Local limits for orthogonal polynomials for varying measures via universality
type: Item
instance of: scholarly article
title: Local limits for orthogonal polynomials for varying measures via universality
zbMATH Open document ID: 1442.42063
author: Eli Levin
author: Doron S. Lubinsky
published in: Journal of Approximation Theory
publication date: 2020-04-07T00:00:00Z
review text: In the setting of varying measures, the authors establish local asymptotics from universality limits. More precisely, they consider orthogonal polynomials \(\left\{ p_n \left(e^{-2nQ_n}, x \right) \right\}\) and use universality limits to prove ``local limits'' \[ \lim_{n \to +\infty} \frac{ p_n \left(e^{-2nQ_n}, y_{jn} + \frac{z}{\tilde{K}_{n} \left(y_{jn}, y_{jn} \right)} \right)}{ p_n \left(e^{-2nQ_n}, y_{jn} \right)} \, e^{{}^{-\frac{nQ_{n}'(y_{jn})}{\tilde{K}_{n} \left(y_{jn}, y_{jn} \right)}z}} = \cos(\pi z), \] where \(y_{jn}\) is a local maximum point of \(|p_n| e^{-n Q_n}\) in the ``bulk'' of the support, \(\tilde{K}_{n} \left(y_{jn}, y_{jn} \right)\) is the normalized reproducing kernel, and the limit holds uniformly for \(z\) in compact subsets of the plane. They also consider local limits at the ``soft edge'' of the spectrum, which involve the Airy function. Unlike from earlier works (see e.g. [\textit{D. S. Lubinsky}, Proc. Am. Math. Soc. 147, No. 9, 3877--3886 (2019; Zbl 1422.42037)]), in this paper the authors consider extra factors in the limits.
reviewed by: P. Rocha
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 41A60
zbMATH DE Number: 7187599
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: universality limits
zbMATH Keywords: varying weights
zbMATH Keywords: exponential weights
zbMATH Keywords: asymptotics
MaRDI profile type: Publication
full work available at URL: j.jat.2020.105394
OpenAlex ID: W3010289128
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DOI: 10.1016/J.JAT.2020.105394
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description: scientific article; zbMATH DE number 7187599