Moment vanishing of piecewise solutions of linear ODEs

TL;DR

This paper investigates the maximum number of zero moments for piecewise solutions of linear ODEs with polynomial coefficients, linking the problem to analytic ODE theory, approximation, and signal processing, and providing bounds based on recurrence relations.

Contribution

It introduces a recurrence relation approach to bound the moment vanishing index for solutions of linear ODEs with polynomial coefficients.

Findings

Established a general bound for the moment vanishing index for any given operator.

Provided uniform bounds for several families of operators.

Linked the moment vanishing problem to analytic ODE theory and approximation theory.

Abstract

We consider the "moment vanishing problem" for a general class of piecewise-analytic functions which satisfy on each continuity interval a linear ODE with polynomial coefficients. This problem, which essentially asks how many zero first moments can such a (nonzero) function have, turns out to be related to several difficult questions in analytic theory of ODEs (Poincare's Center-Focus problem) as well as in Approximation Theory and Signal Processing ("Algebraic Sampling"). While the solution space of any particular ODE admits such a bound, it will in the most general situation depend on the coefficients of this ODE. We believe that a good understanding of this dependence may provide a clue for attacking the problems mentioned above. In this paper we undertake an approach to the moment vanishing problem which utilizes the fact that the moment sequences under consideration satisfy a recurrence relation of fixed length, whose coefficients are polynomials in the index. For any given operator, we prove a general bound for its moment vanishing index. We also provide uniform bounds for several operator families.