Moment inversion problem for piecewise D-finite functions

TL;DR

This paper presents a method for exact reconstruction of piecewise D-finite functions with unknown discontinuities from their moments, deriving explicit recurrence relations and proposing an algorithm tested through numerical simulations.

Contribution

The authors derive explicit recurrence relations for moments of piecewise D-finite functions and develop a generic reconstruction algorithm accounting for unknown discontinuities.

Findings

Recurrence relations explicitly derived for moments of piecewise D-finite functions.

Reconstruction algorithm formulated and analyzed for solvability.

Numerical simulations demonstrate algorithm sensitivity to noise.

Abstract

We consider the problem of exact reconstruction of univariate functions with jump discontinuities at unknown positions from their moments. These functions are assumed to satisfy an a priori unknown linear homogeneous differential equation with polynomial coefficients on each continuity interval. Therefore, they may be specified by a finite amount of information. This reconstruction problem has practical importance in Signal Processing and other applications. It is somewhat of a ``folklore'' that the sequence of the moments of such ``piecewise D-finite''functions satisfies a linear recurrence relation of bounded order and degree. We derive this recurrence relation explicitly. It turns out that the coefficients of the differential operator which annihilates every piece of the function, as well as the locations of the discontinuities, appear in this recurrence in a precisely controlled manner. This leads to the formulation of a generic algorithm for reconstructing a piecewise D-finite function from its moments. We investigate the conditions for solvability of the resulting linear systems in the general case, as well as analyze a few particular examples. We provide results of numerical simulations for several types of signals, which test the sensitivity of the proposed algorithm to noise.