Prony's method on the sphere
For researchers in signal processing and approximation theory, it offers a theoretical guarantee for measure reconstruction on spheres, but the results are incremental extensions of existing eigenvalue-based methods.
The paper extends Prony's method to reconstruct finitely supported measures on the unit sphere from their spherical harmonic moments, providing an interpolation condition for uniqueness and a certificate for semidefinite relaxations.
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.