A new Representation of $\alpha$-Bernstein Operators

TL;DR

This paper introduces a new recursive-based representation of $oldsymbol{ extalpha}$-Bernstein operators, providing deeper insights and simpler proofs of their properties, advancing approximation theory.

Contribution

It presents a novel recursive technique for constructing Bernstein-like bases, offering a new perspective and properties for $oldsymbol{ extalpha}$-Bernstein operators.

Findings

New recursive representation of $ extalpha$-Bernstein operators

Simplified proofs of existing theorems

Enhanced understanding of operator properties

Abstract

The $\alpha$-Bernstein operators were initially introduced in the paper by Chen, X., Tan, J., Liu, Z., Xie, J. (2017) titled "Approximation of Functions by a New Family of Generalized Bernstein Operators" (Journal of Mathematical Analysis and Applications, 450(1), 244-261). Since their introduction, these operators have served as a source of inspiration for numerous research endeavors. In this study, we propose a novel technique, founded on a recursive relation, for constructing Bernstein-like bases. A special case of this new representation yields a novel portrayal of Chen's operators. This innovative representation enables the discovery of additional properties of $\alpha$-Bernstein operators and facilitates alternative and more straightforward proofs for certain theorems.