Prony's method on the sphere

Eigenvalue analysis based methods are well suited for the reconstruction of finitely supported measures from their moments up to a certain degree. We give a precise description when Prony's method succeeds in terms of an interpolation condition. In particular, this allows for the unique reconstruction of a measure from its trigonometric moments whenever its support is separated and also for the reconstruction of a measure on the unit sphere from its moments with respect to spherical harmonics. Both results hold in arbitrary dimensions and also yield a certificate for popular semidefinite relaxations of these reconstruction problems.