Orthogonal polynomials on the circumference and arcs of the circumference

In this paper we study measures and orthogonal polynomials with asymptotica lly periodic reflection coefficients. Il's known that the support of the or thogonality measure of such polynomials consists of several arcs. We show h ow the measure of orthogonality can be approximated (resp. described) by th e aid of the related orthonormal polynomials if the reflection coefficients are additionally of bounded variation (mod N). As an interesting byproduct we obtain that the orthogonality measure is (up to N points) absolutely co ntinuous on the whole circumference. if the reflection coefficients { a(n) } are of bounded variation (mod N) and satisfy lim(n --> infinity) a(n) = 0 . Furthermore, it is demonstrated that the reflection coefficients remain a symptotically periodic if point measures are added on the support. Finally, wr prove that under certain conditions on the arcs orthogonality measures which satisfy a generalized Szego condition have asymptotically periodic re flection coefficients. (C) 2000 Academic Press.