  Prony's method is a prototypical eigenvalue analysis based method for the
reconstruction of a finitely supported complex measure on the unit circle from
its moments up to a certain degree. In this note, we give a generalization of
this method to the multivariate case and prove simple conditions under which
the problem admits a unique solution. Provided the order of the moments is
bounded from below by the number of points on which the measure is supported as
well as by a small constant divided by the separation distance of these points,
stable reconstruction is guaranteed. In its simplest form, the reconstruction
method consists of setting up a certain multilevel Toeplitz matrix of the
moments, compute a basis of its kernel, and compute by some method of choice
the set of common roots of the multivariate polynomials whose coefficients are
given in the second step. All theoretical results are illustrated by numerical
experiments.
