Description of Pompeiu sets in terms of approximations of their indicator functions

Let $H$ be an open upper half-space in $mathbb R^n$, $ngeq2$,and assume that $A$ is a non-empty, open, bounded subset of$mathbb R^n$ such that $overline{A}subset H$ and theexterior of $A$ is connected. Let $pin[2, +infty).$ It isproved that there is a nonzero function with zero integralsover all sets in $mathbb R^n$ congruent to $A$ if and only ifthe indicator function of $A$ is the limit in $L^p(H)$ of asequence of linear combinations of indicator functions of ballsin $H$ with radii proportional to positive zeros of the Besselfunction $J_{n/2}$. The proportionality coefficient here is thesame for all balls and depends only on $A$.