From local to global asymptotic behaviour of orthogonal polynomials

Abstract

Let $\{{\varphi }_n^{\ast }\}$ be the sequence of reflected orthogonal polynomials on the unit circle $\partial \mathbb{D}$ generated by a measure $\mu$ of Szegő class, and let $D_{\mu }$ be the Szegő function of $\mu$. We prove the uniform Cesàro asymptotics $$\underset{z\in {\Gamma }_{\zeta }}{\sup }\left(\frac{1}{n}\sum _{k=0}^{n-1}\right||{\varphi }_k^{\ast }(z)D_{\mu }(z){|}^2-1|)\to 0,n\to \infty ,$$ for almost all Stolz angles ${\Gamma }_{\zeta }$, $\zeta \in \partial \mathbb{D}$. 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.

36 pages; 3 figures