Yosef Yomdin

Dmitry Batenkov, Yosef Yomdin

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.

Gil Goldman, Yehonatan Salman, Yosef Yomdin

We start a systematic study of the topology, geometry and singularities of the Prony varieties $S_q(μ)$, defined by the first $q+1$ equations of the classical Prony system $$\sum_{j=1}^d a_j x_j^k = μ_k, \ k= 0,1,\ldots \ .$$ Prony varieties, being a generalization of the Vandermonde varieties, introduced in [5,21], present a significant independent mathematical interest (compare [5,19,21]). The importance of Prony varieties in the study of the error amplification patterns in solving Prony system was shown in [1-4,19]. In [19] a survey of these results was given, from the point of view of Singularity Theory. In the present paper we show that for $q\ge d$ the variety $S_q(μ)$ is diffeomerphic to an intersection of a certain affine subspace in the space ${\cal V}_d$ of polynomials of degree $d$, with the hyperbolic set $H_d$. On the Prony curves $S_{2d-2}$ we study the behavior of the amplitudes $a_j$ as the nodes $x_j$ collide, and the nodes escape to infinity. We discuss the behavior of the Prony varieties as the right hand side $μ$ varies, and possible connections of this problem with J. Mather's result in [23] on smoothness of solutions in families of linear systems.

Gil Goldman, Yehonatan Salman, Yosef Yomdin

This is a survey paper discussing one specific (and classical) system of algebraic equations - the so called "Prony system". We provide a short overview of its unusually wide connections with many different fields of Mathematics, stressing the role of Singularity Theory. We reformulate Prony System as the problem of reconstruction of "Spike-train" signals of the form $F(x)=\sum_{j=1}^d a_jδ(x-x_j)$ from the noisy moment measurements. We provide an overview of some recent results of [1-3, 6, 8, 9, 11, 12, 5] on the "geometry of the error amplification" in the reconstruction process, in situations where the nodes $x_j$ near-collide. Some algebraic-geometric structures, underlying the error amplification, are described (Prony, Vieta, and Hankel mappings, Prony varieties), as well as their connection with Vandermonde mappings and varieties. Our main goal is to present some promising fields of possible applications of Singulary Theory.

Dmitry Batenkov, Yosef Yomdin

Many reconstruction problems in signal processing require solution of a certain kind of nonlinear systems of algebraic equations, which we call Prony systems. We study these systems from a general perspective, addressing questions of global solvability and stable inversion. Of special interest are the so-called "near-singular" situations, such as a collision of two closely spaced nodes. We also discuss the problem of reconstructing piecewise-smooth functions from their Fourier coefficients, which is easily reduced by a well-known method of K.Eckhoff to solving a particular Prony system. As we show in the paper, it turns out that a modification of this highly nonlinear method can reconstruct the jump locations and magnitudes of such functions, as well as the pointwise values between the jumps, with the maximal possible accuracy.