Analytic Versus Algebraic Density of Polynomials

TL;DR

This paper investigates conditions under which polynomial spans are dense in L^2 spaces with respect to certain measures, providing multiple proofs and extending results to polynomial ideals and measures with infinite index of determinacy.

Contribution

It offers new proofs of polynomial density in L^2 spaces and extends the theory to polynomial ideals and measures with infinite index of determinacy.

Findings

Polynomial spans are dense in L^2(μ) under mild conditions.

Two different proofs of polynomial density are provided.

Polynomial ideals are dense in L^2(μ) for measures with infinite index of determinacy.

Abstract

We show that under very mild conditions on a measure $\mu$ on the interval $[0,\infty)$, the span of $\{x^k\}_{k=n}^{\infty}$ is dense in $L^2(\mu)$ for any $n=0,1,\ldots$. We present two different proofs of this result, one based on the density index of Berg and Thill and one based on the Hilbert space $L^2(\mu)\oplus \mathbb{C}^{n+1}$. Using the index of determinacy of Berg and Dur\'an we prove that if the measure $\mu$ on $\mathbb{R}$ has infinite index of determinacy then the polynomial ideal $R(x)\mathbb{C}[x]$ is dense in $L^2(\mu)$ for any polynomial $R$ with zeros having no mass under $\mu$.