Injectivity Sets for Spherical Radon Transform

Abstract

Throughout in this chapter we assume that n ⩾ 2. Let $\mathcal{U}$ be a domain in ℝⁿ and let $f \in L_{loc} \left( \mathcal{U} \right)$. For any $x \in \mathcal{U}$ and almost all $r \in \left( {0,dist\left( {x,\partial \mathcal{U}} \right)} \right)$ the spherical Radon transform of f is defined by

$$ \mathcal{R}f\left( {x,r} \right) = \frac{1} {{\omega _{n - 1} }}\int\limits_{\mathbb{S}^{n - 1} } {f\left( {x + r\eta } \right)d\omega \left( \eta \right)} . $$

(1.1)

(The reader should be warned that there does not seem to be a standard terminology in this area. Some authors use spherical Radon transform to refer to the transform $\widehat f(\omega ,t)$ defined below in Section 1.2, which we have called the spherical Radon transform on spheres).