Collaborative Research: Topics in Abstract, Applied, and Computational Harmonic Analysis

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The underlying theoretical mechanisms supporting the digital world that our societies benefit from today result from sophisticated mathematics developed over many centuries. Among the mathematical tools employed in modern signal processing, Fourier analysis stands as one of the key players. In particular, the methods developed in Fourier analysis are instrumental in decomposing complex signals into their elementary building blocks. This project will push the current understanding of several modern tools related to Fourier analysis in applications such as data science, signal processing, and quantum information theory beyond their current frontiers. Moreover, this project's educational component will allow the investigators to continue training students in the underlying mathematics fields it covers. The investigators will also integrate the outcomes of this research program into graduate and advanced undergraduate courses offered at their respective institutions. The project will solve some fundamental and unresolved problems in time-frequency analysis, especially the Heil-Ramanathan-Topiwala (HRT) conjecture (which asserts that every finite collection of time-frequency shifts of a square-integrable function must be linearly independent) and several other related unresolved problems. These problems arise in time-frequency analysis and are at the intersection of many areas of mathematics, applied mathematics, and even engineering. The investigators will attack these problems from a multi-field approach, bringing to bear techniques from abstract, applied computational harmonic analysis, ergodic theory, Lie group, Lie algebra, complex, functional, and real analysis. This research will build on recent successes of applied and pure harmonic analysis, which include the wavelet-based JPEG standard, advances in phaseless reconstruction, and the fundamental role played by Gabor (or Weyl-Heisenberg) systems in the detection of the gravitational waves. A standard paradigm in many of these applications consists of decomposing arbitrary signals into redundant elementary building blocks. While the redundancy of these systems might seem counterintuitive for their use, it is nonetheless responsible for the robustness of certain algorithms for data transmission using unreliable channels. It will play a vital role in noise reduction algorithms. Wavelets and Gabor systems are examples of redundant systems, and such systems can represent many natural signals.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.