The Pompeiu problem and discrete groups |
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Authors: | Michael J Puls |
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Institution: | 1. Department of Mathematics, John Jay College-CUNY, 524 West 59th Street, New York, NY, 10019, USA
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Abstract: | We formulate a version of the Pompeiu problem in the discrete group setting. Necessary and sufficient conditions are given for a finite collection of finite subsets of a discrete abelian group, whose torsion free rank is less than the cardinal of the continuum, to have the Pompeiu property. We also prove a similar result for nonabelian free groups. A sufficient condition is given that guarantees the harmonicity of a function on a nonabelian free group if it satisfies the mean-value property over two spheres. |
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