Uncertainty Principles in Reproducing Kernel Hilbert Spaces

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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.