On the Pompeiu problem and its generalizations (Q1344976)

The well-known Pompeiu problem is to describe compact sets \(K \subset \mathbb{R}^ n\) such that any function \(f \in L_{\text{loc}} (\mathbb{R}^ N)\) satisfying \[ \int_{AK} f(u)du = 0 \tag{1} \] for all sets \(AK\) congruent to \(K\), is equal to zero a.e. in \(\mathbb{R}^ n\). The author proves that there exists a nonzero slowly increasing function \(f\) which satisfies (1) if and only if the indicator \(\chi_ K\) of \(K\) can be approximated in \(L (\mathbb{R}^ n)\) by linear combinations of indicators of balls with radii proportional to positive zeros of the Bessel function \(I_{n/2}\). Some generalizations of the problem are considered for \(K\) either an ellipsoid or a parallelepiped. The case where a function \(f\) is defined on a bounded domain is considered, too.