Research Statement Overview: My Research Interests Lie in Harmonic Analysis, Focusing on Sparse and Op- Timally Redundant Representation Systems. Recent Discoveries in Compressed Sensing By

Overview: My research interests lie in harmonic analysis, focusing on sparse and optimally redundant representation systems. Recent discoveries in compressed sensing by Emmanuel Candès, Justin Romberg, Terence Tao, and others [CRT06] have put a new emphasis on specialized bases and dictionaries. Wavelets generate some of the most renowned such systems, with the first wavelet basis appearing 100 years ago in work by Alfréd Haar [Haa09] [Haa10]. Some 70 years later, wavelets were resurrected by Jean Morlet and Alex Grossmann to analyze geophysical measurements and other physical phenomena [GGM84] [Gro88] [GHKMM87]. Since then, the field has grown immensely and wavelets have surged in popularity. Many new systems have sprung up recently, such as the contourlet systems of Do and Vetterli [DV03], the curvelet systems of Candès and Donoho [CD99], and the shearlet systems of Kutyniok, Labate, and Weiss [LLKW05]. Such systems are used for data compression, pattern recognition, noise reduction, transient recognition, and associated algorithms work in such varied areas as applied statistics, numerical PDEs, and image processing. In 1952, Richard Duffin and Albert Schaeffer [DS52] synthesized the earlier ideas of a number of illustrious mathematicians, including Ralph Boas Jr., Raymond Paley, and Norbert Wiener into a unified theory, the theory of redundant representation systems called frames. In some sense, frames may be thought of generalizations of bases which serve as powerful tools in such fields as pseudodifferential operators [GS07], signal processing [KV95], and wireless communication [SH03]. One aspect of this theory that is particularly alluring to me is the simple beauty of the mathematics behind it, combined with the utility of the many applications.

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