Abstract Rochberg, Krantz, Wickerhauser, Weiss, Taibleson, McCarthy The principal investigators propose to form a Research Group in Analysis. They propose to investigate a broad range of areas in harmonic analysis, operator theory, function theory, and differential equations. Specific areas of research include analysis of Bergman and Hardy spaces and the operators which act on them, the relationship between the geometry of complex domains and the associated function theory and operator theory, the theory of wavelets and related constructions and the applications of those tools and ideas in theoretical and computational problems, and harmonic analysis on trees and the exploration of the analogies between analysis on trees and continuous harmonic analysis. The principal investigators propose research programs in harmonic analysis (in the broad sense). One of the basic unifying viewpoints in harmonic analysis is the development of systematic techniques to decompose complicated objects into simple, well understood, pieces put together in a straightforward way. In many cases these same techniques, which were first developed to understand theoretical mathematical constructs, can also be adapted productively to the analysis and manipulation of data and images. An important recent example of this is the development of new waveform technologies which have provided important new tools in signal processing. The detailed research proposed includes both theoretical and applied components. The latter center on expanding the range of applicability of the new adapted waveforms techniques such as wavelets. Those techniques, originally developed for use in theoretical mathematics, have already provided important new tools in signal processing. It appears that they these ideas have great potential for further development as tools for computation and for data analysis. That development is one of the themes of the proposed research.
