Universality Limits Involving Orthogonal Polynomials on a Smooth Closed Contour

We establish universality limits for measures on a smooth closed contour Γ in the plane. Assume that μ is a regular measure on Γ, in the sense of Stahl, Totik, and Ullmann. Let Γ1 be a closed subarc of Γ, such that μ is absolutely continuous in an open arc containing Γ1, and μ′ is positive and continuous in that open subarc. Then universality for μ holds in Γ1, in the sense that the reproducing kernels {Kn (z, t)} for μ satisfy lim n→∞ Kn ( z0 + 2πis n Φ(z0) Φ′(z0) , z0 + 2πit̄ n Φ(z0) Φ′(z0) ) Kn (z0, z0) = eS (s− t) , uniformly for z0 ∈ Γ1, and s, t in compact subsets of the complex plane. Here S (z) = sinπz πz is the sinc kernel, and Φ is a conformal map of the exterior of Γ onto the exterior of the unit ball.