Comparative Asymptotics for Perturbed Orthogonal Polynomials

Let {Φn}n∈N0 and {Φ̃n}n∈N0 be such systems of orthonormal polynomials on the unit circle that the recurrence coefficients of the perturbed polynomials Φ̃n behave asymptotically like those of Φn. We give, under weak assumptions on the system {Φn}n∈N0 and the perturbations, comparative asymptotics as for Φ̃n(z)/Φ ∗ n(z) etc., Φ ∗ n(z) := z Φ̄n( 1 z ), on the open unit disk and on the circumference mainly off the support of the measure σ with respect to which the Φn’s are orthonormal. In particular these results apply if the comparative system {Φn}n∈N0 has a support which consists of several arcs of the unit circumference, as in the case when the recurrence coefficients are (asymptotically) periodic.