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International Conference on Difference Equations and Applications

ICDEA 2012: Difference Equations, Discrete Dynamical Systems and Applications pp 15–28Cite as

Moment Vanishing of Piecewise Solutions of Linear ODEs

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 180))

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.

Batenkov is supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities.

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Notes

  1. 1.

    For consistency of notation, the sequence \(\left\{ m_{k}\right\} \) is understood to be extended with zeros for negative k.

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Acknowledgments

The authors would like to thank Y. Yomdin for useful discussions.

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Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA

    Dmitry Batenkov

  2. Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel

    Gal Binyamini

Authors
  1. Dmitry Batenkov
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  2. Gal Binyamini
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Corresponding author

Correspondence to Dmitry Batenkov .

Editor information

Editors and Affiliations

  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, Cerdanyola del Vallès, Barcelona, Spain

    Lluís Alsedà i Soler

  2. Department of Mathematics, University of Arizona, Tucson, Arizona, USA

    Jim M. Cushing

  3. Department of Mathematics, Trinity University, San Antonio, Texas, USA

    Saber Elaydi

  4. Faculty of Sciences, Department of Mathematics, University of Porto, Porto, Portugal

    Alberto A. Pinto

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Batenkov, D., Binyamini, G. (2016). Moment Vanishing of Piecewise Solutions of Linear ODEs. In: Alsedà i Soler, L., Cushing, J., Elaydi, S., Pinto, A. (eds) Difference Equations, Discrete Dynamical Systems and Applications. ICDEA 2012. Springer Proceedings in Mathematics & Statistics, vol 180. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-52927-0_2

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