Dissertation
Spectral density functions and their applications
Doctor of Philosophy (Ph.D.), Drexel University
Aug 2016
DOI:
https://doi.org/10.17918/etd-7370
Abstract
The Bernstein-Szegő measure moment problem asks when a given finite list of complex numbers form the Fourier coefficients of the spectral density function of a stable polynomial in the one-variable case. Szegő proved that it is possible if and only if the Toeplitz matrix form by these numbers is positive definite. Bernstein later proved a real line analog of the problem. The question remained open in two variables until Geronimo and Woerdeman stated and proved the necessary and sufficient conditions. Unlike the solution in one variable, it does not suffice to write down a single matrix and check whether it is positive definite. A positive definite completion condition is also required. In this thesis, we further pursue the moment problem in two variables and beyond. We first enhance the two-variable results by identifying the eigenstructure of matrices that arise from the theory. We then create a method that allows us to compute the Fourier coefficients in a given infinite region by using a finite portion of the coefficients. Use is made of determinantal representations of stable polynomials. In addition, we compute the asymptotics for the Fourier coefficients and later generalize the result to higher dimensions. In the final chapter, we draw a connection between offset words and a particular type of spectral density functions and compute the asymptotics of the number of offset words as different parameter changes.
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Details
- Title
- Spectral density functions and their applications
- Creators
- Chung Yuen Wong - DU
- Contributors
- Hugo J. Woerdeman (Advisor) - Drexel University (1970-)
- Awarding Institution
- Drexel University
- Degree Awarded
- Doctor of Philosophy (Ph.D.)
- Publisher
- Drexel University; Philadelphia, Pennsylvania
- Number of pages
- vii, 95 pages
- Resource Type
- Dissertation
- Language
- English
- Academic Unit
- College of Arts and Sciences; Drexel University; Mathematics
- Other Identifier
- 7370; 991014632316604721