Approximation of functions with zero integrals over balls by linear combinations of the Laplace operator eigenfunctions

It is proved that the special linear combinations of Bessel functions are dense in the C ^∞-topology in the space of functions with zero integrals over balls of fixed radii on an arbitrary open domain U ì \mathbbRⁿ U \subset {\mathbb{R}^n} . Some generalizations of this result for solutions of some convolution equations of the form f * T = 0, where T is radial, are obtained. Analogous results for rank-one symmetric spaces of the noncompact type are considered.