Strong Converse Inequalities for Bernstein Polynomials with Explicit Asymptotic Constants

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.