Journal of Operator Theory

Volume 54, Issue 1, Summer 2005 pp. 189-226.

Truncated $K$-moment problems in several variables

Authors: Ra\'ul E. Curto (1) and Lawrence A. Fialkow (2)
Author institution: (1) Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242, USA, (2) Department of Computer Science, State University of New York, New Paltz, NY 12561, USA

Summary: Let $\beta\equiv\beta^{(2n)}$ be an $N$-dimensional real multi-sequence of degree $2n$, with associated moment matrix $\mathcal{M}(n)\equiv \mathcal{M}(n)(\beta)$, and let $r:=\operatorname*{rank}\mathcal{M}(n)$. 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 $\mu$, which is $r$-atomic, with $\operatorname*{supp}\mu$ equal to $\mathcal{V} (\mathcal{M}(n+1))$, the algebraic variety of $\mathcal{M}(n+1)$. Further, $\beta$ has an $r$-atomic (minimal) representing measure supported in a semi-algebraic set $K_{\mathcal{Q}}$ subordinate to a family $\mathcal{Q} \equiv\{q_{i}\}_{i=1}^{m}\subseteq\mathbb{R}[t_{1},\ldots,t_{N}]$ 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 $\mathcal{M}_{q_{i}}\big(n+\big[\frac{1+\deg q_{i}}{2}\big]\big)$ are positive semidefinite, $1\leqslant i\leqslant m$; in this case, $\mu$ (as above) satisfies $\operatorname*{supp}\mu\subseteq K_{\mathcal{Q}}$, and $\mu$ has precisely $\operatorname*{rank}\mathcal{M}(n)-\operatorname*{rank}\mathcal{M}_{q_{i} }\big(n+\big[\frac{1+\deg q_{i}}{2}\big]\big)$ atoms in $\mathcal{Z}(q_{i})\equiv\{ t\in\mathbb{R}^{N}:q_{i}(t)=0\} $, $1\leqslant i\leqslant m$.

Keywords: Truncated moment problem, several variables, moment matrix extension, flat extensions of positive matrices, semi-algebraic sets, localizing matrix.

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