Weak convergence of CD kernels and applications

TL;DR

This paper establishes a general result on the weak limits of zero counting measures of orthogonal polynomials and their relation to Christoffel-Darboux kernels, with applications to measures with singular parts.

Contribution

It introduces a new general result linking zero counting measures and Christoffel-Darboux kernels, extending previous bounds and convergence results.

Findings

Weak limits of zero counting measures match the scaled Christoffel-Darboux kernel measures.

Results on convergence of integrals involving the kernel and measure densities.

Applications to measures with singular parts and equilibrium measure densities.

Abstract

We prove a general result on equality of the weak limits of the zero counting measure, $d\nu_n$, of orthogonal polynomials (defined by a measure $d\mu$) and $\frac{1}{n} K_n(x,x) d\mu(x)$. By combining this with Mate--Nevai and Totik upper bounds on $n\lambda_n(x)$, we prove some general results on $\int_I \frac{1}{n} K_n(x,x) d\mu_s\to 0$ for the singular part of $d\mu$ and $\int_I |\rho_E(x) - \frac{w(x)}{n} K_n(x,x)| dx\to 0$, where $\rho_E$ is the density of the equilibrium measure and $w(x)$ the density of $d\mu$.