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arXiv · 1302.0991

Moment vanishing of piecewise solutions of linear ODEs

Dmitry Batenkov·Gal Binyamini
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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.

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Dmitry Batenkov, Gal Binyamini. 2013-02-05. Moment vanishing of piecewise solutions of linear ODEs. https://arxiv.org/abs/1302.0991

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