Universality Limits Involving Orthogonal Polynomials on the Unit Circle

We establish universality limits for measures on the unit circle. Assume that 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 . Assume, moreover, that 0 is positive and continuous at z. Then universality for holds at z, in the sense that the normalized reproducing kernel ~ Kn (z; t) satis…es lim n!1 1 n ~ Kn exp i + 2 a n ; exp i + 2 b n = e (a b) sin (b a) (b a) ; uniformly for a; b in compact subsets of the real line. 1. Introduction and Results Let be a …nite positive Borel measure on [ ; ) with in…nitely many points in its support. Then we may de…ne orthonormal polynomials n (z) = nz n + :::; n > 0; n = 0; 1; 2; ::: satisfying the orthonormality conditions (1.1) 1 2 Z n (z) m (z)d ( ) = mn; where z = ei . We shall usually assume that is regular in the sense of Stahl and Totik [25], so that (1.2) lim n!1 1=n n = 1: This is true if for example 0 > 0 a.e. in [ ; ), but there are pure jump and pure singularly continuous measures that are regular. The nth reproducing kernel for is