Local asymptotics for orthonormal polynomials on the unit circle via universality

• Published in: Journal d'analyse mathématique (Jerusalem) Vol. 141; no. 1; pp. 285 - 304
• Main Author: Lubinsky, Doron S.
• Format: Journal Article
• Language: English
• Published: Jerusalem The Hebrew University Magnes Press 01.09.2020; Springer Nature B.V
• Subjects: Abstract Harmonic Analysis; Analysis; Dynamical Systems and Ergodic Theory; Functional Analysis; Mathematics; Mathematics and Statistics; Partial Differential Equations; Polynomials
• ISSN: 0021-7670; 1565-8538
• DOI: 10.1007/s11854-020-0121-8

Let µ be a positive measure on the unit circle that is regular in the sense of Stahl, Totik, and Ullmann. Assume that in some subarc J, µ is absolutely continuous, while µ ′ is positive and continuous. Let { φ n } be the orthonormal polynomials for µ . We show that for appropriate ζ n ∈ J , { φ n ( ζ n ( 1 + z n ) ) φ n ( ζ n ) } n ≥ 1 is a normal family in compact subsets of ℂ. Using universality limits, we show that limits of subsequences have the form e z + C ( e z − 1) for some constant C . Under additional conditions, we can set C = 0.