Applications of universality limits to zeros and reproducing kernels of orthogonal polynomials

Eli Levin; Doron S. Lubinsky

doi:10.1016/j.jat.2007.05.003

Applications of universality limits to zeros and reproducing kernels of orthogonal polynomials

Eli Levin
, Doron S. Lubinsky

Mathematics

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

Abstract

We apply universality limits to asymptotics of spacing of zeros {} fenced(x_kn) of orthogonal polynomials, for weights with compact support and for exponential weights. A typical result isunder(lim, n → ∞) fenced(x_kn - x_{k + 1, n}) over(K, ̃)_n fenced(x_kn, x_kn) = 1under minimal hypotheses on the weight, with over(K, ̃)_n denoting a normalized reproducing kernel. Moreover, for exponential weights, we derive asymptotics for the differentiated kernels:K_n^{fenced(r, s)} fenced(x, x) = underover(∑, k = 0, n - 1) p_k^fenced(r) fenced(x) p_k^fenced(s) fenced(x) .

Original language	English
Pages (from-to)	69-95
Number of pages	27
Journal	Journal of Approximation Theory
Volume	150
Issue number	1
DOIs	https://doi.org/10.1016/j.jat.2007.05.003
State	Published - Jan 2008

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10.1016/j.jat.2007.05.003

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title = "Applications of universality limits to zeros and reproducing kernels of orthogonal polynomials",

abstract = "We apply universality limits to asymptotics of spacing of zeros \{\} fenced(xkn) of orthogonal polynomials, for weights with compact support and for exponential weights. A typical result isunder(lim, n → ∞) fenced(xkn - xk + 1, n) over(K,{\~ })n fenced(xkn, xkn) = 1under minimal hypotheses on the weight, with over(K,{\~ })n denoting a normalized reproducing kernel. Moreover, for exponential weights, we derive asymptotics for the differentiated kernels:Knfenced(r, s) fenced(x, x) = underover(∑, k = 0, n - 1) pkfenced(r) fenced(x) pkfenced(s) fenced(x) .",

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doi = "10.1016/j.jat.2007.05.003",

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AU - Levin, Eli

AU - Lubinsky, Doron S.

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N2 - We apply universality limits to asymptotics of spacing of zeros {} fenced(xkn) of orthogonal polynomials, for weights with compact support and for exponential weights. A typical result isunder(lim, n → ∞) fenced(xkn - xk + 1, n) over(K, ̃)n fenced(xkn, xkn) = 1under minimal hypotheses on the weight, with over(K, ̃)n denoting a normalized reproducing kernel. Moreover, for exponential weights, we derive asymptotics for the differentiated kernels:Knfenced(r, s) fenced(x, x) = underover(∑, k = 0, n - 1) pkfenced(r) fenced(x) pkfenced(s) fenced(x) .

AB - We apply universality limits to asymptotics of spacing of zeros {} fenced(xkn) of orthogonal polynomials, for weights with compact support and for exponential weights. A typical result isunder(lim, n → ∞) fenced(xkn - xk + 1, n) over(K, ̃)n fenced(xkn, xkn) = 1under minimal hypotheses on the weight, with over(K, ̃)n denoting a normalized reproducing kernel. Moreover, for exponential weights, we derive asymptotics for the differentiated kernels:Knfenced(r, s) fenced(x, x) = underover(∑, k = 0, n - 1) pkfenced(r) fenced(x) pkfenced(s) fenced(x) .

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EP - 95

JO - Journal of Approximation Theory

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Applications of universality limits to zeros and reproducing kernels of orthogonal polynomials

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