Moment vanishing problem and positivity: some examples
type: Item
instance of: scholarly article
title: Moment vanishing problem and positivity: some examples
zbMATH Open document ID: 1217.44008
published in: Bulletin des Sciences Mathématiques
publication date: 2011-03-03T00:00:00Z
review text: Let \(\Omega\) be a compact domain in \(\mathbb R^n\), let \(\mu\) be an arbitrary measure on \(\Omega\), and let \(P,\,q\) be complex continuous functions on \(\Omega\). The moment vanishing problem means to give necessary and sufficient conditions for vanishing of all moments of the form \(m_k(P,q,\mu)=\int_\Omega P^k(x)q(x)d\mu(x),\,k\geq1\) integer. When \(\Omega\) is a semi-algebraic compact set and \(P,\,q\) are polynomials, the moment vanishing problem is called the \textit{polynomial moment vanishing problem}. One of the authors (W. Zhao) previously stated the following conjecture: For positive \(\mu\) and \(P,q\) polynomials, if \(m_k(P,1,\mu)=0\) for all \(k\geq1\), then \(m_k(P,q,\mu)=0\) for \(k\gg1\) for any \(q\). The authors provide, in some special cases, necessary and sufficient conditions on the measure \(\mu\) for which the conjecture stated above holds.
reviewed by: Florian-Horia Vasilescu
Mathematics Subject Classification ID: 44A60
Mathematics Subject Classification ID: 28A25
zbMATH DE Number: 5860018
zbMATH Keywords: semi-algebraic sets
zbMATH Keywords: multidimensional composition condition
zbMATH Keywords: moment vanishing problem
zbMATH Keywords: polynomial moment vanishing problem
author: Jean-Pierre Francoise
author: F. Pakovich
author: Yosef Yomdin
author: Wenhua Zhao
MaRDI profile type: Publication
full work available at URL: j.bulsci.2010.06.002
OpenAlex ID: W2036453561
cites work: Non-autonomous equations related to polynomial two-dimensional systems
cites work: The Jacobian conjecture: Reduction of degree and formal expansion of the inverse
cites work: Moment inversion problem for piecewise D -finite functions
cites work: An "algebraic" reconstruction of piecewise-smooth functions from integral measurements
cites work: Relative differential forms and complex polynomials
cites work: Center conditions, compositions of polynomials and moments on algebraic curves
cites work: Center conditions. III: Parametric and model center problems
cites work: Q2717357
cites work: Center conditions at infinity for Abel differential equations
cites work: On the center problem for ordinary differential equations
cites work: Tree composition condition and moments vanishing
cites work: Abel Equations: Composition Conjectures and the Model Problem
cites work: Limit cycles of differential equations
cites work: The monodromy problem and the tangential center problem
cites work: Sampling Moments and Reconstructing Signals of Finite Rate of Innovation: Shannon Meets Strang–Fix
cites work: Constant terms in powers of a Laurent polynomial
cites work: Polynomial automorphisms and the Jacobian conjecture
cites work: Q3995206
cites work: Q3520669
cites work: Positivity, sums of squares and the multi-dimensional moment problem
cites work: Q4369156
cites work: Q4001531
cites work: Solution of the polynomial moment problem
cites work: Prime and composite Laurent polynomials
cites work: On polynomials orthogonal to all powers of a Chebyshev polynomial on a segment
cites work: On rational functions orthogonal to all powers of a given rational function on a curve
cites work: Generalized ‘second Ritt theorem’ and explicit solution of the polynomial moment problem
cites work: Cauchy-type integrals of algebraic functions
cites work: Q4285035
cites work: Signal Acquisition from Measurements via Non-Linear Models
cites work: The \(K\)-moment problem for compact semi-algebraic sets
cites work: Q5723265
cites work: Images of commuting differential operators of order one with constant leading coefficients
cites work: Generalizations of the image conjecture and the Mathieu conjecture
DOI: 10.1016/J.BULSCI.2010.06.002
description: scientific article; zbMATH DE number 5860018