Prony's method on the sphere

TL;DR

This paper extends Prony's method to reconstruct measures on the sphere using moments, providing conditions for success, uniqueness, and certificates for semidefinite relaxations across arbitrary dimensions.

Contribution

It offers a precise interpolation-based criterion for Prony's method success on the sphere, enabling unique measure reconstruction from moments in any dimension.

Findings

Successful reconstruction when support is separated

Unique measure recovery from moments on the sphere

Certificates for semidefinite relaxation methods

Abstract

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.