The Pompeiu problem and spherical spectral analysis

Abstract

Let K be a compact subgroup of a locally compact group G. We investigate a Pompeiu type problem for homogeneous spaces G/K. Suppose E is a compact subset of G/K. Using recent work of László Székelyhidi on K-spectral analysis [L. Székelyhidi, Spherical spectral synthesis. Acta Math. Hungar. 153(1), 120–142 (2017)] we are able to give necessary and sufficient conditions for E to have the Pompeiu property when (G, K) is a Gelfand pair.