Bernstein approximation and beyond: proofs by means of elementary probability theory

TL;DR

This paper explores how elementary probability theory can be used to prove approximation theorems involving Bernstein polynomials and extends these methods to other operators, providing error bounds and new insights.

Contribution

It introduces a probabilistic approach to polynomial approximation proofs and extends these techniques to Szász-Mirakjan and Baskakov operators.

Findings

Elementary probability methods prove approximation theorems.

Error bounds for Lipschitz functions are derived.

Extensions to other operators like Szász-Mirakjan and Baskakov are demonstrated.

Abstract

Bernstein polynomials provide a constructive proof for the Weierstrass approximation theorem, which states that every continuous function on a closed bounded interval can be uniformly approximated by polynomials with arbitrary accuracy. Interestingly the proof of this result can be done using elementary probability theory. This way one can even get error bounds for Lipschitz functions. In this note, we present these techniques and show how the method can be extended naturally to other interesting situations. As examples, we obtain in an elementary way results for the Sz\'{a}sz-Mirakjan operator and the Baskakov operator.