On a density problem related to a theorem of Szegő

2025/11/11 by Paulsen, Chiara
#42C05 #60G25 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2511.08739

Abstract

A classical theorem of Szegő states that for any probability measure μ=w(dθ)/(2π)+μ_s on the unit circle the polynomials are dense in L²(\mathbbT,μ) if and only if log(w)∉ L¹(\mathbbT). A related question asks whether the monomials with exponents in some subset Λ⊆ ℕ₀ already span L²(\mathbbT,μ) if log(w)∉ L¹(\mathbbT). A result by Olevskii and Ulanovskii gives an answer if μ belongs to a class of absolutely continuous measures. We investigate the same question for Markoff measures.

Citations

• On Szegö--Kolmogorov Prediction Theorem
• Szegö's Theorem and its Probabilistic Descendants

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• Notes on the Szego minimum problem. I. Measures with deep zeroes
• Coefficients of Orthogonal Polynomials on the Unit Circle and Higher Order Szego Theorems
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• Analytic Versus Algebraic Density of Polynomials