Szego transformations and Nth order associated polynomials on the unit circle

Abstract

In this paper we analyze the Stieltjes functions defined by the Szego inverse transformation of a nontrivial probability measure supported on the unit circle such that the corresponding sequence of orthogonal polynomials is defined by either backward or forward shifts in their Verblunsky parameters. Such polynomials are called anti-associated (respectively associated) orthogonal polynomials. Thus, rational spectral transformations appear in a natural way. (C) 2009 Elsevier Ltd. All rights reserved.