Local and global geometry of Prony systems and Fourier reconstruction of piecewise-smooth functions

TL;DR

This paper analyzes Prony systems' geometry and their application to Fourier-based reconstruction of piecewise-smooth functions, addressing solvability, stability, and near-singular cases to improve accuracy in signal processing tasks.

Contribution

It provides a comprehensive study of Prony systems' global and local properties and introduces a modified method for highly accurate Fourier reconstruction of piecewise-smooth functions.

Findings

Analysis of Prony systems' global solvability and stability.

Identification of challenges in near-singular cases like node collisions.

Development of a modified reconstruction method achieving maximal accuracy.

Abstract

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.