Analytic Versus Algebraic Density of Polynomials

TL;DR

This work addresses when the tail span $\mathrm{span}\{x^k\}_{k=n}^{\infty}$ is dense in $L^2(\mu)$ under mild measure assumptions, notably for measures supported on $[0,\infty)$ with $0$ in the support but not mass at $0$. It develops two complementary proofs: a density-index approach tied to moment problems and a second proof via a Hilbert-space augmentation that avoids the density index. It also extends the density theory to infinite index of determinacy, showing that if $\mathrm{ind}(\mu)=\infty$ then the polynomial ideal $R(x)\mathbb{C}[x]$ is dense in $L^2(\mu)$ for any polynomial $R$ whose zeros carry no mass under $\mu$, connecting density to Nevanlinna parametrization and N-extremal measures. Together, these results deepen understanding of when polynomial spans are dense in weighted $L^2$ spaces and link polynomial density to the determinacy properties of the moment problem, with implications for polynomial approximation beyond classical unweighted spaces.

Abstract

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

Theorems & Definitions (13)

• Theorem 1.1
• Remark 1.2
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Lemma 3.1
• proof
• Remark 3.2
• proof : Second Proof of Theorem \ref{['maindense2']}