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Moment Vanishing of Piecewise Solutions of Linear ODEs

  • Dmitry Batenkov
  • Gal Binyamini
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (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.

Keywords

Moment vanishing Holonomic ODEs Recurrence relations Generalised exponential sums 

Notes

Acknowledgments

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

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Department of MathematicsWeizmann Institute of ScienceRehovotIsrael

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