Comparative asymptotics for perturbed orthogonal polynomials

Abstract:

Let $\{\Phi _n\}_{n\in \mathbb N_0}$ and $\{\widetilde \Phi _n\}_{n\in \mathbb N_0}$ be such systems of orthonormal polynomials on the unit circle that the recurrence coefficients of the perturbed polynomials $\widetilde \Phi _n$ behave asymptotically like those of $\Phi _n$. We give, under weak assumptions on the system $\{\Phi _n\}_{n\in \mathbb N_0}$ and the perturbations, comparative asymptotics as for $\widetilde \Phi _n^*(z)/ \Phi _n^*(z)$ etc., $\Phi _n^*(z):= z^n\bar \Phi _n(\frac 1z)$, on the open unit disk and on the circumference mainly off the support of the measure $\sigma$ with respect to which the $\Phi _n$’s are orthonormal. In particular these results apply if the comparative system $\{\Phi _n\} _{n\in \mathbb N_0}$ has a support which consists of several arcs of the unit circumference, as in the case when the recurrence coefficients are (asymptotically) periodic.