In signal processing, periodic signals can be studied using Fourier series, where the basic building blocks are sine waves. However, when a signal changes substantially over time, such as a piece of music, different methods are needed. Gabor frames provide one such method. A Gabor frame allows for representations of a signal that emphasizes its frequency content at each point in time, similarly to how sheet music is written.
When constructing a Gabor frame, a point set in the time-frequency plane needs to be specified. Gabor frames over lattice point sets enjoy a deep duality theory with connections to operator algebras and the representation theory of groups. For non-lattice point sets, these connections are less clear. The aim of this project is to fill this gap in the existing knowledge, associating operator algebras to Gabor frames over a class of point sets known as quasicrystals. In particular, a major goal is to prove existence results for Gabor frames over quasicrystals which generalize known results for lattices.
In mathematics and signal processing, Gabor frames are structured function systems that allow for basis-like representations of functions in one or several real variables. A Gabor frame is constructed from a window function and a point set in the time-frequency plane. The fundamental problem that we will study in this project is the following: Which point sets admit Gabor frames, possibly with additional regularity assumptions on the window function? We have three main objectives:
A: Determine when a lattice point set admits a Gabor frame with a well-localized window.
B: Study the existence of Gabor frames over quasicrystals in the time-frequency plane.
C: Study objectives A and B in the general setting of frames in the orbit of unitary group representations sampled from approximate lattices.
Techniques from operator algebras have been highly successful in answering existence questions for Gabor frames over lattices as seen by the work of Bekka, Rieffel and Jakobsen-Luef. In Objective A, we aim to resolve an open problem using Rieffel's Heisenberg modules over rational noncommutative tori and higher-dimensional Zak transforms. The significant challenge here will be to obtain a transparent description of the modules in terms of vector bundles. In Objective B, we use groupoids and their operator algebras to approach Gabor frames over quasicrystals as initiated in a paper by Kreisel. In Objective C we study the extent to which our methods generalize, establishing connections to new developments on approximate lattices for locally compact groups. A challenging task will be to construct operator algebras associated to approximate lattices that generalize group algebras.