Perturbation of orthogonal polynomials on an arc of the unit circle

Details Perturbation of orthogonal polynomials on an arc of the unit circle Perturbation of orthogonal polynomials on an arc of the unit circle

1995-10-08

arxiv.org/abs/math/9510211v1

Mathematics

Classical Analysis and ODEs

Scientific paper

Orthogonal polynomials on the unit circle are completely determined by their reflection coefficients through the Szeg\H{o} recurrences. We assume that the reflection coefficients converge to some complex number a with 0 < |a| < 1. The polynomials then live essentially on the arc {e^(i theta): alpha <= theta <= 2 pi - alpha} where cos alpha/2 = sqrt(1-|a|^2) with 0 <= alpha <= 2 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 arc. In addition, we also prove the unit circle analogue of M. G. Krein's characterization of compactly supported nonnegative Borel measures on the real line whose support contains one single limit point in terms of the corresponding system of orthogonal polynomials.

Affiliated with

Also associated with

No associations

Rating

Perturbation of orthogonal polynomials on an arc of the unit circle does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Perturbation of orthogonal polynomials on an arc of the unit circle, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Perturbation of orthogonal polynomials on an arc of the unit circle will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-656946