Orthogonal polynomials on the unit circle and chain sequences

Abstract

Szegő has shown that real orthogonal polynomials on the unit circle can be mapped to orthogonal polynomials on the interval [−1,1] by the transformation 2x=z+z−1. In the 80’s and 90’s Delsarte and Genin showed that real orthogonal polynomials on the unit circle can be mapped to symmetric orthogonal polynomials on the interval [−1,1] using the transformation 2x=z1/2+z−1/2. We extend the results of Delsarte and Genin to all orthogonal polynomials on the unit circle. The transformation maps to functions on [−1,1] that can be seen as extensions of symmetric orthogonal polynomials on [−1,1] satisfying a three-term recurrence formula with real coefficients {cn} and {dn}, where {dn} is also a positive chain sequence. Via the results established, we obtain a characterization for a point w(|w|=1) to be a pure point of the measure involved. We also give a characterization for orthogonal polynomials on the unit circle in terms of the two sequences {cn} and {dn}.