On the Constant in The Lower Estimate for the Bernstein Operator

TL;DR

This paper establishes explicit lower bounds for the approximation error of Bernstein polynomials for smooth functions, using the second order Ditzian-Totik modulus of smoothness, with applications to specific functions.

Contribution

It provides the first explicit constant in the lower estimate for Bernstein polynomial approximation for functions in $C^{2}[0,1]$ and $C^{3}[0,1]$, enhancing understanding of approximation quality.

Findings

Explicit lower bounds with constants for Bernstein approximation.

Application of bounds to concrete functions.

Improved understanding of approximation limits for smooth functions.

Abstract

For functions belonging to the classes $C^{2}[0, 1]$ and $C^{3}[0, 1]$, we establish the lower estimate with an explicit constant in approximation by Bernstein polynomials in terms of the second order Ditzian-Totik modulus of smoothness. Several applications to some concrete examples of functions are presented.