A new approach to the asymptotics for Sobolev orthogonal polynomials

Abstract

In this paper we deal with polynomials orthogonal with respect to an inner product involving derivatives, that is, a Sobolev inner product. Indeed, we consider Sobolev type polynomials which are orthogonal with respect to $$(f,g)=\int fg d\mu +\sum_{i=0}^r M_i f^{(i)}(0) g^{(i)}(0), \quad M_i \ge 0,$$ where $\mu$ is a certain probability measure with unbounded support. For these polynomials, we obtain the relative asymptotics with respect to orthogonal polynomials related to $\mu$, Mehler--Heine type asymptotics and their consequences about the asymptotic behaviour of the zeros. To establish these results we use a new approach different from the methods used in the literature up to now. The development of this technique is highly motivated by the fact that the methods used when $\mu$ is bounded do not work.