Mathematical Sciences: Research in Fourier Analysis

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9307781 Benedetto This project will develop several themes within the general framework of harmonic analysis. One is a continuation of work on weighted inequalities. These are inequalities relating norms of functions with the norms of their transforms; i.e. through a partial differential operator or the Fourier transform. The norms are computed against weighted measures and their comparisons provide fundamental information about the range of the transformation, that is, about the set of solutions. Particular emphasis will be placed on the derivation of two- weight estimates of convolution operators. Work will also be done on sampling problems. There is an extensive theory which treats regular sampling but only scattered results are currently available for irregular sampling - which is often the only feasible means for gathering information. This research continues efforts to expand on recent irregular sampling results by using either coefficients from inverse operators or by varying the representing functions. A specific goal is that of constructing a genuine multidimensional version of existing theory. Additional work on wavelet theory will also be pursued. The object is the development of Riesz products from sets of quadrature mirror filters for the construction of wavelet packets and the construction of an L-1 theory of multiresolution analysis. Harmonic analysis is a branch of mathematics which seeks to analyze complex phenomena through processes of decomposition and synthesis. The classical models for such analysis can be found in Fourier series and Fourier integral representations. Recent innovations have led to wavelet theory whic h has proved successful in yielding much a refined capability in analyzing local time and frequency characteristics. New developments have had a major impact on studies related to signal analysis, data compression and sampling theory. ***