A characterization of the Radon transform's range by a system of PDEs - Kurusa Árpád (1991)
Let $g$ be a compactly supported function of $d$-planes in ${opr}^n$. We prove that then $g$ is in the range of the Radon transform if and only if $g$ satisfies an ultrahyperbolic system of PDEs. We parameterize the $d$-planes by $d+1$ points $x_0,x_1,ldots,x_d$ on them and get the PDE $$left( {partial ^2over partial x^k_ipartial x^l_j} -{partial ^2over partial x^l_ipartial x^k_j} ight) {g(x_0,x_1,ldots,x_d)over Vol{x_i-x_0}_{i=1,d}} = 0, $$ where $x^k_i$ denotes the $k-th$ coordinate of $x_i$. At the end we analyze in detail the case of $d=1$.