Recovering the laplacian from centered means on balls and spheres of fixed radius

Various issues related to restrictions on radii in mean-value formulas are well-known in the theory of harmonic functions. In particular, using the Brown-Schreiber-Taylor theorem on spectral synthesis for motion-invariant subspaces in $C(\mathbb{R}^n)$, one can obtain the following strengthening of the classical mean-value theorem for harmonic functions: if a continuous function on $\mathbb{R}^n$ satisfies the mean-value equations for all balls and spheres of a fixed radius $r$, then it is harmonic on $\mathbb{R}^n$. In connection with this result, the following problem arises: recover the Laplacian from the deviation of a function from its average values on balls and spheres of a fixed radius. The aim of this work is to solve this problem. The article uses methods of harmonic analysis, as well as the theory of entire and special functions. The key step in the proof of the main result is expansion of the Dirac delta function in terms of a system of radial distributions supported in a fixed ball, biorthogonal to some system of spherical functions. A similar approach can be used to invert a number of convolution operators with compactly supported radial distributions.