Universality Type Limits for Bergman Orthogonal Polynomials

We establish universality type limits for Bergman orthogonal polynomials on simply connected regions in the complex plane with smooth boundary. Keywords: Universality Limits, Bergman Polynomials AMS Classi…cation: 41A55, 65D30, 65B99, 42C99 Research supported by NSF grant DMS0700427 and US-Israel BSF grant 2004353 1. Introduction and Results Let G be a bounded simply connected domain in the complex plane, bounded by a Jordan curve . Let be a …nite positive Borel measure on G. We may de…ne, for n 0; orthonormal polynomials pn (z) = nz n + :::; n > 0 satisfying Z G pnpmd = mn: We shall usually assume that is regular in the sense of Stahl and Totik [27, p. 60], so that (1.1) lim n!1 1=n n = 1 cap (G) : Here cap (G) denotes the logarithmic capacity of G. Moreover, we shall also assume that is absolutely continuous with respect to planar Lebesgue measure dA near given points on @G. In this sense, the polynomials fpng fall within the framework of Bergman polynomials. Throughout, we let w (z) = d dA (z) , a.e. z 2 G: Moreover, for u 2 @G, we de…ne w (u) = lim z!u;z2G w (z) ; whenever the limit is de…ned. We note that when w > 0 a.e. on G, then it follows from Widom’s criterion for regularity, that is regular in the sense of Stahl and Totik [27, pp. 106-107]. Date : November 3, 2009. 1