Analytic Versus Algebraic Density of Polynomials

Abstract

We show that under very mild conditions on a measure $\hat{I}¼$ on the interval $[0,\infty )$, the span of $\{x^k{\}}_{k=n}^{\infty }$ is dense in $L^2(\hat{I}¼)$ for any $n=0,1,\dots$. 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 $L^2(\hat{I}¼)\oplus {\mathbb{C}}^{n+1}$. Using the index of determinacy of Berg and Durán we prove that if the measure $\hat{I}¼$ on $\mathbb{R}$ has infinite index of determinacy then the polynomial ideal $R(x)\mathbb{C}[x]$ is dense in $L^2(\hat{I}¼)$ for any polynomial $R$ with zeros having no mass under $\hat{I}¼$.

Significant overlap with arXiv:2406.18353