Date: Friday, November 20th

Time: 12:00pm (EST)

Zoom link: https://gatech.zoom.us/j/4217842249

Committee members:

Dr. Christopher Heil (Advisor)

Dr. Michael Lacey

Dr. Michael Damron

Dr. Plamen Iliev

Dr. Pu-Ting Yu

Abstract:

The first chapter, based on joint work with Shahaf Nitzan, focuses on mobile sampling in Paley–Wiener spaces. In many applications, data is collected by sensors moving along continuous paths rather than at isolated points. This viewpoint was introduced by Unnikrishnan and Vetterli (2013), and has since been developed by Gröchenig, Romero, Unnikrishnan and Vetterli (2015), and by Rashkovskii, Ulanovskii and Zlotnikov (2023). We study when a union of parallel lines, determined by a discrete sequence, forms a mobile sampling set for a two-dimensional Paley–Wiener space. We establish necessary and sufficient conditions under which this mobile sampling property is equivalent to the discrete sequence being a stable sampling sequence for the one-dimensional Paley–Wiener space associated with the projection of the spectrum. We also extend these results to families of tilted lines, where the sampling bounds are rescaled by a factor depending on the slope.

In the second chapter, based on join work with Rohit Pai and Pu-Ting Yu, we study woven weighted exponential systems in L^2[0,1]. The basis properties of a single weighted exponential system {g e_n} are classical. For instance, it is a Riesz basis exactly when |g| is bounded above and away from zero. Woven frames were introduced by Bemrose, Casazza, Gröchenig, Lammers and Lynch (2016), and weavings of Riesz bases were studied more recently by Cabrelli, Molter and Negreira (2025). Given two weights f and g, a weaving is a system in which, for each integer n, the n-th vector is chosen to be either f e_n or g e_n. We obtain a complete characterization of when every weaving is complete. This happens exactly when f and g are nonzero almost everywhere, and there is no nonzero function p for which pf and pg have non-overlapping Fourier spectra. We also characterize woven Bessel systems and orthonormal bases and give examples showing that woven completeness and woven minimality are genuinely different properties. For frames, we show that if f and g each generate a frame and the ratio g/f is strictly positive (or strictly negative), then every weaving is a Riesz basis. 

The third chapter, based on joint work with Pu-Ting Yu, studies exponential systems with polynomial windows. Boas and Pollard (1948) showed that deleting finitely many exponentials from the trigonometric system can always be compensated, in the sense of completeness, by multiplying the remaining exponentials by a suitable weight. Yoon and Heil (2012) studied how many exponentials must be removed from a system weighted by a polynomial-type window. We refine this question from how many frequencies to which ones, i.e., given a polynomial window w and a finite set A of integers, we determine exactly when the system {w e_n}_{n not in A} is complete, minimal, or exact. The answer is encoded in an evaluation matrix built from the values and derivatives of the exponentials at the zeros of w. A key feature is that completeness is governed by the width of A, rather than by its size.