Uncertainty Principles in Reproducing Kernel Hilbert Spaces

信号和信息处理理论的基本原理之一是所谓的Gabor不确定性原理，它说信号不可能在时间和频率上同时完美地定位。也许与直觉相反，正是这种不确定性原理使我们能够将模拟信号数字化；也就是说，在某种意义上，从（适当的）离散时间组记录的样本中完全恢复带宽有限的模拟（连续）信号。保证稳定恢复所需的精确采样率主要取决于信号的频带限制性质。该项目的主要目标是精确地量化这种微妙的关系，用于非常一般的信号带宽限制类型。此外，该项目还为研究生提供了研究培训机会。本项目旨在发展不确定原理的一般算子理论处理，目的是统一谐波分析中出现的许多不同形式的不确定原理，并发现新的不确定原理。鉴于谐波分析的先进发展，令人惊讶的是，测不准原理的表现通常仍然是一次研究一种，彼此孤立。本项目旨在将这些结果系统化，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。

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.