Inversion of a convolution operator associated with spherical means

An obvious property of an arbitrary nonzero smooth antiperiodic function is that its derivative has no corresponding period. In other words, if $r$ is a fixed positive number, $f(x+r)+f(x-r)=0$ and $f'(x+r)-f'(x-r)=0$ on the real axis, then $f=0$. This fact admits non-trivial generalizations to multidimensional spaces. One general method for such generalizations is the following Brown-Schreiber-Taylor theorem on spectral analysis: any non-zero subspace $\mathcal{U}$ in $C(\mathbb{R}^n)$ invariant under all motions of $\mathbb{R}^n$ contains for some $\lambda\in \mathbb{C}$, the radial function $(\lambda|x|)^{1-\frac{n}{2}}J_{\frac{n}{2}-1}(\lambda|x|)$, where $J_{\nu}$ is the Bessel function of the first kind of order $\nu$. In particular, if a function $f\in C^1(\mathbb{R}^n)$ and its normal derivative have zero integrals over all spheres of fixed radius $r$ in $\mathbb{R}^n$, then $f=0$. In terms of convolution, this means that the operator $\mathcal{P}f =(f\ast \Delta \chi_r, f\ast \sigma_r)$, $f\in C(\mathbb{R}^n)$, is injective, where $\Delta$ is the Laplace operator, $\chi_{r}$ is the indicator of the ball $B_r=\{x\in\mathbb{R}^n: |x|<r\}$, $\sigma_{r}$ is the surface delta function centered on the sphere $S_r= \{x\in\mathbb{R}^n: |x|=r\}$. In this paper, we study the inversion problem for the operator $\mathcal{P}$ on the class of distributions. A new formula for reconstruction a distribution $f\in \mathcal{D}'(\mathbb{R}^n)$ from known convolutions $f\ast \Delta \chi_r$ and $f\ast \sigma_r$ is obtained. The paper uses the 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 the expansion of the Dirac delta function in terms of a system of radial distributions supported in $\overline{B}_r$, biorthogonal to some system of spherical functions. A similar approach can be used to invert other convolution operators with radial distributions in $\mathcal{E}'(\mathbb{R}^n)$.