07-182 Barry Simon

Weak Convergence of CD Kernels and Applications (283K, PDF) Jul 17, 07

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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 $d\mu$.

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