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On Moment Problems with Holonomic Functions

Florent Bréhard 1, 2, 3, 4 Mioara Joldes 1 Jean-Bernard Lasserre 1, 5
1 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
LAAS - Laboratoire d'analyse et d'architecture des systèmes
3 ARIC - Arithmetic and Computing
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
4 PLUME - Preuves et Langages
LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : Many reconstruction algorithms from moments of algebraic datawere developed in optimization, analysis or statistics. Lasserre andPutinar proposed an exact reconstruction algorithm for the algebraicsupport of the Lebesgue measure, or of measures with density equalto the exponential of a known polynomial. Their approach relies onlinear recurrences for the moments, obtained using Stokes theorem.In this article, we extend this study to measures with holonomicdensities and support with real algebraic boundary. In the frameworkof holonomic distributions (i.e. they satisfy a holonomic system oflinear partial or ordinary differential equations with polynomial coef-ficients), an alternate method to creative telescoping is proposed forcomputing linear recurrences for the moments. When the coefficientsof a polynomial vanishing on the support boundary are given as pa-rameters, the obtained recurrences have the advantage of staying linearwith respect to them.This property allows for an efficient reconstruction method. Givena finite number of numerically computed moments for a measure withholonomic density, and assuming a real algebraic boundary for thesupport, we propose an algorithm for solving the inverse problem ofobtaining both the coefficients of a polynomial vanishing on the bound-ary and those of the polynomials involved in the holonomic operatorswhich annihilate the density
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Florent Bréhard, Mioara Joldes, Jean-Bernard Lasserre. On Moment Problems with Holonomic Functions. 44th International Symposium on Symbolic and Algebraic Computation (ISSAC 2019), Jul 2019, Pékin, China. pp.66-73. ⟨hal-02006645⟩

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