Deconvolution for the Pompeiu Problem on the Heisenberg Group, I
We consider variations on the Pompeiu transform for the Heisenberg group H n and focus on cases where the transform is known to be injective; in particular the cases of averages over a sphere and a ball, or two balls of appropriate radii. In these cases we develop a method which provides for the reconstruction of the function f from its integrals.
In addition, we consider these issues in connection with the Weyl calculus and group Fourier transform. We furthermore explore issues of convergence for this method of deconvolution and related issues of size of the Gelfand transform near the zero sets. Finally, given a set of deconvolvers which work for Euclidean space C n , we consider the problem of how to extend the deconvolution to the Heisenberg group, and we provide the extension in special cases.
KeywordsHeisenberg Group Spherical Function Real Hypersurface Common Zero Schwartz Space
Part of the research work for this paper was completed during the visits of the three authors at the National Center for Theoretical Sciences in Hsinchu, Taiwan. They would like to thank NCTS (Hsinchu) for partial support of this research. They take great pleasure in expressing their thanks to Professor Winnie Li, the director of NCTS, for the invitation and the warm hospitality during their visits in Taiwan. The second author would also like to use this opportunity to thank the Academia Sinica in Taipei for the invitation during the summer of 2007. He would also like to thank the colleagues at the Sinica for the warm hospitality during his visit in Taipei.
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