A new proof of the support theorem and the range characterization for the Radon transform (Q796781)

The Radon transform R associates with a function f on \({\mathbb{R}}^ n\) the set of its integrals over hyperplanes. The support theorem states that a function \(f\in {\mathcal S}({\mathbb{R}}^ n)\) whose Radon transform vanishes for all hyperplanes which do not meet some ball B is supported in B. The author gives a simple proof for the support theorem. The range of the Radon transform is characterized by the following theorem of Helgason: If \(g\in {\mathcal S}({\mathbb{R}}^ 1\times S^{n-1}),\quad g(s,\omega)=0\) for \(| s| \geq A\) and \(\omega \in S^{n-1},\) and \(\int_{R^ 1}g(s,\omega)s^ mds\) is a homogeneous polynomial of degree m in \(\omega\), then \(g=Rf\) with a \(C^{\infty}\)-function f whose support is in the ball \(| x|<A.\) The author gives a proof for \(n=2\).