The problem of finding a function by its ball means values

A classical property of a non-constant $2r$-periodic function on the real axis is that it has no period incommensurable with $r$. One of the multidimensional analogues of this statement is the following well-known theorem of L. Zalcman on two radii: for the existence of a nonzero locally summable function $f:\mathbb{R}^n\to \mathbb{C}$ with nonzero integrals over all balls of radii $r_1$ and $r_2$ in $\mathbb{R}^n$ it is necessary and sufficient that $r_1/r_2\in E_n$, where $E_n$ is the set of all possible ratios of positive zeros of the Bessel function $J_{n/2}$. The condition $r_1/r_2\notin E_n$is equivalent to the equality $\mathcal{Z}_{+}\big(\widetilde{\chi}_{r_1}\big)\cap\mathcal{Z}_{+}\big(\widetilde{\chi}_{r_2}\big)=\varnothing$, where $\chi_{r}$ is the indicator of the ball $B_r=\{x\in\mathbb{R}^n: |x|<r\}$, $\widetilde{\chi}_{r}$ is the spherical transform (Fourier-Bessel transform) of the indicator $\chi_{r}$, $\mathcal{Z}_{+}(\widetilde{\chi}_{r})$ is the set of all positive zeros of even entire function $\widetilde{\chi}_{r}$. In terms of convolutions, L. Zalcman's theorem means that the operator
$$\mathcal{P}f=(f\ast \chi_{r_1}, f\ast \chi_{r_2}), f\in L^{1,\mathrm{loc}}(\mathbb{R}^n) $$
is injective if and only if $r_1/r_2\notin E_n$. In this paper, a new formula for the inversion of the operator $\mathcal{P}$ is found under the condition $r_1/r_2\notin E_n$. The result obtained significantly simplifies the previously known procedures for recovering a function $f$ from given ball means values $f\ast \chi_{r_1}$ и $f\ast \chi_{r_2}$. The proofs use the methods of harmonic analysis, as well as the theory of entire and special functions.