Orthogonal polynomials on the unit circle: Symmetrization and quadratic decomposition

Abstract

Problems related to the symmetrization of sequences of orthogonal polynomials on the real line play an important role, as shown in Chihara (“An Introduction to Orthogonal Polynomials,” Gordon & Breach, New York, 1978), pages 40–43. On the other hand, the study done in Chihara and Chihara ( J. Math. Anal. Appl., 126 (1987) , 275–291) corresponds to a particular quadratic decomposition of a sequence of orthogonal polynomials on the real line, as a constructive method of a class of nonsymmetric orthogonal polynomials. In this paper we present some results concerning the symmetrization and quadratic decomposition of sequences of orthogonal polynomials, related to a quasi-definite functional on the unit circle.