A range theorem for the Radon transform
Wolodymyr R. Madych, Donald C. Solmon · Proceedings of the American Mathematical Society · 1988
Conditions are prescribed for a function g g which are sufficient to ensure that it is the Radon transform of a continuous function f f on R n {{\mathbf {R}}^n} such that f ( x ) = O ( | x | − n − k − 1 ) f(x) = O({\left | x \right |^{ - n - k - 1}}) as | x | → ∞ \left | x \right | \to \infty . Roughly speaking, these criteria involve smoothness and the classical polynomial consistency conditions up to order k k on g g . In particular, the result implies Helgason’s Schwartz theorem for the Radon transform [Acta Math. 113 (1965)].