The truncated moment problem via homogenization and flat extensions
type: Item
instance of: scholarly article
title: The truncated moment problem via homogenization and flat extensions
zbMATH Open document ID: 1262.47018
published in: Journal of Functional Analysis
publication date: 2012-09-21T00:00:00Z
reviewed by: Adhemar Bultheel
Mathematics Subject Classification ID: 47A57
Mathematics Subject Classification ID: 44A60
zbMATH DE Number: 6084765
zbMATH Keywords: flat extension
zbMATH Keywords: homogenization
zbMATH Keywords: moment
zbMATH Keywords: representing measure
zbMATH Keywords: Riesz functional
zbMATH Keywords: semidefinite programming
zbMATH Keywords: truncated moment problem
author: Lawrence Fialkow
author: Jiawang Nie
describes a project that uses: SeDuMi
MaRDI profile type: Publication
full work available at URL: j.jfa.2012.06.004
OpenAlex ID: W2021129860
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DOI: 10.1016/J.JFA.2012.06.004
review text: The paper studies the truncated moment problem in \(n\) variables, namely, to find a positive measure on \(\mathbb{R}^n\) such that matches the prescribed moments \(y_\alpha\) in NEWLINE\[NEWLINE\int x^\alpha d\mu=\int x_1^{\alpha_1}\dots x_n^{\alpha_n}\,d\mu=y_\alpha,\quad \alpha\in\mathbb{N}^n,\quad |\alpha|=\alpha_1+\dots+\alpha_n\leq d.NEWLINE\]NEWLINE The problem is reformulated in homogeneous form, which means to rewrite \(x^\alpha=x_1^{\alpha_1}\dots x_n^{\alpha_n}\) as \(x_0^{d-|\alpha|}x^\alpha\). The original problem admits a solution supported in \(\mathbb{R}^n\) if and only if the homogeneous problem admits a solution supported on the unit sphere of \(\mathbb{R}^{n+1}\). Furthermore, by associating with this truncated moment problem also a moment matrix and a positive definite linear functional (the Riesz functional) acting on the space of polynomials, it is shown that the existence of a solution and, in fact, a solution itself (if it exists) can be obtained by solving a sequence of semidefinite programming problems (a sequence depending on the degree of the polynomials involved). This increase of degree corresponds to extensions of the moment problem. If a flat extension is reached, i.e., if it does not increase the rank of the moment matrix, a solution is found. Some examples illustrate the ideas.
description: scientific article; zbMATH DE number 6084765