Diagonal spherical means

Abstract

We introduce a mean for functions and distributions of two vector variables, $k\in {\mathcal{D}}^{\mathrm{\prime }}({\mathbb{R}}^n\times {\mathbb{R}}^n)$, the diagonal spherical mean K, defined as $K(x,y)={\int }_{{\mathbb{S}}^{n-1}}k(x-\xi ,y-\xi )\mathrm{d}\xi .$ We study several properties of these means as well as identities satisfied by them. We show that the Dirac delta function in the diagonal of the unit ball $\mathbb{B}$ in ${\mathbb{R}}^n$ admits a representation as the diagonal spherical mean of certain kernels κ, $\delta (x-y)={\int }_{{\mathbb{S}}^{n-1}}\kappa (x-\xi ,y-\xi )\mathrm{d}\xi ,$ distributionally for $(x,y)\in \mathbb{B}\times \mathbb{B}$. These representations of the delta function solve the problem of reconstruction of a distribution with compact support inside $\mathbb{B}$ if its standard spherical Radon transform is known in the boundary ${\mathbb{S}}^{n-1}=\mathrm{\partial }\mathbb{B}$.

Keywords:

• generalized functions
• diagonal means
• spherical mean Radon transform
• thermoacoustic tomography

2000 Mathematics Subject Classification:

• Primary: 46F10
• Secondary: 44A12
• 92C55
• 65R32

Funding

The research was supported in part by the NSF grant 0968448.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. One may also consider normalized diagonal spherical means, that is, the integrals $\left(1/{\sigma }_{n-1}\right){\int }_{{\mathbb{S}}^{n-1}}f(x-\xi ,y-\xi )\mathrm{d}\xi$, where ${\sigma }_{n-1}$ is given by (3.4(3.4) ${\sigma }_{n-1}=\frac{2{\pi }^{n/2}}{\mathrm{\Gamma }\left(n/2\right)},$(3.4) ).

2. If $U\subset {\mathbb{R}}^n$ is an open set and $f,g\in {\mathcal{D}}^{\mathrm{\prime }}\left({\mathbb{R}}^n\right)$, then $f\left(x\right)=g\left(x\right)$ distributionally for $x\in U$ if $⟨f,\phi ⟩=⟨g,\phi ⟩$ for all test functions with $\mathrm{supp}\phi \subset U$.

3. These spaces are very similar to the spaces ${\mathcal{R}}_n$ introduced in [13]. The spaces ${\mathcal{R}}_n[0,\mathrm{\infty })$ were introduced in [14].