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arXiv · 1911.11280

Zero sets, entropy, and pointwise asymptotics of orthogonal polynomials

Roman Bessonov·Sergey Denisov
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Abstract

Let $μ$ be a measure from Szegő class on the unit circle $\mathbb T$ and let $\{f_n\}$ be the family of Schur functions generated by $μ$. In this paper, we prove a version of the classical Szegő's formula which controls the oscillation of $f_n$ on $\mathbb T$ for all $n \ge 0$. Then, we focus on an analog of Lusin's conjecture for polynomials $\{φ_n\}$ orthogonal with respect to measure $μ$ and prove that pointwise convergence of $\{|φ_n|\}$ almost everywhere on $\mathbb T$ is equivalent to a certain condition on zeroes of $φ_n$.

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Roman Bessonov, Sergey Denisov. 2019-12-24. Zero sets, entropy, and pointwise asymptotics of orthogonal polynomials. https://doi.org/10.1016/j.jfa.2021.109002

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