On Christoffel Functions and Related Quantities for Compactly Supported Measures

Let be a compactly supported positive measure on the real line, with associated orthogonal polynomials fpng. Without any global restrictions such as regularity, we discuss convergence in measure for (i) ratio asymptotics for Christo¤el functions; (ii) the Nevai operators (aka the Nevai condition); (iii) universality limits in the bulk. We also establish convergence a.e. for su¢ cently sparse subsequences of Christo¤el function ratios. Orthogonal Polynomials on the real line, Christo¤el functions, universality limits in the bulk. 42C05 1. Introduction Let be a positive measure on the real line, with compact support supp[ ], and in…nitely many points in its support. Then we may de…ne orthonormal polynomials pn (x) = nx n + :::, n > 0; satisfying Z pnpmd = mn: The measure is said to be regular in the sense of Stahl, Totik and Ullmann [29] if (1.1) lim n!1 1=n n = 1 cap (supp [ ]) ; where cap (supp [ ]) is the logarithmic capacity of the support of . In particular, if the support is an interval [a; b], the requirement is that lim n!1 1=n n = 4 b a For de…nitions of logarithmic capacity, and the associated potential theory, see [22], [23], [29]. At …rst this particular de…nition seems technical and obscure to the extent that one might doubt the utility of the concept. There are numerous equivalent de…nitions of regularity, but (1.1) is used because it is relatively direct. An important monograph by Stahl and Totik [29] comprehensively explores regular measures and the asymptotics in of their orthogonal polynomials. More recent analysis appears in [24]. Regularity of a measure is a very weak global requirement. Thus the Erd1⁄2osTurán criterion asserts that if 0 > 0 a.e. in supp[ ], then is regular. But far Date : May 14, 2010. 1Research supported by NSF grant DMS0700427 and US-Israel BSF grant 2004353 1