Asymptotics on the support for sobolev orthogonal polynomials on a bounded interval

Abstract

In the present paper we give sufficient conditions on the measures of orthogonality in order to establish the asymptotic behavior, on the support of the measures, for the Sobolev orthogonal polynomials with respect to a Sobolev inner product ( f ( x ) , g ( x ) ) s μ = ∫ − 1 1 f ( x ) g ( x ) d μ o ( x ) + ∫ − 1 1 f ' ( x ) g ' ( x ) d μ 1 ( x ) with μ 0 and μ 1 two finite positive Borel measures on [−1, 1]. In this situation we prove that the monic Sobolev orthogonal polynomials behave, inside the support, like the monic orthogonal polynomials with respect to a rational modification of the measure μ 1. We remark that we obtain the results using the connection between Sobolev orthogonal Laurent polynomials on the circle and Sobolev orthogonal polynomials on the interval and using the asymptotic theory of Sobolev orthogonal Laurent polynomials on the circle, that we develop in this paper, and which is important by itself.