Matrix-like Representations in Time-Frequency and Applied Harmonic Analysis

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KrishtalDMS-0908239 The project is dedicated to creation, development, andapplication of a unified theory of matrix-like representationsfor frames, fusion frames, and fuzzy fusion frames; and causalitytheory and regular factorizations of pseudodifferentialoperators. These topics touch upon a large variety of areas offundamental and applied analysis, including frame theory,pseudodifferential operator theory, Banach modules andrepresentations of locally compact Abelian groups, non-standardmultiresolution analysis, and signal processing. The problemsstudied in this project address interesting and importantquestions regarding certain mathematical tools on which modernsignal processing is based. The project is also expected to leadto new questions in more fundamental analysis, thus promotingcross-fertilization of the fields. The investigator and his colleagues develop a mathematicalframework aimed to address various problems posed by the delugeof data in the modern world. Potential applications includeimage processing, wireless communications, and others. Particular attention is given to an application for protoncomputed tomography. Broader impact of the proposed activitiesincludes integration of research and education, dissemination ofknowledge, and enhancement of interdisciplinary collaboration,both domestic and international.