Zeros of orthogonal polynomials near an algebraic singularity of the measure

TL;DR

This paper investigates the local zero distribution of orthogonal polynomials near algebraic singularities in the measure, revealing detailed asymptotics and zero spacing behavior through Bessel functions and Sturm-Liouville theory.

Contribution

It refines previous results by describing zero behavior near singularities using Bessel functions and provides estimates for zero locations with Sturm-Liouville analysis.

Findings

Zero spacing unravels near algebraic singularities.

Zeros asymptotically relate to zeros of a Bessel function combination.

Provides estimates for zeros using Sturm-Liouville theory.

Abstract

In this paper we study the local zero behavior of orthogonal polynomials around an algebraic singularity, that is, when the measure of orthogonality is supported on $ [-1,1] $ and behaves like $ h(x)|x - x_0|^\lambda dx $ for some $ x_0 \in (-1,1) $, where $ h(x) $ is strictly positive and analytic. We shall sharpen the theorem of Yoram Last and Barry Simon and show that the so-called fine zero spacing (which is known for $ \lambda = 0$) unravels in the general case, and the asymptotic behavior of neighbouring zeros around the singularity can be described with the zeros of the function $ c J_{\frac{\lambda - 1}{2}}(x) + d J_{\frac{\lambda + 1}{2}}(x) $, where $ J_a(x) $ denotes the Bessel function of the first kind and order $ a $. Moreover, using Sturm-Liouville theory, we study the behavior of this linear combination of Bessel functions, thus providing estimates for the zeros in question.