Singular Measures on the Unit Circle and Their Reflection Coefficients

Abstract

Orthogonal polynomials on the unit circle are determined by their reflection coefficients through the Szegő recurrences. In the present paper we examine two particular classes of measures on the unit circle. The first one consists of measures whose reflection coefficients tend to the unit circle. For such measures we give complete description of their supports (up to the set of isolated masspoints) in terms of reflection coefficients. The supports of measures from the second class have finitely many limit points. We prove the unit circle analogue of M. G. Krein's characterization for the similar class of measures on the real line. The examples of measures from both classes are given.