Weak Convergence of CD Kernels: A New Approach on the Circle and Real Line

TL;DR

This paper introduces a new proof for the convergence of certain measures associated with orthogonal polynomials on the circle and real line, refining understanding of their asymptotic behavior.

Contribution

It provides a novel proof of a 2009 theorem on the asymptotic difference between moments of two measures related to orthogonal polynomials on specific supports.

Findings

Proves the convergence rate of moments is O(1/n)

Applies to measures supported on the real line and the circle

Enhances understanding of zero distribution of orthogonal polynomials

Abstract

Let m be a probability measure supported on some infinite and compact set K in the complex plane and let p_n(z) be the corresponding degree n orthonormal polynomial with positive leading coefficient. Let v_n be the normalized zero counting measure for the polynomial p_n and let u_n be the probability measure given by (n+1)u_n=K_n(z,z)m, where K_n(z,w) is the reproducing kernel for polynomials of degree at most n. If m is supported on a compact subset of the real line or the unit circle, we provide a new proof of a 2009 theorem due to Simon, that for any fixed natural number k, the k^{th} moment of u_n and v_{n+1} differ by at most O(1/n) as n tends to infinity.