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

合作研究：抽象、应用和计算谐波分析主题

• 批准号: 2205852
• 负责人: Vignon Oussa
• 金额: $ 15万
• 依托单位: BRIDGEWATER STATE UNIVERSITY
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-06-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2205852&HistoricalAwards=false
• 关键词: Collaborative; Research; Topics; Abstract; Applied

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.

支持当今我们的社会受益的数字世界的基本理论机制源于多个世纪以来发展的复杂数学。在现代信号处理中使用的数学工具中，傅里叶分析是关键人物之一。特别是，傅立叶分析中开发的方法有助于将复杂信号分解为其基本构建块。该项目将推动目前对数据科学、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种现代工具的理解超越当前的前沿。此外，该项目的教育部分将使研究人员能够继续在其所涵盖的基础数学领域对学生进行培训。研究人员还将将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的一些基本且未解决的问题，特别是 Heil-Ramanathan-Topiwala (HRT) 猜想（该猜想断言平方可积函数的时频偏移的每个有限集合必须是线性独立的）以及其他几个相关的未解决问题。这些问题出现在时频分析中，并且是数学、应用数学甚至工程学许多领域的交叉点。研究人员将从多领域方法来解决这些问题，运用抽象技术、应用计算调和分析、遍历理论、李群、李代数、复分析、泛函分析和实分析。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、无相重建的进展以及 Gabor（或 Weyl-Heisenberg）系统在引力波探测中发挥的基本作用。许多此类应用中的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其使用来说似乎违反直觉，但它仍然对使用不可靠通道进行数据传输的某些算法的鲁棒性负责。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，此类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价值和更广泛的影响审查标准进行评估，被认为值得支持。