New theorems on the mean for solutions of the Helmholtz equation

It is proved that the solutions of the equation $\Delta u+u=0$ are characterized by vanishing of integrals over all balls in $R^n$ with radii belonging to the zero set of the Bessel function $J_{n/2}$. This result enables us to get a solution of the Pompeiu problem on the class of functions of slow growth in terms of approximation in $L(R^n)$ by linear combinations with special radii.