Fine Structure of the Zeros of Orthogonal Polynomials, I. A Tale of Two Pictures

TL;DR

This paper investigates the detailed zero distribution of orthogonal polynomials, revealing universal bulk behaviors and asymptotic zero spacing under specific coefficient conditions, extending prior understanding of their fine structure.

Contribution

It introduces new results on the fine zero structure of orthogonal polynomials with random Verblunsky coefficients and under the BLS condition, including asymptotic zero spacing.

Findings

Universal bulk zero behavior for large degree polynomials

Asymptotically equal spacing of zeros under BLS condition

Results extend understanding of zero distribution in orthogonal polynomials

Abstract

Mhaskar-Saff found a kind of universal behavior for the bulk structure of the zeros of orthogonal polynomials for large $n$. Motivated by two plots, we look at the finer structure for the case of random Verblunsky coefficients and for what we call the BLS condition: $\alpha_n = Cb^n + O((b\Delta)^n)$. In the former case, we describe results of Stoiciu. In the latter case, we prove asymptotically equal spacing for the bulk of zeros.