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Universality limits involving orthogonal polynomials on the unit circle. (English) Zbl 1214.42045

Summary: 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.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
30C10 Polynomials and rational functions of one complex variable
Full Text: DOI

References:

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