Weak convergence of CD kernels and applications

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 $d\mu$.