A survey on the Weierstrass approximation theorem

TL;DR

This survey reviews the Weierstrass approximation theorem, its historical development, recent advancements, and open problems, highlighting its foundational role in approximation theory of continuous functions.

Contribution

It provides a comprehensive overview of the theorem's history, recent results, and open questions, offering insights into ongoing research in approximation theory.

Findings

Historical overview of the theorem

Recent developments and extensions

Open problems in the field

Abstract

The celebrated and famous Weierstrass approximation theorem characterizes the set of continuous functions on a compact interval via uniform approximation by algebraic polynomials. This theorem is the first significant result in Approximation Theory of one real variable and plays a key role in the development of General Approximation Theory. Our aim is to investigate some new results relative to such theorem, to present a history of the subject, and to introduce some open problems.