Computational Methods and Function Theory 7 (2007), No. 2, 543--561
Copyright Heldermann Verlag 2007
Eli Levin
elile@openu.ac.il , The Open University of Israel, Mathematics Department, P.O. Box 808, Raanana 43107, Israel.
Doron S. Lubinsky
lubinsky@math.gatech.edu , Georgia Institute of Technology, School of Mathematics, Atlanta, GA 30332-0160, U.S.A.
We establish universality limits for measures on the unit circle. Assume that $\mu$ 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 $z=e^{i\theta}$. Assume, moreover, that $\mu'$ is positive and continuous at $z$. Then universality for $\mu$ holds at $z$, in the sense that the normalized reproducing kernel $\tilde{K}_{n}(z,t)$ satisfies $$ \lim_{n\to \infty }\frac{1}{n} \tilde{K}_{n}\!\left( e^{i(\theta+2\pi a/n)},e^{i(\theta+2\pi b/n)} \right) = e^{i\pi(a-b)}\frac{\sin\pi(b-a)}{\pi(b-a)}, $$ uniformly for $a,b$ in compact subsets of the real line.
Keywords: Universality limits.
MSC 2000: 42C05, 30C10.
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