Sobolev orthogonal polynomials: balance and asymptotics

Abstract

Let μ0 and μ1 be measures supported on an unbounded interval and Sn,λn the extremal varying Sobolev polynomial which minimizes $$<P,P>_\lambda_n=\int P^2 d\mu_0+\lambda_n \int P'^2 d\mu_1, \lambda_n>0$$ 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 (λn) such that both measures μ0 and μ1 play a role in the asymptotics of (Sn,λn). On the other, we apply such ideas to the case when both μ0 and μ1 are Freud weights. Asymptotics for the corresponding Sn,λn are computed, illustrating the accuracy of the choice of λn.