Zero sets, entropy, and pointwise asymptotics of orthogonal polynomials

TL;DR

This paper extends Szeg\

Contribution

It proves a Szeg\

Findings

Controls oscillation of Schur functions for all n

Establishes equivalence between zero distribution and pointwise convergence of orthogonal polynomials

Provides new insights into asymptotic behavior of orthogonal polynomials

Abstract

Let $\mu$ be a measure from Szeg\H{o} class on the unit circle $\mathbb T$ and let $\{f_n\}$ be the family of Schur functions generated by $\mu$. In this paper, we prove a version of the classical Szeg\H{o}'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 $\{\varphi_n\}$ orthogonal with respect to measure $\mu$ and prove that pointwise convergence of $\{|\varphi_n|\}$ almost everywhere on $\mathbb T$ is equivalent to a certain condition on zeroes of $\varphi_n$.