Sobolev orthogonal polynomials: balance and asymptotics

Abstract

Let $μ_0$ and $μ_1$ be measures supported on an unbounded interval and $S_{n,λ_n}$ the extremal varying Sobolev polynomial which minimizes \begin{equation*} < P, P >_{λ_n}=\int P^2 dμ_0 + λ_n \int P&#39;^2 dμ_1, \quad λ_n >0 \end{equation*} \noindent in the class of all monic polynomials of degree $n$. The goal of this paper is twofold. On one hand, we discuss how to balance both terms of this inner product, that is, how to choose a sequence $(λ_n)$ such that both measures $μ_0$ and $μ_1$ play a role in the asymptotics of $(S_{n, λ_n}).$ On the other, we apply such ideas to the case when both $μ_0$ and $μ_1$ are Freud weights. Asymptotics for the corresponding $S_{n, λ_n}$ are computed, illustrating the accuracy of the choice of $λ_n .$