On asymptotic properties of Freud-Sobolev orthogonal polynomials

Abstract

In this paper we consider a Sobolev inner product (f, g)s = ∫ fg dμ + λ ∫ f′ g′ dμ and we characterize the measures μ for which there exists an algebraic relation between the polynomials, {Pn}, orthogonal with respect to the measure μ 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 dμ = 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 orthongal polynomials Qn. Finally, we obtain some asymptotics for {Qn}. More precisely, we give the relative asymptotics {Qn(x)/Pn (x)} on compact subsets of ℂ\ℝ as well as the outer Plancherel-Rotach-type asymptotics {Qn(4 nx)/Pn(4nx)} on compact subsets of ℂ\[-a,a] being a = 4 4/3. © 2003 Elsevier Inc. All rights reserved.