Geometry and Singularities of the Prony mapping
For researchers in signal reconstruction and applied mathematics, this work provides a geometric understanding of singularities in Prony's method, but is incremental as it builds on prior work.
The paper studies singularities and global geometry of the Prony mapping, which inverts moment measurements to recover parameters of a sum of delta functions. It establishes connections with Vieta and Vandermonde mappings, and analyzes collision singularities of the points x_i.
Prony mapping provides the global solution of the Prony system of equations \[ Σ_{i=1}^{n}A_{i}x_{i}^{k}=m_{k},\ k=0,1,...,2n-1. \] 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 $g(x)=\sum_{i=1}^{n}a_{i}δ(x-x_{i})$, with the unknown parameters $a_{i},\ x_{i},\ i=1,...,n,$ from the "moment measurements" $m_{k}=\int x^{k}g(x)dx.$ Global solution of the Prony system, i.e. 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 \cite{yom2009Singularities} 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.