The paley-wiener theorem for the radon transform

We prove that the support of a complex-valued function f in ?^k is contained in a convex set K if and only if the support of its Radon transform k(s, ω) is, for each ω, contained in s ≦ S_K (ω); here S_K is the support function of the set K. This theorem is used to determine the propagation speeds of hyperbolic differential equations with constant coefficients, to prove the nonexistence of point spectrum for a certain class of partial differential operators, and to give a simple reduction of Lions' convolution theorem to the one-dimensional convolution theorem of Titchmarsh.