In the first five sections, we deal with the class of probability measures
with asymptotically periodic Verblunsky coefficients of p-type bounded
variation. The goal is to investigate the perturbation of the Verblunsky
coefficients when we add a pure point to a gap of the essential spectrum.
For the asymptotically constant case, we give an asymptotic formula for the
orthonormal polynomials in the gap, prove that the perturbation term converges
and show the limit explicitly. Furthermore, we prove that the perturbation is
of bounded variation. Then we generalize the method to the asymptotically
periodic case and prove similar results.
In the last two sections, we show that the bounded variation condition can be
removed if a certain symmetry condition is satisfied. Finally, we consider the
special case when the Verblunsky coefficients are real with the rate of
convergence being c_n . We prove that the rate of convergence of the
perturbation is in fact O(c_n). In particular, the special case c_n = 1/n will
serve as a counterexample to the possibility that the convergence of the
perturbed Verblunsky coefficients should be exponentially fast when a point is
added to a gap.Comment: Published in the Journal of Approximation Theor