Orthogonal Polynomials on the Circumference and Arcs of the Circumference

Abstract

In this paper we study measures and orthogonal polynomials with asymptotically periodic reflection coefficients. It's known that the support of the orthogonality measure of such polynomials consists of several arcs. We show how the measure of orthogonality can be approximated (resp. described) by the aid of the related orthonormal polynomials if the reflection coefficients are additionally of bounded variation (modN). As an interesting byproduct we obtain that the orthogonality measure is (up to N points) absolutely continuous on the whole circumference, if the reflection coefficients {an} are of bounded variation (modN) and satisfy limn→∞an=0. Furthermore, it is demonstrated that the reflection coefficients remain asymptotically periodic if point measures are added on the support. Finally, we prove that under certain conditions on the arcs orthogonality measures which satisfy a generalized Szegő condition have asymptotically periodic reflection coefficients.