Gr\"uss and Gr\"uss-Voronovskaya-type estimates for some Bernstein-type polynomials of real and complex variables

TL;DR

This paper establishes Grüss and Grüss-Voronovskaya estimates for Bernstein-type polynomials in real and complex variables, including extensions to Bernstein-Faber polynomials, enhancing understanding of their approximation properties.

Contribution

It provides new Grüss and Grüss-Voronovskaya estimates for Bernstein and Bernstein-Durrmeyer polynomials in both real and complex settings, including for Bernstein-Faber polynomials.

Findings

Derived Grüss-Voronovskaya estimates for Bernstein polynomials on [0,1]

Extended estimates to complex variables on compact disks

Generalized results to Bernstein-Faber polynomials for compact sets in the complex plane

Abstract

The first aim of this paper is to prove a Gr\"uss-Voronovskaya estimate for Bernstein and for a class of Bernstein-Durrmeyer polynomials on $[0, 1]$. Then, Gr\"uss and Gr\"uss-Voronovskaya estimates for their corresponding operators of complex variable on compact disks are obtained. Finally, the results are extended to Bernstein-Faber polynomials attached to compact sets in the complex plane.