Simultaneous approximation by Bernstein polynomials with integer coefficients

TL;DR

This paper demonstrates that certain Bernstein polynomials with integer coefficients can simultaneously approximate functions and their derivatives, providing both error estimates and necessary conditions for such approximation.

Contribution

It introduces new results on simultaneous approximation by Bernstein polynomials with integer coefficients, including error bounds and necessary conditions.

Findings

Bernstein polynomials with integer coefficients can approximate functions and derivatives simultaneously

Established direct error estimates using moduli of smoothness

Identified necessary and sufficient conditions for approximation accuracy

Abstract

We prove that several forms of the Bernstein polynomials with integer coefficients possess the property of simultaneous approximation, that is, they approximate not only the function but also its derivatives. We establish direct estimates of the error of that approximation in uniform norm by means of moduli of smoothness. Moreover, we show that the sufficient conditions under which those estimates hold are also necessary.