On asymptotic properties of Freud-Sobolev orthogonal polynomials

In this paper we consider a Sobolev inner product (f, g)(S) = integral fg dmu + lambda integral f' g' dmu and we characterize the measures mu for which there exists an algebraic relation between the polynomials, {P-n}, orthogonal with respect to the measure P and the polynomials, {Qn}, orthogonal with respect to (*), 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 dmu = e(-x4) dx supported on the full real axis and we analyze the connection between the so-called Nevai polynomials (associated with the Freud weight e(-x4) and the Sobolev orthogonal polynomials Q(n). Finally, we obtain some asymptotics for {Q(n)}. More precisely, we give the relative asymptotics {Q(n)(x)/P-n(x)} on compact subsets of C\R as well as the outer Plancherel-Rotach-type asymptotics {Qn((4)rootnx)/P-n((4)rootnx)} on compact subsets of C\[-a, a] being a = (4)root4/3 (C) 2003 Elsevier Inc. All rights reserved.