Identifiability of harmonic parameterization in N dimensions

In 1911, Carathéodory et al published a result that is a cornerstone of line spectra (harmonic) analysis and modern parametric harmonic retrieval. This result was later popularized by Pisarenko, and is widely known in the spectral analysis community as "Carathéodory's Parameterization". The uniqueness part of Carathéodory's result specifies the condition under which one can uniquely recover the frequencies (spectral lines) in a finite sum of one-dimensional harmonics, given a finite set of measurements. The multidimensional case is of interest in a variety of problems, including joint multiuser / multipath carrier offset, angle, and delay estimation, yet the associated model identifiability problem has not been thoroughly addressed. This is the subject of the main Theorem in this paper. The proof relies on a recent result regarding the uniqueness of low-rank decomposition of N-way arrays.