Exponential systems and related topics

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This research project is in the field of harmonic analysis, a mathematical area that grew from the fact that many functions defined over an interval can be decomposed as sums of the simple sine and cosine functions. The principal investigator plans to study cases where such a decomposition is not possible, or is not efficient enough, for example because the functions in question are no longer defined over intervals. The question is whether similar decompositions are possible in such cases, with the sines and cosines replaced by other functions of a simple structure. The goal is to use functions that mimic the structure of the sines and cosines, in one way or another, as good replacements for the trigonometric functions, which provide an excellent tool for studying properties of functions and the interrelationships between them. In particular, this project includes the development of efficient ways to sample, interpolate, and approximate signals.The principal investigator and collaborators have recently shown that any finite union of intervals admits a Riesz basis of exponentials. While a big advance in the area, there remain many aspects of the question that the principal investigator will study, such as clarifying whether this is a generic property of all sets of finite measure and understanding whether the result holds for Riesz bases with "good bounds." Further, she plans to study the benefits of using even weaker notions (e.g., frames, Riesz sequences) for the same purposes. Of particular interest is the relation between the study of such systems and the development of techniques to analyze sampling and interpolating sequences. Finally, the project will explore applications to Gabor systems in time-frequency analysis, for instance, possible extensions of the Balian-Low theorem and linear independence of Gabor elements.