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 $(1/n)K_n(x,x)d\mu (x)$. By combining this with the asymptotic upper bounds of Máté and Nevai [16] and Totik [33] on $n{\lambda }_n(x)$, we prove some general results on ${\int }_I(1/n)K_n(x,x)d{\mu }_s\to 0$ for the singular part of $d\mu$ and ${\int }_I|{\rho }_E(x)-(w(x)/n)K_n(x,x)|\mathrm{dx}\to 0$, where ${\rho }_E$ is the density of the equilibrium measure and $w(x)$ the density of $d\mu$