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We study the decomposition of a multivariate Hankel matrix H\_$\sigma$ as asum of Hankel matrices of small rank in correlation with the decomposition ofits symbol $\sigma$ as a sum of polynomial-exponential series. We present a newalgorithm to compute the low rank decomposition of the Hankel operator and thedecomposition of its symbol exploiting the properties of the associatedArtinian Gorenstein quotient algebra A\_$\sigma$. A basis of A\_$\sigma$ iscomputed from the Singular Value Decomposition of a sub-matrix of the Hankelmatrix H\_$\sigma$. The frequencies and the weights are deduced from thegeneralized eigenvectors of pencils of shifted sub-matrices of H $\sigma$.Explicit formula for the weights in terms of the eigenvectors avoid us to solvea Vandermonde system. This new method is a multivariate generalization of theso-called Pencil method for solving Prony-type decomposition problems. Weanalyse its numerical behaviour in the presence of noisy input moments, anddescribe a rescaling technique which improves the numerical quality of thereconstruction for frequencies of high amplitudes. We also present a new Newtoniteration, which converges locally to the closest multivariate Hankel matrix oflow rank and show its impact for correcting errors on input moments.

Structured low rank decomposition of multivariate Hankel matrices