Moment vanishing of piecewise solutions of linear ODEs

Dmitry Batenkov*, Gal Binyamini

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

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.

Original languageEnglish
Title of host publicationDifference Equations, Discrete Dynamical Systems and Applications - ICDEA 2012
EditorsJim M. Cushing, Alberto A. Pinto, Saber Elaydi, Lluis Alseda i Soler
PublisherSpringer New York LLC
Pages15-28
Number of pages14
ISBN (Print)9783662529263
DOIs
StatePublished - 2016
Externally publishedYes
Event18th International Conference on Difference Equations and Applications, ICDEA 2012 - Barcelona, Spain
Duration: 23 Jul 201227 Jul 2012

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume180
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Conference

Conference18th International Conference on Difference Equations and Applications, ICDEA 2012
Country/TerritorySpain
CityBarcelona
Period23/07/1227/07/12

Funding

FundersFunder number
Israel Academy of Sciences and Humanities

    Keywords

    • Generalised exponential sums
    • Holonomic ODEs
    • Moment vanishing
    • Recurrence relations

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