The Pompeiu problem and spherical spectral analysis

TL;DR

The paper extends the Pompeiu problem to homogeneous spaces $G/K$ under a Gelfand pair $(G,K)$ with $K$-spectral analysis, providing a precise, testable criterion: a compact $E\subset G/K$ has the Pompeiu property iff the associated ideal $I$ in $\mathcal{M}_c(G//K)$ has empty zero set $Z(I)$. The main tool is the $K$-spectral framework and $K$-spherical functions, which yield a spectral criterion via transforms $\Phi_f$ and the kernel/maximal ideals they generate. The results unify a convolution-analytic characterization with ideal-theoretic conditions, and specialize to the Euclidean case $G=M(n)$ with $K=SO(n)$ to recover and relate to classical Fourier-analytic criteria (e.g., Brown–Schreiber–Taylor) and known Euclidean cases such as polytopes having the Pompeiu property. The approach provides a practical method to certify the Pompeiu property by examining zero sets of spherical transforms, linking harmonic analysis on homogeneous spaces to geometric Pompeiu questions with potential implications for related conjectures and finite-group analogues.

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 \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.

Theorems & Definitions (18)

• Lemma 2.1
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.3
• proof
• Theorem 4.1
• proof