A range theorem for the Radon transform

Abstract:

Conditions are prescribed for a function $g$ which are sufficient to ensure that it is the Radon transform of a continuous function $f$ on ${{\mathbf {R}}^n}$ such that $f(x) = O({\left | x \right |^{ - n - k - 1}})$ as $\left | x \right | \to \infty$. Roughly speaking, these criteria involve smoothness and the classical polynomial consistency conditions up to order $k$ on $g$. In particular, the result implies Helgason’s Schwartz theorem for the Radon transform [Acta Math. 113 (1965)].