Algebraic reconstruction of piecewise-smooth functions from integral measurements
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 -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.
Keywords
Cite
@article{arxiv.1103.3969,
title = {Algebraic reconstruction of piecewise-smooth functions from integral measurements},
author = {Dmitry Batenkov and Niv Sarig and Yosef Yomdin},
journal= {arXiv preprint arXiv:1103.3969},
year = {2013}
}