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We show how localization and smoothing techniques can be used to establish universality in the bulk of the spectrum for a fixed positive measure $\mu $ on $[-1,1] $. Assume that $\mu $ is a regular measure, and is absolutely continuous in an open interval containing some point $x$. Assume moreover, that $\mu ^{\prime }$ is positive and continuous at $x$. Then universality for $\mu $ holds at $x$. If the hypothesis holds for $x$ in a compact subset of $\left( -1,1\right) $, universality holds uniformly for such $x$. Indeed, this follows from universality for the classical Legendre weight. We also establish universality in an $L_{p}$ sense under weaker assumptions on $\mu .$
@article{10_4007_annals_2009_170_915,
author = {Doron S. Lubinsky},
title = {A new approach to universality limits involving orthogonal polynomials},
journal = {Annals of mathematics},
pages = {915--939},
publisher = {mathdoc},
volume = {170},
number = {2},
year = {2009},
doi = {10.4007/annals.2009.170.915},
mrnumber = {2552113},
zbl = {1176.42022},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2009.170.915/}
}
TY - JOUR AU - Doron S. Lubinsky TI - A new approach to universality limits involving orthogonal polynomials JO - Annals of mathematics PY - 2009 SP - 915 EP - 939 VL - 170 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2009.170.915/ DO - 10.4007/annals.2009.170.915 LA - en ID - 10_4007_annals_2009_170_915 ER -
%0 Journal Article %A Doron S. Lubinsky %T A new approach to universality limits involving orthogonal polynomials %J Annals of mathematics %D 2009 %P 915-939 %V 170 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2009.170.915/ %R 10.4007/annals.2009.170.915 %G en %F 10_4007_annals_2009_170_915
Doron S. Lubinsky. A new approach to universality limits involving orthogonal polynomials. Annals of mathematics, Tome 170 (2009) no. 2, pp. 915-939. doi: 10.4007/annals.2009.170.915
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