Date of Award
8-2026
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
School of Mathematical and Statistical Sciences
Committee Chair/Advisor
Mishko Mitkovski
Committee Member
Shitao Liu
Committee Member
Jeong Rock Yoon
Committee Member
Cody Bullett Stockdale
Abstract
In this dissertation, we will introduce the concept of pseudo-Riesz bases, extending the influential notion of near-Riesz bases first proposed by J. Holub in the 1990s. Our goal is to develop an analogous theory for pseudo-Riesz bases, obtaining results parallel to those known for near-Riesz bases, with suitable modifications.
We will develop the foundational theory of pseudo-Riesz bases. Unlike classical Riesz bases, these sequences need not be complete or independent; however, they still retain meaningful expansion properties. We will investigate the conditions under which a Bessel sequence can be transformed into a Riesz basis through finite modifications. In particular, we will characterize pseudo-Riesz bases in terms of the Fredholm properties of their associated synthesis operators. We will also introduce and study dual-type structures that naturally arise in this setting.
Next, we will establish perturbation results for pseudo-Riesz bases, extending well-known stability results from the theory of near-Riesz bases. These results demonstrate that pseudo-Riesz bases remain stable under suitable perturbations, thereby reinforcing their structural robustness.
Finally, we will consider systems of non-harmonic complex exponentials. A classical result due to Hruščev, Nikol'skiǐ, and Pavlov characterizes such systems forming Riesz bases in terms of the invertibility of an associated Toeplitz operator. We will extend this framework to pseudo-Riesz bases, obtaining analogous characterizations. Furthermore, we will analyze the behavior of these systems in the case when the corresponding Toeplitz operator has a unimodular symbol, highlighting the role of index theory in this context.
Recommended Citation
Biswas, Deborpita, "Pseudo-Riesz Bases" (2026). All Dissertations. 4368.
https://open.clemson.edu/all_dissertations/4368