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Uncertainty Principles in Reproducing Kernel Hilbert Spaces

再现核希尔伯特空间的不确定性原理

基本信息

批准号：
2000236
负责人：
Mishko Mitkovski
金额：
$ 30.01万
依托单位：
Clemson University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-06-15 至 2025-05-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2000236&HistoricalAwards=false
关键词：
Uncertainty Principles Reproducing Kernel Hilbert

项目摘要

One of the fundamental principles of the theory of signal and information processing is the so-called Gabor uncertainty principle, which says that it is impossible for a signal to be perfectly localized simultaneously in time and in frequency. Perhaps counterintuitively, it is exactly this uncertainty principle that allows us to digitize analog signals; that is, to completely recover analog (continuous) signals that are band-limited, in some sense, from their samples recorded at (appropriate) discrete set of times. The precise rate at which the sampling needs to be done to guarantee stable recovery depends crucially on the band-limited nature of the signal. The main goal of this project is to quantify this subtle relationship precisely for very general types of restriction on signal bandwidth. In addition the project provides research training opportunities for graduate students. This project aims to develop a general operator-theoretic treatment of uncertainty principles, with a goal of unifying many different forms of the uncertainty principle that appear in harmonic analysis, and to discover new ones. In view of the advanced development of harmonic analysis, it is quite surprising that manifestations of the uncertainty principle are still usually studied one at a time, isolated one from another. This project aims to systematize these results, further explore the connections between them, and pave the way for new applications to control theory, partial differential equations, and other areas of mathematics.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.

信号和信息处理理论的基本原理之一是所谓的伽柏不确定性原理，即信号不可能在时间和频率上同时完美定位。也许与直觉相反，正是这种不确定性原理使我们能够对模拟信号进行重构;也就是说，从在（适当的）离散时间组记录的样本中完全恢复在某种意义上是带限的模拟（连续）信号。为了保证稳定的恢复，需要进行采样的精确速率关键取决于信号的带限性质。这个项目的主要目标是量化这种微妙的关系，精确地为非常一般类型的信号带宽的限制。此外，该项目还为研究生提供研究培训机会。该项目的目的是开发一个通用的算子理论处理的不确定性原则，与统一的目标出现在谐波分析中的不确定性原则的许多不同形式，并发现新的。鉴于谐波分析的先进发展，令人惊讶的是，不确定性原理的表现形式仍然通常一次一个地研究，彼此孤立。该项目旨在将这些成果系统化，进一步探索它们之间的联系，并为控制理论、偏微分方程和其他数学领域的新应用铺平道路。该奖项反映了NSF的法定使命，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。