Geometry and Singularities of the Prony mapping

Journal of Singularities
volume 10 (2014), 1-25

Received 7 January 2013. Received in revised form 12 May 2013.

DOI: 10.5427/jsing.2014.10a

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Abstract:

The Prony mapping provides the global solution of the Prony system of equations. This system appears in numerous theoretical and applied problems arising in Signal Reconstruction. The global solution of the Prony system, i.e., the inversion of the Prony mapping, encounters several types of singularities. One of the most important ones is a collision of some of the points. The investigation of this type of singularities has been started By Yomdin in 2010, where the role of finite differences was demonstrated. In the present paper we study this and other types of singularities of the Prony mapping, and describe its global geometry. We show, in particular, close connections of the Prony mapping with the "Vieta mapping" expressing the coefficients of a polynomial through its roots, and with hyperbolic polynomials and "Vandermonde mapping" studied by V. Arnold.

Keywords:

Singularities, Signal acquisition, Non-linear models, Moments inversion

Mathematical Subject Classification:

94A12, 62J02, 14P10, 42C99

Author(s) information: