Weak convergence of CD kernels and applications

Abstract

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