From local to global asymptotic behaviour of orthogonal polynomials

Abstract

Let $\{ϕ^*_n\}$ be the sequence of reflected orthogonal polynomials on the unit circle $\partial \mathbb{D}$ generated by a measure $μ$ of Szegő class, and let $D_μ$ be the Szegő function of $μ$. We prove the uniform Cesàro asymptotics $$ \sup_{z \in Γ_ζ}\Biggl(\frac{1}{n}\sum_{k = 0}^{n-1}\Bigl||ϕ_k^*(z) D_μ(z)|^2 - 1\Bigr|\Biggr) \to 0, \qquad n \to \infty, $$ for almost all Stolz angles $Γ_ζ$, $ζ\in \partial \mathbb{D}$. This extends a well-known asymptotic result of Máté, Nevai, and Totik (1991) from the local scale $O(1/n)$ near $\partial \mathbb{D}$ to the global scale $O(1)$. We also study asymptotic behavior of arguments of orthogonal polynomials and extend a classical theorem due to Grenander and Szegő using a new technique. As an application, we derive global asymptotic results for polynomial reproducing kernels under various assumptions on the orthogonality measure.