On asymptotic properties of Freud–Sobolev orthogonal polynomials

Abstract

In this paper we consider a Sobolev inner product $(f,g)_S=\int fg\,d\mu+ \lambda \int f'g'\,d\mu (*)$, 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 $d\mu=e -x }dx$ 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 Qn. Finally, we obtain some asymptotics for {Qn}.