COMPARATIVE ASYMPTOTICS FOR PERTURBED ORTHOGONAL POLYNOMIALS

Let {Phi(n)}(n is an element of N0) and {<(Phi)over tilde>(n)}(n is an element of N0) be such systems of orthonormal polynomials on the unit circle that the recurrence coefficients of the perturbed polynomials <(Phi)over tilde>(n) behave asymptotically like those of Phi(n). We gi ve, under weak assumptions on the system {Phi(n)}(n is an element of N 0) and the perturbations, comparative asymptotics as for <(Phi)over ti lde>(n)(z)/Phi*(n)(z) etc., Phi*(n)(z) := z(n)<(Phi)over bar>(n)(1/z) , on the open unit disk and on the circumference mainly off the suppor t of the measure sigma with respect to which the Phi(n)'s are orthonor mal. In particular these results apply if the comparative system {Phi( n)}(n is an element of N0) has a support which consists of several arc s of the unit circumference, as in the case when the recurrence coeffi cients are (asymptotically) periodic.