Universality Limits involving Orthogonal Polynomials on an Arc of the Unit Circle
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- 作者：Doron S. Lubinsky (1)
Vy Nguyen (1) - 关键词：Orthogonal polynomials ; Subarc of unit circle ; Universality limits ; Primary 41A10 ; 42C05 ; 42C99 ; Secondary 30C15 ; 30E10
- 刊名：Computational Methods and Function Theory
- 出版年：2013
- 出版时间：May 2013
- 年：2013
- 卷：13
- 期：1
- 页码：91-106
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Vy Nguyen (1)
1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 聽30332-0160, USA - ISSN：2195-3724
文摘
We establish universality limits for measures on a subarc of the unit circle. Assume that $\mu $ is a regular measure on such an arc, in the sense of Stahl, Totik, and Ullmann, and is absolutely continuous in an open arc containing some point $z_{0}={\text{ e }}^{i\theta _{0}}$ . Assume, moreover, that $\mu ^{\prime }$ is positive and continuous at $z_{0}$ . Then universality for $\mu $ holds at $z_{0}$ , in the sense that the reproducing kernel $K_{n}\left( z,t\right) $ for $\mu $ satisfies $$\begin{aligned} \lim _{n\rightarrow \infty }\frac{K_{n}\left( z_{0}\exp \left( \frac{2\pi is}{ n}\right) ,z_{0}\exp \left( \frac{2\pi i\bar{t}}{n}\right) \right) }{ K_{n}\left( z_{0},z_{0}\right) }={\text{ e }}^{i\pi \left( s-t\right) }S\left( \left( s-t\right) T\left( \theta _{0}\right) \right) , \end{aligned}$$ uniformly for $s,t$ in compact subsets of the plane, where $S\left( z\right) =\frac{\sin \pi z}{\pi z}$ is the sinc kernel, and $T/2\pi $ is the equilibrium density for the arc.