Problème des moments sur un compact de \({\mathbb{R}}^ n\) et décomposition de polynômes a plusieurs variables
type: Item
instance of: scholarly article
title: Problème des moments sur un compact de \({\mathbb{R}}^ n\) et décomposition de polynômes a plusieurs variables
zbMATH Open document ID: 0556.44006
DOI: 10.1016/0022-1236(84)90042-9
author: Gilles Cassier
published in: Journal of Functional Analysis
publication date: 1984-01-01T00:00:00Z
review text: Given a positive measure \(\mu\) on a compact set K, one has an associated sequence of moments \(a_ k=\int_{k}t^ kd\mu (t),\) with the usual meaning of \(t^ k=t_ 1^{k_ 1}...t_ n^{k_ n},k\in {\mathbb{N}}^ n.\) The moment problem is to characterize the sequences obtained in this way. It is understood in dimension 1, where, for example, \textit{Ch. Berg} and \textit{P. H. Maserick} [Math. Ann. 259, 487-495 (1982; Zbl 0486.44004)] proved that if \(K=P^{-1}({\mathbb{R}})\), then \((s_ k)\) is a sequence of moments of K iff \((s_ k)\) and \(P(s)_ k\) are sequences of positive type. For \(n>1\) they were able to prove the equivalence only if \((s_ k)\) and \((P(s))_ k\) are a sequence of moments, and posed the question of whether the theorem remains true when the sequences \((s_ k)\) and \((P(s))_ k\) satisfy the weaker condition of being of positive type. The author shows that this is true for polynomials P of a special type; either for a polynomial in two variables whose homogeneous part of highest degree is strictly negative on \({\mathbb{R}}^ 2\setminus \{0\}\), or for polynomials whose homogeneous part of highest degree is of the form \(-(\alpha_ 1X_ 1^{2p}+...+\alpha_ nX_ n^{2p})-H_{2p},\) with \(p\in {\mathbb{N}}\), \(\alpha_ k\in {\mathbb{R}}^*_+\) and \(H_{2p}^ a \)sum of squares. He also applies his method to solve the moment problem for p- balls, \(K=\{| x_ 1|^ p+...+| x_ n|^ p\leq 1\},\) generalizing a result for \(p=2\) by \textit{J. L. McGregor} [J. Approximation Theory 30, 315-333 (1980; Zbl 0458.41025)]. He concludes by giving a decomposition theorem for strictly positive polynomials on \(K=R^{- 1}({\mathbb{R}}_+)\) with R one of the previously cited two types. The key to his results is a functional analytic characterization of a necessary and sufficient condition for a moment sequence. Given \((a_ k| k\in {\mathbb{N}}^ n)\), and \(T=\sum_{r}u_ rX^ r\) a polynomial, let \(M_ T(a)=[m_{ij}],\) where \(m_{ij}=\sum u_ ra_{i+j+r}.\) Then \((a_ k)\) is a moment s…
Mathematics Subject Classification ID: 44A60
Mathematics Subject Classification ID: 26C05
Mathematics Subject Classification ID: 12D05
zbMATH DE Number: 3886535
zbMATH Keywords: moment problem
zbMATH Keywords: sequences of positive type
zbMATH Keywords: p-balls
zbMATH Keywords: decomposition
zbMATH Keywords: strictly positive polynomials
MaRDI profile type: Publication
full work available at URL: 0022-1236(84)90042-9
OpenAlex ID: W1482962557
cites work: A moment problem for positive measures on the unit disc
cites work: A remark on the multidimensional moment problem
cites work: Polynomially positive definite sequences
cites work: Q3038004
cites work: Q3671421
cites work: The multidimensional moment problem
cites work: Q4073488
cites work: Q5588717
cites work: Moments of measures on convex bodies
cites work: Solvability criteria for certain N-dimensional moment problems
cites work: Q5508589
description: scientific article; zbMATH DE number 3886535