On asymptotic properties of Freud-Sobolev orthogonal polynomials

Abstract

In this paper we consider a Sobolev inner product (f, g) S = fgd + # f # g # d (1) and we characterize the measures for which there exists an algebraic relation between the polynomials, orthogonal with respect to the measure and the polynomials, orthogonal with respect to (1), such that the number of involved terms does not depend on the degree of the polynomials. Thus, we reach in a natural way the measures associated with a Freud weight. In particular, we study the case d = dx supported on the full real axis and we analyze the connection between the so-called Nevai polynomials (associated with the Freud weight e -x ) and the Sobolev orthogonal polynomials Q n . Finally, we obtain some asymptotics for }