On a density problem related to a theorem of Szeg\H{o}

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Contribution

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Abstract

A classical theorem of Szeg\H{o} states that for any probability measure $\mu=w\frac{\mathrm{d}\theta}{2\pi}+\mu_s$ on the unit circle the polynomials are dense in $L^2(\mathbb{T},\mu)$ if and only if $\log(w)\notin L^1(\mathbb{T})$. A related question asks whether the monomials with exponents in some subset $\Lambda\subseteq \mathbb{N}_0$ already span $L^2(\mathbb{T},\mu)$ if $\log(w)\notin L^1(\mathbb{T})$. A result by Olevskii and Ulanovskii gives an answer if $\mu$ belongs to a class of absolutely continuous measures. We investigate the same question for Markoff measures.