Analytic Versus Algebraic Density of Polynomials
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915157165211648 |
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| author | Berg, Christian Simanek, Brian Wellman, Richard |
| author_facet | Berg, Christian Simanek, Brian Wellman, Richard |
| contents | 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 $μ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_12229 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Analytic Versus Algebraic Density of Polynomials Berg, Christian Simanek, Brian Wellman, Richard Classical Analysis and ODEs 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 $μ$. |
| title | Analytic Versus Algebraic Density of Polynomials |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2502.12229 |