Proceedings of the international seminar on Approximation and optimization
Ratio and relative asymptotics of polynomials orthogonal on an arc of the unit circle
Journal of Approximation Theory
Asymptotic behaviour of solutions of linear recurrences and sequences of Mo¨bius-transformations
Journal of Approximation Theory
Orthogonal polynomials on the circumference and arcs of the circumference
Journal of Approximation Theory
A formula for inserting point masses
Journal of Computational and Applied Mathematics
Some asymptotic properties for orthogonal polynomials with respect to varying measures
Journal of Approximation Theory
Journal of Approximation Theory
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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.