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Computational Methods and Function Theory

, Volume 7, Issue 2, pp 543–561 | Cite as

Universality Limits Involving Orthogonal Polynomials on the Unit Circle

  • Eli Levin
  • Doron S. Lubinsky
Article

Abstract

We establish universality limits for measures on the unit circle. Assume that μ is a regular measure on the unit circle in the sense of Stahl and Totik, and is absolutely continuous in an open arc containing some point z = e iθ. Assume, moreover, that μ′ is positive and continuous at z. then universality for μ holds at z, in the sense that the normalized reproducing kernel ~Kn(z, t) satisfies
$${\rm lim}_{n\to \infty}{1\over n}\tilde{K}_{n}(e^{i(\theta + 2\pi a/n)},e^{i(\theta + 2\pi b/n)}\bigg) = e^{i\pi(a-b)}{{\rm sin} \pi (b-a)\over \pi (b-a)}$$
, uniformly for a, b in compact subsets of the real line.

En]Keywords

Universality limits 

2000 MSC

42C05 30C10 

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Copyright information

© Heldermann  Verlag 2007

Authors and Affiliations

  1. 1.Mathematics DepartmentThe Open University of IsraelRaananaIsrael
  2. 2.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

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