Truncated K-moment problems in several variables

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dc.creatorCurto, Raul E.
dc.creatorFialkow, Lawrence A.
dc.date2005-07-04
dc.date.accessioned2026-07-07T06:33:40Z
dc.date.available2026-07-07T06:33:40Z
dc.descriptionLet $β\equivβ^{(2n)}$ be an N-dimensional real multi-sequence of degree 2n, with associated moment matrix $\mathcal{M}(n)\equiv \mathcal{M}(n)(β)$, and let $r:=rank \mathcal{M}(n)$. We prove that if $\mathcal{M}(n)$ is positive semidefinite and admits a rank-preserving moment matrix extension $\mathcal{M}(n+1)$, then $\mathcal{M}(n+1)$ has a unique representing measure μ, which is r-atomic, with supp μ$ equal to $\mathcal{V}(\mathcal{M}(n+1))$, the algebraic variety of $\mathcal{M}(n+1)$. Further, βhas an r-atomic (minimal) representing measure supported in a semi-algebraic set $K_{\mathcal{Q}}$ subordinate to a family $\mathcal{Q}% \equiv\{q_{i}\}_{i=1}^{m}\subseteq\mathbb{R}[t_{1},...,t_{N}]$ if and only if $\mathcal{M}(n)$ is positive semidefinite and admits a rank-preserving extension $\mathcal{M}(n+1)$ for which the associated localizing matrices $\mathcal{M}_{q_{i}}(n+[\frac{1+°q_{i}}{2}])$ are positive semidefinite $(1\leq i\leq m)$; in this case, μ(as above) satisfies supp μ\subseteq K_{\mathcal{Q}}$, and μhas precisely rank \mathcal{M}(n)-rank \mathcal{M}_{q_{i}}(n+[\frac{1+°q_{i}}{2}])$ atoms in $\mathcal{Z}(q_{i})\equiv {t\in\mathbb{R}^{N}:q_{i}(t)=0}$, $1\leq i\leq m$.
dc.description33 pages; to appear in J. Operator Theory
dc.identifierhttps://arxiv.org/abs/math/0507067
dc.identifierhttp://arxiv.org/abs/math/0507067
dc.identifierJ. Operator Theory 54(2005), 189-226
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/99272
dc.subjectFunctional Analysis
dc.subject47A57, 44A60, 30D05
dc.titleTruncated K-moment problems in several variables
dc.typetext

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