Asymptotics of orthogonal polynomials and point perturbation on the unit circle

In the first five sections, we deal with the class of probability measures with asymptotically periodic Verblunsky coefficients of p-type bounded variation. The goal is to investigate the perturbation of the Verblunsky coefficients when we add a pure point to a gap of the essential spectrum. For the asymptotically constant case, we give an asymptotic formula for the orthonormal polynomials in the gap, prove that the perturbation term converges and show the limit explicitly. Furthermore, we prove that the perturbation is of bounded variation. Then we generalize the method to the asymptotically periodic case and prove similar results. In the last two sections, we show that the bounded variation condition can be removed if a certain symmetry condition is satisfied. Finally, we consider the special case when the Verblunsky coefficients are real with the rate of convergence being c_n . We prove that the rate of convergence of the perturbation is in fact O(c_n). In particular, the special case c_n = 1/n will serve as a counterexample to the possibility that the convergence of the perturbed Verblunsky coefficients should be exponentially fast when a point is added to a gap.Comment: Published in the Journal of Approximation Theor

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 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

Orthogonal polynomials on the unit circle: New results

We announce numerous new results in the theory of orthogonal polynomials on the unit circle

OPUC on One Foot

We present an expository introduction to orthogonal polynomials on the unit circle

Necessary and Sufficient Conditions in the Spectral Theory of Jacobi Matrices and Schr\"odinger Operators

We announce three results in the theory of Jacobi matrices and Schr\"odinger operators. First, we give necessary and sufficient conditions for a measure to be the spectral measure of a Schr\"odinger operator -\f{d^2}{dx^2} +V(x) on $L^2 (0,\infty)$ with $V\in L^2 (0,\infty)$ and $u(0)=0$ boundary condition. Second, we give necessary and sufficient conditions on the Jacobi parameters for the associated orthogonal polynomials to have Szeg\H{o} asymptotics. Finally, we provide necessary and sufficient conditions on a measure to be the spectral measure of a Jacobi matrix with exponential decay at a given rate.Comment: 10 page

Jost asymptotics for matrix orthogonal polynomials on the real line

We obtain matrix-valued Jost asymptotics for block Jacobi matrices under an L1-type condition on Jacobi parameters, and give a necessary and sufficient condition for an analytic matrix-valued function to be the Jost function of a block Jacobi matrix with exponentially converging parameters. This establishes the matrix-valued analogue of Damanik-Simon-II paper [6]. The above results allow us to fully characterize the matrix-valued Weyl-Titchmarsh m-functions of block Jacobi matrices with exponentially converging parameters