Range descriptions for the spherical mean Radon transform (Q2372177)

The authors consider Radon type transforms obtained by averaging compactly supported functions on \(\mathbb{R}^n\) over spheres. More precisely, if \(B \subset \mathbb{R}^n\) denotes the ball of radius 1 and \(S = \partial B\) is the sphere, the spherical Radon transform of a function \(f \in C_0^\infty(B)\) is the function \[ Rf(p,t) = \int_{S} f(p+ty) \,dS(y), \] where \(dS\) denotes surface measure on the sphere, normalized to one, the center \(p\) ranges over all of \(S\), and the radius \(t\) over the relevant interval \([0,2]\). The authors characterize those functions \(g \in C_0^\infty(S \times [0,2])\) that arise as Radon transforms of some \(f \in C_0^\infty(B)\), via three equivalent criteria (Theorems 10, 11). The first criterion involves properties of the solution \(G(x,t)\) to the Darboux equation with boundary values given by \(g(x,t)\). The second is a sequence of orthogonality relations, involving eigenfunctions of the Dirichlet Laplacian. The third criterion refers to properties of the expansion of the partial Fourier-Hankel transform \(\widehat{g}(x,\lambda)\) of \(g(x,t)\) into spherical harmonics. In even dimensions, an additional moment condition has to be met (Theorem 10).