Algebraic reconstruction of piecewise-smooth functions from integral measurements

TL;DR

This paper develops algebraic methods for reconstructing piecewise-smooth functions from integral measurements like moments and Fourier coefficients, focusing on specific classes of functions and analyzing the stability of the solution systems.

Contribution

It introduces algebraic reconstruction techniques for piecewise-smooth functions, extending solutions of Prony-type systems to multi-dimensional cases and analyzing their stability.

Findings

Reconstruction reduces to solving non-linear algebraic systems.

Extensions of known methods to multi-dimensional cases are provided.

The local stability of solving Prony-type systems is investigated.

Abstract

This paper presents some results on a well-known problem in Algebraic Signal Sampling and in other areas of applied mathematics: reconstruction of piecewise-smooth functions from their integral measurements (like moments, Fourier coefficients, Radon transform, etc.). Our results concern reconstruction (from the moments or Fourier coefficients) of signals in two specific classes: linear combinations of shifts of a given function, and "piecewise $D$-finite functions" which satisfy on each continuity interval a linear differential equation with polynomial coefficients. In each case the problem is reduced to a solution of a certain type of non-linear algebraic system of equations ("Prony-type system"). We recall some known methods for explicitly solving such systems in one variable, and provide extensions to some multi-dimensional cases. Finally, we investigate the local stability of solving the Prony-type systems.