Graduation Year
2022
Document Type
Dissertation
Degree
Ph.D.
Degree Name
Doctor of Philosophy (Ph.D.)
Degree Granting Department
Mathematics and Statistics
Major Professor
Catherine Bénéteau, Ph.D.
Committee Member
Dmitry Khavinson, Ph.D.
Committee Member
Myrto Manolaki, Ph.D.
Committee Member
Evguenii A. Rakhmanov, Ph.D.
Committee Member
Boris Shekhtman, Ph.D.
Keywords
digital filter, Hardy space, Lp space, Moore-Penrose inverse, Shanks' Conjecture
Abstract
The history of optimal polynomial approximants (OPAs) dates back to the engineering literature of the 1970s. Here, these polynomials were studied in the context of the Hardy space H^2(X), where X denotes the open unit disk D or the bidisk D^2. Under certain conditions, it was thought that these polynomials had all of their zeros outside the closure of X. Hence, it was suggested that these polynomials could be used to design a stable digital filter. In recent mathematics literature, OPAs have been studied in many different function spaces. In these settings, numerous papers have been devoted to studying the properties of their zeros. In this dissertation, we begin by introducing the notion of optimal polynomial approximant in the space L^p(T), where T denotes the unit circle and p ranges from 1 to infinity. Here, we shed light on an orthogonality condition that allows us to study OPAs in L^p(T) with the additional tools from the L^2(T) setting. We later use this orthogonality condition to compute the coefficients of some OPAs in L^p(T); this will give us insight into the location of their zeros. We continue the dissertation by discussing the connection of OPAs to 1D digital filter design; a majority of these discussions will be devoted to surveying the design process of Chui and Chan. Toward the end of this dissertation, we extend the operator theoretic approach of Izumino to the L^2(T) setting; in light of the orthogonality condition, this provides us additional tools to study the zeros of OPAs in L^p(T).
Scholar Commons Citation
Centner, Raymond, "Advances and Applications of Optimal Polynomial Approximants" (2022). USF Tampa Graduate Theses and Dissertations.
https://digitalcommons.usf.edu/etd/9315
