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arXiv:1202.2398 [math.FA]AbstractReferencesReviewsResources

The Pompeiu Problem and Discrete Groups

Michael J. Puls

Published 2012-02-11, updated 2013-05-19Version 3

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.

Comments: Version two fixes some typos. Also a new section was added concerning the harmonicity of a function with regards to the mean-value property on two spheres. Corrected some more typos
Categories: math.FA, math.GR
Subjects: 43A15
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