PERTURBATION OF ORTHOGONAL POLYNOMIALS ON AN ARC OF THE UNIT-CIRCLE

Orthogonal polynomials on the unit circle are completely determined by their reflection coefficients through the Szego recurrences. We assume that the reflection coefficients converge to some complex number a with 0 < \a\ < 1. The polynomials then live essentially on the are {e(io) : a less than or equal to theta less than or equal to 2 pi - alpha) where cos(a/2) (def) root 1 - \a\(2) with a is an element of (0, pi). We analyze the orthogonal polynomials by comparing them with the orthogonal polynomials with constant reflection coefficients, which were studied earlier by Ya. L. Geronimus and N. I. Akhiezer. In particular, we show that under certain assumptions on the rate of convergence of the reflection coefficients the orthogonality measure will be absolutely continuous on the are. In addition, we also prove the unit circle analogue of M. G. Krein's characterization of compactly.