A multivariate generalization of Prony's method

Autor(en): Kunis, Stefan 
Peter, Thomas
Roemer, Tim 
von der Ohe, Ulrich
Stichwörter: Exponential sum; FOURIER; Frequency analysis; INTERPOLATION; Mathematics; Mathematics, Applied; Moment problem; PARAMETER-ESTIMATION; RECONSTRUCTION; Spectral analysis; Super-resolution
Erscheinungsdatum: 2016
Herausgeber: ELSEVIER SCIENCE INC
Journal: LINEAR ALGEBRA AND ITS APPLICATIONS
Volumen: 490
Startseite: 31
Seitenende: 47
Zusammenfassung: 
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. (C) 2015 Elsevier Inc. All rights reserved.
ISSN: 00243795
DOI: 10.1016/j.laa.2015.10.023

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