On the zeros of orthogonal polynomials on the unit circle

TL;DR

This paper investigates the zeros of orthogonal polynomials on the unit circle, exploring how the asymptotic behavior of their zeros influences the properties of the associated orthogonality measure, with special focus on periodic zero sequences.

Contribution

It characterizes the relationship between zero sequences and orthogonality measures, especially for periodic zeros of period two and three, advancing understanding of polynomial asymptotics.

Findings

Asymptotic behavior of zeros determines measure characteristics

Periodic zero sequences influence polynomial asymptotics

Explicit analysis for zeros of period two and three

Abstract

Let ${z_n}$ be a sequence in the unit disk ${z\in\mathbb{C}:|z|<1}$. It is known that there exists a unique positive Borel measure in the unit circle ${z\in\mathbb{C}:|z|=1}$ such that the orthogonal polynomials ${\Phi_n}$ satisfy [\Phi_n(z_n)=0] for each $n=1,2,...$. Characteristics of the orthogonality measure and asymptotic properties of the orthogonal polynomial are given in terms of asymptotic behavior of the sequence ${z_n}$. Particular attention is paid to periodic sequence of zeros ${z_n}$ of period two and three.