Citation
Kozhan, Rostyslav (2010) Asymptotics for orthogonal polynomials, exponentially small perturbations and meromorphic continuations of Herglotz functions. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:06072010004607725
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 matrixvalued measures under the assumption that the absolutely continuous part satises Szego's condition and the mass points satisfy a Blaschketype condition. This generalizes the scalar analogue of PeherstorferYuditskii [PY01] and the matrixvalued result of AptekarevNikishin [AN83], which handles only a finite number of mass points. 2. We obtain matrixvalued Jost asymptotics for a block Jacobi matrix under an L^1type condition on parameters, and give a necessary and sufficient condition for an analytic matrixvalued function to be the Jost function of a block Jacobi matrix with exponentially converging parameters. This establishes the matrixvalued analogue of DamanikSimon [DS06b]. 3. The latter results allow us to fully characterize the matrixvalued WeylTitchmarsh mfunctions 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 mfunction 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 matrixvalued 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 L^1type condition on the Jacobi coefficients, equivalent Jacobi matrices of type 1, 2, and 3 are pairwise asymptotic.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  orthogonal polynomials, Jacobi matrices, matrixvalued functions 
Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Awards:  Scott Russell Johnson Graduate Dissertation Prize in Mathematics, 2010 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  2 June 2010 
NonCaltech Author Email:  rostysla (AT) caltech.edu 
Record Number:  CaltechTHESIS:06072010004607725 
Persistent URL:  http://resolver.caltech.edu/CaltechTHESIS:06072010004607725 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  5919 
Collection:  CaltechTHESIS 
Deposited By:  Rostyslav Kozhan 
Deposited On:  08 Jun 2010 15:55 
Last Modified:  22 Aug 2016 21:20 
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