The Pompeiu problem and spherical spectral analysis

TL;DR

This paper explores a Pompeiu problem for homogeneous spaces G/K, providing necessary and sufficient conditions for a set to have the Pompeiu property using spectral analysis in the context of Gelfand pairs.

Contribution

It introduces a characterization of the Pompeiu property on G/K spaces via spectral analysis, extending previous work to Gelfand pairs.

Findings

Established conditions for the Pompeiu property in G/K spaces.

Connected spectral analysis with the Pompeiu problem in the context of Gelfand pairs.

Utilized recent spectral analysis results to advance understanding of the Pompeiu problem.

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 L\'{a}szl\'{o} Sz\'{e}kelyhidi on $K$-spectral analysis \cite{Szekelyhidi17} we are able to give necessary and sufficient conditions for $E$ to have the Pompeiu property when $(G, K)$ is a Gelfand pair.