Pointwise weighted approximation of functions with inner singularities by Bernstein operators

TL;DR

This paper studies how Bernstein operators can approximate functions with inner singularities using weighted norms, providing direct and inverse approximation results.

Contribution

It introduces a new approach to weighted approximation of functions with inner singularities using Bernstein operators, including direct and inverse theorems.

Findings

Established bounds for approximation errors with weighted Bernstein operators.

Derived inverse theorems linking smoothness and approximation quality.

Extended classical approximation theory to functions with inner singularities.

Abstract

We consider the pointwise weighted approximation by Bernstein operators with inner singularities. The related weight functions are weights $\bar{w}(x)=|x-\xi|^\alpha(0<\xi<1,\ \alpha>0).$ In this paper we give direct and inverse results of this type of Bernstein polynomials.