Connections between Interval and Unit Circle for Sobolev Orthogonal Polynomials. Strong Asymptotics on the Real Line

Abstract

In this paper we show the connection between Sobolev orthogonal Laurent polynomials on the unit circle and Sobolev orthogonal polynomials on a bounded interval of the real line. As a consequence we deduce the strong outer asymptotics for Sobolev orthogonal polynomials with respect to the inner product $$\langle f(x),g(x)\rangle_{s_{\mu}}=\int_{-1}^{1}f(x)g(x)\,\mathrm{d}\mu _{0}(x)+\int_{-1}^{1}f'(x)g'(x)\,\mathrm{d}\mu _{1}(x),$$ assuming that μ1 belongs to the Szegő class as well as (1−x2)−1∈L1(μ1).