Quantitative convergence theorems for a class of Bernstein–Durrmeyer operators preserving linear functions

Abstract

We supplement recent results on a class of Bernstein-Durrmeyer operators preserving linear functions. This is done by discussing two limiting cases and proving quantitative Voronovskaya-type assertions involving the first-order and second-order moduli of smoothness. The results generalize and improve earlier statements for Bernstein and genuine Bernstein-Durrmeyer operators.

Approximation by Bernstein–Durrmeyer-type operators in compact disks

“…In [3] a Voronovskaja-type result with a quantitative estimate for complex Bernstein polynomials in compact disks was obtained. In [4][5][6][7][8][9][10][11]1,12,13] similar results for Bernstein-Stancu, Kantorovich-Stancu and q-Sancu polynomials were obtained while in [14] similar results for Bernstein-Schurer polynomials were proved. Very recently, Anastassiou and Gal [15], Gal [7] studied the order of simultaneous approximation and Voronovskaja-kind results with quantitative estimates for complex Bernstein-Durrmeyer and genuine Durrmeyer polynomials attached to analytic functions on compact disks.…”

Quantitative Estimates for Positive Linear Operators in terms of the Usual Second Modulus

“…, +1, given by the constant defined in (9). The second equality in (8) follows from the first one and the equalities…”

“…The first equality in (8) follows from the fact that the two functions involved have the same Radon-Nikodym derivative in ( −1 , ), = − , . .…”

“…For instance, Adell and Sangüesa [7] [8] showed that ≃ 3/2 for a certain class of Bernstein-Durrmeyer operators. More generally, in Pȃltȃnea's book [5, Corollary 2.2.1, p. 31] it is shown that ≃ 3/2 for a large class of positive linear operators reproducing linear functions.…”

“…Let̃∈ L ( ) be the function having representation (8), whose set of nodes is N , , as defined in (37), and satisfying the following properties: These properties, together with (8) and (39), allow us to writẽ(…”