Analytic Versus Algebraic Density of Polynomials

We show that under very mild conditions on a measure $\mu$ on the interval $[0,\mathrm{\infty })$, the span of $\{x^k{\}}_{k=n}^{\mathrm{\infty }}$ is dense in $L^2(\mu )$ 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 $L^2(\mu )\oplus {\mathbb{C}}^{n+1}$. Using the index of determinacy of Berg and Durán 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 $\mu$.