Sobolev orthogonal polynomials: balance and asymptotics

Abstract

Let ${\mu }_0$ and ${\mu }_1$ be measures supported on an unbounded interval and $S_{n,{\lambda }_n}$ the extremal varying Sobolev polynomial which minimizes \begin{equation*} < P, P >_{λ_n}=\int P^2 dμ_0 + λ_n \int P'^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 $({\lambda }_n)$ such that both measures ${\mu }_0$ and ${\mu }_1$ play a role in the asymptotics of $(S_{n,{\lambda }_n}).$ On the other, we apply such ideas to the case when both ${\mu }_0$ and ${\mu }_1$ are Freud weights. Asymptotics for the corresponding $S_{n,{\lambda }_n}$ are computed, illustrating the accuracy of the choice of ${\lambda }_n.$

20 pages. Changed content