Non-uniform non-asymptotical sharp estimate for the rate of convergence for Bernstein's polynomial approximation, with bilateral constant evaluation

TL;DR

This paper provides precise, non-asymptotic error bounds for Bernstein polynomial approximation of continuous functions, including derivatives and multivariate cases, using modern probabilistic methods.

Contribution

It introduces non-uniform, sharp error estimates for Bernstein approximation with bilateral constant evaluation, extending to derivatives and multivariate functions.

Findings

Established non-asymptotic error bounds for Bernstein approximation

Analyzed convergence of derivatives of Bernstein polynomials

Extended results to multivariate functions

Abstract

We derive the non-asymptotical non-uniform sharp error estimation for Bernstein's approximation of continuous function based on the modern probabilistic apparatus. We investigate also the convergence of derivative of these polynomials and we will consider briefly also the multivariate case.