A new approach to the asymptotics for Sobolev orthogonal polynomials

TL;DR

This paper introduces a novel approach to analyze the asymptotic behavior of Sobolev orthogonal polynomials with derivatives in the inner product, especially for measures with unbounded support, providing new insights into their zeros and asymptotics.

Contribution

The paper develops a new method for studying Sobolev orthogonal polynomials with unbounded support, overcoming limitations of previous techniques used for bounded measures.

Findings

Derived relative asymptotics for Sobolev orthogonal polynomials

Established Mehler--Heine type asymptotics for these polynomials

Analyzed the asymptotic behavior of zeros

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.