TY - GEN

T1 - Moment vanishing of piecewise solutions of linear ODEs

AU - Batenkov, Dmitry

AU - Binyamini, Gal

N1 - Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 2016.

PY - 2016

Y1 - 2016

N2 - 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.

AB - 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.

KW - Generalised exponential sums

KW - Holonomic ODEs

KW - Moment vanishing

KW - Recurrence relations

UR - http://www.scopus.com/inward/record.url?scp=84994905135&partnerID=8YFLogxK

U2 - 10.1007/978-3-662-52927-0_2

DO - 10.1007/978-3-662-52927-0_2

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AN - SCOPUS:84994905135

SN - 9783662529263

T3 - Springer Proceedings in Mathematics and Statistics

SP - 15

EP - 28

BT - Difference Equations, Discrete Dynamical Systems and Applications - ICDEA 2012

A2 - Cushing, Jim M.

A2 - Pinto, Alberto A.

A2 - Elaydi, Saber

A2 - i Soler, Lluis Alseda

PB - Springer New York LLC

T2 - 18th International Conference on Difference Equations and Applications, ICDEA 2012

Y2 - 23 July 2012 through 27 July 2012

ER -