Algebraic Versus Analytic Density of Polynomials

Abstract

We show that under very mild conditions on a measure $\hat{I}¼$ on the real line, the span of $\{x^n{\}}_{n=j}^{\infty }$ is dense in $L^2(\hat{I}¼)$ for any $j\in \mathbb{N}$. We also present a slightly weaker result with an interesting proof that uses Sobolev orthogonality.