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

支持数字世界的基本理论机制，使我们的社会受益于今天，源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，傅里叶分析是其中的关键之一。特别是，在傅里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建块。该项目将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里叶分析相关的几个现代工具的理解。此外，该项目的教育部分将允许研究人员继续在其涵盖的基础数学领域培训学生。研究人员还将把该研究项目的成果整合到各自机构提供的研究生和高级本科课程中。该项目将解决时频分析中一些基本的和未解决的问题，特别是heil - ramanana - topiwala （HRT）猜想（它断言平方可积函数的时频移的每一个有限集合必须是线性无关的）和其他几个相关的未解决的问题。这些问题出现在时频分析中，是数学、应用数学甚至工程许多领域的交叉点。研究人员将从多领域的方法来解决这些问题，带来从抽象、应用计算调和分析、遍历理论、李群、李代数、复、泛函和实分析等技术。这项研究将建立在最近成功的应用和纯谐波分析的基础上，其中包括基于小波的JPEG标准，无相位重建的进展，以及Gabor（或Weyl-Heisenberg）系统在引力波探测中所起的基本作用。在许多这类应用中，标准范例包括将任意信号分解为冗余的基本构建块。虽然这些系统的冗余对于它们的使用似乎是违反直觉的，但它仍然负责使用不可靠通道进行数据传输的某些算法的鲁棒性。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，这样的系统可以表示许多自然信号。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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 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.