Mathematical Sciences: Wavelets and Time-Frequency Analysis

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9401340 Heil This award supports a mathematical research program in a range of related areas in harmonic analysis, concentrating on time-frequency analysis and wavelet theory, with potential applications to signal processing. There are three main themes of the program. The first concerns the analysis and reconstruction of signals from nonuniform samples of their Gabor transforms. The nonuniformity of the sampling set requires the introduction of new techniques with results formulated in terms of the properties of coherent states. Questions such as whether a given collection of time-frequency shifts of a fixed window function are complete, are minimal, form a frame or form a Riesz basis will be taken up. The second line of investigation studies singular values of Weyl operators. These operators are closely related to Gabor transform with good localizing properties, i.e. operators which concentrate the energy of a function in a given region of space. The localization properties of these operators can be inferred from the behavior of their singular values. Studies of the asymptotic decay of these values will be carried out. Work on wavelets and dilation equations continues earlier work which developed important applications of an older notion of joint spectral radius to the existence and smoothness characterizations of solutions to dilations equations. In this work, properties of non-canonical solutions will be studied. These include non-compactly supported or distributional solutions. Connections with random matrices will be considered. Harmonic analysis combines those elements of mathematics best exemplifying the ideas of synthesis. One seeks to decompose complex problems into fundamental components. These components are then analyzed for their basic characteristics. Finally, the solution is reconstructed through a recombination of the components. The Fourier series and Fourier transform are examples of tools used in this context; one dis crete , the other representing a continuous decomposition. More recently the wavelet theory added new dimensions to some of the more classical approaches to harmonic analysis.