Publication Date

2025

Document Type

Dissertation/Thesis

First Advisor

Krishtal, Ilya

Degree Name

Ph.D. (Doctor of Philosophy)

Legacy Department

Department of Mathematical Sciences

Abstract

Dynamical sampling is an area of mathematics which studies the theory and applications of signals evolving through time, as well as the stable recovery of said signals from spatiotemporal measurements. This manuscript is devoted to problems which arise as theoretical considerations in signal recovery and as applications of using space-time measurements as opposed to static measurements.

For the former, we consider the problem of recovering an evolving signal via space-time measurements at irregular time intervals. To do so, we generalize the classic Kadec $1/4$ theorem using techniques of Banach $L^1(\R)$-modules. For the latter, we show how dynamical samples can be used to recover source terms in diffusion models. These models are commonly used in to describe scientific phenomena, and are of the form \begin{equation*} \begin{cases} \dot{u}(t)=Au(t)+ \eta(t) + \sum_{j} h_je^{\rho_j(t_j-t)}\chi_{[t_j,\infty)}(t) \\ u(0)=u_0, \end{cases} \quad t\in\mathbb R_+,\ u_0\in\HH. \end{equation*} We use spatiotemporal samples of the solution to this IVP to recover the data $h_j,\rho_j,t_j$, which represent the intensity, decay rate, and activation time for each source in the system.

Extent

92 pages

Language

en

Publisher

Northern Illinois University

Rights Statement

In Copyright

Rights Statement 2

NIU theses are protected by copyright. They may be viewed from Huskie Commons for any purpose, but reproduction or distribution in any format is prohibited without the written permission of the authors.

Media Type

Text

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Included in

Mathematics Commons

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