Mathematical Sciences: Harmonic Analysis Methods in OperatorTheory and Modern Analysis

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9401009 Berkson The investigations seek to develop a broad abstract operator theory which expands the role of harmonic analysis in the structures of general analysis. In this program, the powerful methods of harmonic analysis will be harnessed to operator theory by expanding the theory and scope of transference methods. These investigations constitute a fertile ongoing program of fundamental research which has been successfully applied to numerous disciplines in recent years. The directions and themes will include the following mainstream topics: multiplier theory and transference of maximal estimates for Fourier multipliers; wavelet decompositions for Fourier multipliers; interpolation methods and transference; ergodic theory; abstract spectral decompositions. The planned research will seek a wide unified overview of mathematical applications. Historically such an approach is crucial to disciplines such as quantum physics, which rely for their explication on broad abstract principles of mathematics. ***