On the smallest singular value of multivariate Vandermonde matrices with clustered nodes

Autor(en): Kunis, Stefan 
Nagel, Dominik
Stichwörter: Colliding nodes; Condition number; ESPRIT; Frequency analysis; Mathematics; Mathematics, Applied; PARAMETERS; Restricted Fourier matrices; STABILITY; Super-resolution; SUPERRESOLUTION; Vandermonde matrix
Erscheinungsdatum: 2020
Herausgeber: ELSEVIER SCIENCE INC
Journal: LINEAR ALGEBRA AND ITS APPLICATIONS
Volumen: 604
Startseite: 1
Seitenende: 20
Zusammenfassung: 
We prove lower bounds for the smallest singular value of rectangular, multivariate Vandermonde matrices with nodes on the complex unit circle. The nodes are ``off the grid'', groups of nodes cluster, and the studied minimal singular value is bounded below by the product of inverted distances of a node to all other nodes in the specific cluster. By using known and new upper bounds for the smallest singular value, this completely settles the univariate case and pairs of nodes in the multivariate case, both including reasonable sharp constants. For larger clusters, we show that the smallest singular value depends also on the geometric configuration within a cluster. (C) 2020 Elsevier Inc. All rights reserved.
ISSN: 00243795
DOI: 10.1016/j.laa.2020.06.003

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