Direct and Inverse Estimates for Combinations of Bernstein Polynomials with Endpoint Singularities

TL;DR

This paper develops direct and inverse approximation theorems for functions with endpoint singularities using combinations of Bernstein polynomials, employing the Ditzian-Totik modulus of smoothness to measure approximation quality.

Contribution

It introduces new theorems that connect weighted approximation of endpoint singularities with Bernstein polynomial combinations and the Ditzian-Totik smoothness measure.

Findings

Established direct approximation theorems for endpoint singularities.

Proved inverse theorems linking smoothness and approximation quality.

Enhanced understanding of weighted Bernstein polynomial approximation methods.

Abstract

We give direct and inverse theorems for the weighted approximation of functions with endpoint singularities by combinations of Bernstein polynomials by the $r$th Ditzian-Totik modulus of smoothness $\omega_\phi^{r}(f,t)_w$ where $\phi$ is an admissible step-weight function.