Mathematical Sciences: Pointwise Fourier Inversion in Several Variables

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ABSTRACT Proposal: DMS-9623082 PI: Pinsky Pinsky plans to study pointwise Fourier inversion in rank-one symmetric spaces of the compact and non-compact type. At the same time he plans to continue the wave-equation approach to Fourier inversion, with special emphasis on the connection between focusing phenomena and the speed of convergence of Fourier transforms in two and three dimensions. He will also investigate Hermite polynomial expansions, which are closely related to the Schrodinger equation for the quantum oscillator, in the same fashion that Fourier transforms are related to the (d'Alembert) wave equation. The subject of pointwise Fourier inversion has been a central concern in mathematical analysis for nearly a century. In the case of one variable, Fourier analysis is essential to understanding filtering and sampling in signal processing and other time series phenomena in electrical engineering and the applied sciences. In the case of several variables, Fourier analysis is closely related to solving the partial differential equations that occur in physics, geology, and mechanical engineering. A detailed study of Gibbs' phenomenon and related higher-dimensional examples of divergence and oscillation of Fourier partial sums will be made by the principal investigator. It will be interesting to contrast the results of this study with the currently popular "wavelet" transforms, which have found many applications in recent years.