Methods and Applications of Analysis

Volume 25 (2018)

Number 3

In Memory of Professor John N. Mather: Part 1 of 3

Guest Editors: Sen Hu, University of Science and Technology, China; Stanisław Janeczko, Polish Academy of Sciences, Poland; Stephen S.-T. Yau, Tsinghua University, China; and Huaiqing Zuo, Tsinghua University, China.

Geometry and singularities of Prony varieties

Pages: 257 – 276

DOI: https://dx.doi.org/10.4310/MAA.2018.v25.n3.a5

Authors

Gil Goldman (Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel)

Yehonatan Salman (Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel)

Yosef Yomdin (Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel)

Abstract

We start a systematic study of the topology, geometry and singularities of the Prony varieties $S_q(\mu)$, defined by the first $q +1$ equations of the classical Prony system $\sum^{d}_{j=1} a_j x^k_j = \mu_k , k = 0, 1, \dotsc$

Prony varieties, being a generalization of the Vandermonde varieties, introduced in [5,22], present a significant independent mathematical interest (compare [5, 20, 22]). The importance of Prony varieties in the study of the error amplification patterns in solving Prony system was shown in [1–4, 20]. In [20] a survey of these results was given, from the point of view of Singularity Theory.

In the present paper we show that for $q \geq d$ the variety $S_q (\mu)$ is diffeomerphic to the intersection of a certain affine subspace in the space $\mathcal{V}_d$ of polynomials of degree $d$, with the hyperbolic set $H_d$.

In case of 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 $\mu$ varies, and possible connections of this problem with J. Mather’s result in [24] on smoothness of solutions in families of linear systems.

Keywords

Prony system, spike-train signals, Prony and Vandermonde varieties, nodal collision singularities

2010 Mathematics Subject Classification

32S05, 58K20, 65H10, 94A12

Full Text (PDF format)

Received 4 June 2018

Accepted 12 October 2018

Published 1 November 2019