A new approach to universality limits involving orthogonal polynomials

• 成果类型: Article
• 署名作者: Lubinsky, Doron S.
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2009.170.915
• 发表日期: 2009
• 页码: 915-939
• 关键词: christoffel functions; uniform asymptotics; exponential weights; Respect

摘要

We show how localization and smoothing techniques can be used to establish universality in the bulk of the spectrum for a fixed positive measure mu on [-1, 1]. Assume that mu is a regular measure, and is absolutely continuous in an open interval containing some point x. Assume moreover, that mu' is positive and continuous at x. Then universality for mu holds at x. If the hypothesis holds for x in a compact subset of (-1, 1), universality holds uniformly for such x. Indeed, this follows from universality for the classical Legendre weight. We also establish universality in an L-p sense under weaker assumptions on mu.