2013/02/05 by Dmitry Batenkov, Gal Binyamini, Batenkov, Dmitry +1
Economics, Econometrics and Finance · Mathematics · #Advanced Differential Equations and Dynamical Systems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Stochastic processes and financial applications #math.CA
paper · pdf · doi:10.48550/arxiv.1302.0991
arxiv created 2013/02/05 · openalex publication_date 2013/02/05 · arxiv updated 2013/02/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.