Bernstein's problem on weighted polynomial approximation

TL;DR

This paper establishes a necessary and sufficient condition for polynomial density in weighted continuous function spaces and $L^p$ spaces with respect to Bernstein's weighted uniform norm.

Contribution

It provides a new criterion characterizing when polynomials are dense in weighted function spaces on the real line, extending Bernstein's approximation theory.

Findings

Derived a criterion for polynomial density in weighted uniform norm spaces.

Established conditions for polynomial approximation in $L^p$ spaces with finite measures.

Unified understanding of polynomial approximation in weighted contexts.

Abstract

We formulate and discuss a necessary and sufficient condition for polynomials to be dense in a space of continuous functions on the real line, with respect to Bernstein's weighted uniform norm. Equivalently, for a positive finite measure $\mu$ on the real line we give a criterion for density of polynomials in $L^p(\mu)$.