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

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 An 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.