Generalized Bernstein operators on the classical polynomial spaces

TL;DR

This paper explores generalized Bernstein operators on polynomial spaces, demonstrating their diverse behaviors and proving their existence and convergence properties in higher dimensions.

Contribution

It introduces a broader class of Bernstein operators fixing a polynomial and the constant function, extending classical theory and providing existence and convergence results.

Findings

Existence of generalized Bernstein operators for large dimensions

Diverse behaviors demonstrated through examples

Convergence to the identity operator

Abstract

We study generalizations of the classical Bernstein operators on polynomial spaces, where instead of fixing $\mathbf{1}$ and $x$, we require that $\mathbf{1}$ and a strictly increasing polynomial $f_1$ be fixed. Via several examples, we exhibit the diversity of behaviours in this more general setting. We also prove that for sufficiently large dimensions, there always exist generalized Bernstein operators fixing $\mathbf{1}$ and $f_1$, and converging to the identity.