Sobolev orthogonal polynomials: Balance and asymptotics

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

$$\langle P, P \rangle_{\lambda_n}=\int P^2 \, d\mu_0 + \lambda_n \int P'^2 \, d\mu_1, \quad \lambda_n >0,$$

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 $\lambda_n\, .$