Converse estimates for the simultaneous approximation by Bernstein polynomials with integer coefficients

TL;DR

This paper establishes a weak converse estimate for simultaneous approximation by Bernstein polynomials with integer coefficients, characterizing the approximation rate and saturation properties in terms of smoothness moduli.

Contribution

It introduces a novel weak converse estimate for integer-coefficient Bernstein polynomial approximation and characterizes its saturation behavior.

Findings

Provides a big O-characterization of the approximation rate.

Shows the approximation process is saturated with a specific saturation rate.

Identifies the trivial class in the saturation context.

Abstract

We prove a weak converse estimate for the simultaneous approximation by several forms of the Bernstein polynomials with integer coefficients. It is stated in terms of moduli of smoothness. In particular, it yields a big $O$-characterization of the rate of that approximation. We also show that the approximation process generated by these Bernstein polynomials with integer coefficients is saturated. We identify its saturation rate and the trivial class.