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

支持数字世界的基本理论机制，使我们的社会今天受益于许多世纪以来发展起来的复杂数学。在现代信号处理中使用的数学工具中，傅立叶分析是关键的参与者之一。特别是，傅立叶分析中开发的方法有助于将复杂信号分解成它们的基本构件。该项目旨在推动当前对数据科学、信号处理和量子信息理论等应用中与傅立叶分析相关的几种现代工具的理解，使其超越目前的边界。此外，该项目的教育部分将允许调查人员继续在其涵盖的基本数学领域培训学生。研究人员还将把这一研究项目的成果纳入各自机构提供的研究生和高级本科课程。该项目旨在解决时频分析中一些基本的和尚未解决的问题，特别是Heil-Ramanathan-Topiwala(HRT)猜想(该猜想断言一个平方可积函数的每个有限的时频移位集合必须是线性独立的)以及其他一些相关的尚未解决的问题。这些问题出现在时频分析中，涉及数学、应用数学甚至工程学的许多领域。研究人员将从多领域的方法来解决这些问题，利用抽象的、应用的计算调和分析、遍历理论、李群、李代数、复数、泛函和实数分析的技术。这项研究将建立在应用和纯谐分析的最新成功的基础上，包括基于小波的JPEG标准，无相重构的进展，以及Gabor(或Weyl-Heisenberg)系统在探测引力波中所起的基础作用。其中许多应用中的标准范例包括将任意信号分解成冗余的基本构建块。虽然这些系统的冗余对于它们的使用来说似乎是违反直觉的，但它仍然是使用不可靠的信道进行数据传输的某些算法的健壮性的原因。它将在降噪算法中发挥至关重要的作用。小波和Gabor系统是冗余系统的例子，这样的系统可以代表许多自然信号。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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 aims to 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 aims to 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.