Asymptotics for Orthogonal Polynomials, Exponentially Small Perturbations and Meromorphic Continuations of Herglotz Functions

Citation

Kozhan, Rostyslav (2010) Asymptotics for Orthogonal Polynomials, Exponentially Small Perturbations and Meromorphic Continuations of Herglotz Functions. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/KK1A-Z663. https://resolver.caltech.edu/CaltechTHESIS:06072010-004607725

Abstract

The thesis consists of a series of results on the theory of orthogonal polynomials on the real line.

1. We establish Szego asymptotics for matrix-valued measures under the assumption that the absolutely continuous part satises Szego's condition and the mass points satisfy a Blaschke-type condition. This generalizes the scalar analogue of Peherstorfer-Yuditskii [PY01] and the matrix-valued result of AptekarevNikishin [AN83], which handles only a finite number of mass points.

2. We obtain matrix-valued Jost asymptotics for a block Jacobi matrix under an L¹-type condition on 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 [DS06b].

3. The latter results allow us to fully characterize the matrix-valued Weyl-Titchmarsh m-functions of block Jacobi matrices with exponentially converging parameters.

4. We find a necessary and sufficient condition for a finite gap Herglotz function m to be the m-function of a Jacobi matrix with the prescribed "distance" from the isospectral torus of periodic Jacobi matrices associated with a given finite gap set (with all gaps open). The condition is in terms of meromorphic continuations of the function m to a natural Riemann surface, and the structure of poles and zeros of m.

5. The results from parts 3 and 4 give certain corollaries on the point perturbations of measures. Namely, we find conditions on when adding or removing a pure point preserves the exponential rate of convergence of Jacobi parameters. The method applies in the matrix-valued case of exponential convergence to the free block Jacobi matrix, and in the scalar case of exponential convergence to a periodic Jacobi matrix. This extends Geronimo's results from [Ger94].

6. We obtain two results on the equivalence classes of block Jacobi matrices: first, that the Jacobi matrix of type 2 in the Nevai class has A_n coefficients converging to 1, and second, that under an L¹-type condition on the Jacobi coefficients, equivalent Jacobi matrices of type 1, 2, and 3 are pairwise asymptotic.

Thesis Files