Some remarks on the Pompeiu problem for groups

Scott, David; Sitaram, Alladi

doi:10.1090/S0002-9939-1988-0931747-0

Some remarks on the Pompeiu problem for groups
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by David Scott and Alladi Sitaram

Proc. Amer. Math. Soc. 104 (1988), 1261-1266

DOI: https://doi.org/10.1090/S0002-9939-1988-0931747-0

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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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