Weak convergence of CD kernels and applications
Barry Simon
Source: Duke Math. J. Volume 146, Number 2
(2009), 305-330.
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 $\int_I ({1}/{n}) K_n(x,x) d\mu_\rm{s}\to 0$ for the singular part of $d\mu$ and $\int_I \vert\rho_E(x) - ({w(x)}/{n}) K_n(x,x)\vert dx\to 0$, where $\rho_E$ is the density of the equilibrium measure and $w(x)$ the density of $d\mu$
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Permanent link to this document: http://projecteuclid.org/euclid.dmj/1231170942
Digital Object Identifier: doi:10.1215/00127094-2008-067
Mathematical Reviews number (MathSciNet): MR2477763
Zentralblatt MATH identifier: 1158.33003
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