Algebraic Versus Analytic Density of Polynomials

TL;DR

This paper compares algebraic and analytic approaches to polynomial density in $L^2$ spaces, establishing conditions under which polynomial spans are dense, and introduces a novel proof involving Sobolev orthogonality.

Contribution

It provides new conditions for polynomial density in $L^2( ext{measure})$ spaces and presents a novel proof technique using Sobolev orthogonality.

Findings

Polynomial spans are dense in $L^2( ext{measure})$ under mild conditions.

A new proof method using Sobolev orthogonality is introduced.

The results extend understanding of polynomial approximation in measure spaces.

Abstract

We show that under very mild conditions on a measure $\mu$ on the real line, the span of $\{x^n\}_{n=j}^{\infty}$ is dense in $L^2(\mu)$ for any $j\in\mathbb{N}$. We also present a slightly weaker result with an interesting proof that uses Sobolev orthogonality.