Local limits for orthogonal polynomials for varying measures via universality

Eli Levin; Doron S. Lubinsky

doi:10.1016/j.jat.2020.105394

Local limits for orthogonal polynomials for varying measures via universality

Eli Levin, Doron S. Lubinsky

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We consider orthogonal polynomials p_ne^−2nQ_n,x for varying measures and use universality limits to prove ”local limits” limn→∞[Formula presented]e^{−[Formula presented]z}=cosπz.Here y_jn is a local maximum point of p_ne^−nQ_n in the ”bulk” of the support, K̃_ny_jn,y_jn is the normalized reproducing kernel, and the limit holds uniformly for z in compact subsets of the plane. We also consider local limits at the ”soft edge” of the spectrum, which involve the Airy function.

Original language	English
Article number	105394
Journal	Journal of Approximation Theory
Volume	254
DOIs	https://doi.org/10.1016/j.jat.2020.105394
State	Published - Jun 2020

Bibliographical note

Publisher Copyright:
© 2020 Elsevier Inc.

Keywords

Asymptotics
Exponential weights
Orthogonal Polynomials
University Limits
Varying Weights

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10.1016/j.jat.2020.105394

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title = "Local limits for orthogonal polynomials for varying measures via universality",

abstract = "We consider orthogonal polynomials pne−2nQn ,x for varying measures and use universality limits to prove ”local limits” limn→∞[Formula presented]e−[Formula presented]z=cosπz.Here yjn is a local maximum point of pne−nQn in the ”bulk” of the support, {\~K}nyjn,yjn is the normalized reproducing kernel, and the limit holds uniformly for z in compact subsets of the plane. We also consider local limits at the ”soft edge” of the spectrum, which involve the Airy function.",

keywords = "Asymptotics, Exponential weights, Orthogonal Polynomials, University Limits, Varying Weights",

author = "Eli Levin and Lubinsky, {Doron S.}",

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year = "2020",

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AB - We consider orthogonal polynomials pne−2nQn ,x for varying measures and use universality limits to prove ”local limits” limn→∞[Formula presented]e−[Formula presented]z=cosπz.Here yjn is a local maximum point of pne−nQn in the ”bulk” of the support, K̃nyjn,yjn is the normalized reproducing kernel, and the limit holds uniformly for z in compact subsets of the plane. We also consider local limits at the ”soft edge” of the spectrum, which involve the Airy function.

KW - Asymptotics

KW - Exponential weights

KW - Orthogonal Polynomials

KW - University Limits

KW - Varying Weights

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ER -

Local limits for orthogonal polynomials for varying measures via universality

Abstract

Bibliographical note

Keywords

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