Analytic Versus Algebraic Density of Polynomials

2025/02/17 by Christian Berg, Berg, Christian, Brian Simanek +3
Mathematics · #Mathematical functions and polynomials #Meromorphic and Entire Functions #Functional Equations Stability Results

paper · pdf · doi:10.48550/arxiv.2502.12229

Abstract

We show that under very mild conditions on a measure μ on the interval [0,∞), the span of \x^k\_k=n^∞ is dense in L²(μ) for any n=0,1,…. 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 L²(μ)⊕ ℂⁿ⁺¹. Using the index of determinacy of Berg and Durán we prove that if the measure μ on ℝ has infinite index of determinacy then the polynomial ideal R(x)ℂ[x] is dense in L²(μ) for any polynomial R with zeros having no mass under μ.

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