Geometry and singularities of the prony mapping

The Prony mapping provides the global solution of the Prony system of equations (formula presented) This system appears in numerous theoretical and applied problems arising in Signal Reconstruction. The simplest example is the problem of reconstruction of linear combination of δ functions of the form (formula presented) with the unknown parameters a_i, x_i, i=1,..., n from the “moment measurements” (formula presented).

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 x_i: The investigation of this type of singularities has been started in [21] 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.