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*} \langle P, P \rangle _{\lambda _n}=\int P^2 d\mu _0 + \lambda _n \int P’^2 d\mu _1, \quad \lambda _n >0, \end{equation*} in the class of all monic polynomials of degree $n$. The goal of this paper is twofold. On the 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 $\left (S_{n, \lambda _n} \right ).$ On the other hand, 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 $\lambda _n .$