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Sobolev orthogonal polynomials: balance and asymptotics

M. Alfaro·J. J. Moreno-Balcazar·A. Pena·M. L. Rezola
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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'^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 $λ_n .$

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M. Alfaro, J. J. Moreno-Balcazar, A. Pena, M. L. Rezola. 2007-05-02. Sobolev orthogonal polynomials: balance and asymptotics. https://arxiv.org/abs/math/0606589

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