈ I, i ∈ N and where F : E(I) → R is a positive linear functional, F(e 0 ) = 1 and b F := F(e 1 ). Let I ⊂ N and A k : E(I) → R and B k : E(I) → R be positive linear functionals such that A k (e 0 ) = B k (e 0 ) = 1, k ∈ I. Consider p k ∈ C(I) such that ∑ k∈K p k = e 0 and p k ≥ 0, Consider the positive linear operators P : D(I) → C(I) and Q : D(I) → C(I) defined, for f ∈ D(I) as follows Let ω s ( f , ·), s = 1, 2 . . . , be the usual moduli of smoothness of a function f ∈ C(I).
we obtain Proposition 5. The difference between Bernstein-Durrmeyer operators and modified Bernstein-Durrmeyer operators verifies the following inequalities: Proof. Using

Conclusions and Perspectives
The preservation of affine functions is an important property of positive linear operators in Approximation Theory. It implies the interpolation property at extreme points. Several classical operators are suitable in this sense, but other ones are not, and consequently, they were modified in order to obtain the preservation property. Our paper is devoted to simpler modifications that guarantee interpolation at extreme points without implying the preservation of affine functions. The modified operators provide a better approximation near the extreme points. This can be seen in terms of moments and Voronovskaya-type formulas. Despite the fact that the images of monomials are rather complicated, we are able to give useful results concerning the limit of iterates of the modified operators, as well as estimates for differences involving classical and modified operators. We present general results formulated for operators defined on C(X), the space of all real-valued, continuous functions on a compact Hausdorff space X. The illustrating examples are mainly concerned with operators defined on C[0, 1]. A forthcoming paper will be devoted to the case when X is a simplex in R d or even a more general compact convex set.
Author Contributions: These authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.