Zero sets, entropy, and pointwise asymptotics of orthogonal polynomials

Abstract

Let $\mu$ be a measure from Szegő 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 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 $\{{\phi }_n\}$ orthogonal with respect to measure $\mu$ and prove that pointwise convergence of $\{|{\phi }_n|\}$ almost everywhere on $\mathbb{T}$ is equivalent to a certain condition on zeroes of ${\phi }_n$.

26 pages

References in corpus (2)

• A spectral Szego theorem on the real line
• De Branges canonical systems with finite logarithmic integral

Cited by in corpus (1)

• Entropy function and orthogonal polynomials