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Some remarks on the Pompeiu problem for groups

Authors: David Scott and Alladi Sitaram
Journal: Proc. Amer. Math. Soc. 104 (1988), 1261-1266
MSC: Primary 43A60; Secondary 22E30, 43A80
MathSciNet review: 931747
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Abstract: A Borel set $ E$ in a topological group $ G$ is said to be a $ P$-set for the space of integrable functions on $ G$ if the zero function is the only integrable function whose integral over all left and right translates of $ E$ by elements of $ G$ is zero. For a "sufficiently nice" group $ G$ and a Borel set $ E$ of finite Haar measure a certain condition on the Fourier transform of a function related to $ E$ is shown to be a sufficient condition for $ E$ to be a $ P$-set. This condition is then applied to several classes of groups including certain compact groups, certain semisimple Lie groups, the Heisenberg groups and the Euclidean motion group of the plane.

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