Strong Converse Inequalities for Bernstein Polynomials with Explicit Asymptotic Constants

TL;DR

This paper establishes strong converse inequalities for Bernstein polynomials, providing explicit asymptotic constants and different estimation methods for central and non-central regions of [0,1], using probabilistic and polynomial derivative representations.

Contribution

It introduces new estimation procedures with explicit asymptotic constants for Bernstein polynomials, utilizing probabilistic representations and derivatives in terms of Krawtchouk polynomials.

Findings

Derived strong converse inequalities with explicit constants

Provided estimation methods for different regions of [0,1]

Connected derivatives of Bernstein polynomials to Krawtchouk polynomials

Abstract

We obtain strong converse inequalities for the Bernstein polynomials with explicit asymptotic constants. We give different estimation procedures in the central and non-central regions of [0,1]. The main ingredients in our approach are the following: representation of the derivatives of the Bernstein polynomials in terms of the Krawtchouk polynomials, estimates of different inverse moments of various random variables, sharp estimates of both absolute central moments of Bernstein polynomials and the total variation distance between binomial and Poisson distributions, and iterates of the Bernstein polynomials, together with their probabilistic representations.