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Strong Converse Inequalities for Bernstein Polynomials with Explicit Asymptotic Constants

José A. Adell, Daniel Cárdenas-Morales

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.

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.
Paper Structure (7 sections, 19 theorems, 182 equations)

This paper contains 7 sections, 19 theorems, 182 equations.

Table of Contents

  1. Introduction and statement of the main results
  2. General auxiliary results
  3. An auxiliary result for the central region
  4. Estimates in the central region
  5. Estimates in the non-central region
  6. Proof of Theorem \ref{['th1']}
  7. Proof of Theorem \ref{['th2']}

Key Result

Theorem 1

Let $\mathfrak{F}$ denote the set of all non-affine functions $f\in C[0,1]$. Then,

Theorems & Definitions (19)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 4
  • Theorem 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • ...and 9 more