Injectivity Sets for Spherical Radon Transform

Throughout in this chapter we assume that n ⩾ 2. Let \( \mathcal{U} \) be a domain in ℝ n 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).