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Journal of Operator Theory

Volume 69, Issue 1, Winter 2013  pp. 257-277.

On the closure of positive flat moment matrices

Authors Lawrence Fialkow (1) and Jiawang Nie (2)
Author institution: (1) Department of Computer Science, State University of New York, New Paltz, New York 12561, U.S.A.
(2) Department of Mathematics University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093, U.S.A.

Summary:  Let $y\equiv y^{(2d)} = \{y_{i}\}_{i\in \mathbb{Z}_{+}^{n}, |i|\leqslant 2d}$ denote a real $n$-dimensional multisequence of degree $2d$, $y_{0} > 0$. Let $L_{y}:\mathbb{R}_{2d}[x_{1},\ldots,x_{n}] \mapsto \mathbb{R}$ denote the Riesz functional, defined by $L_{y}( \sum_{|i|\leqslant 2d} a_{i}x^{i}) = \sum a_{i}y_{i}$, and let $M_{d}(y)$ denote the corresponding moment matrix. Positivity of $L_{y}$ plays a significant role in the Truncated Moment Problem and in the Polynomial Optimization Problem, but concrete conditions for positivity are unknown in general. $M_{d}(y)$ is {\it{flat}} if $\mathrm{rank}M_{d}(y) = \mathrm{rank}M_{d-1}(y)$; it is known that if $M_{d}(y)$ is positive semidefinite and flat, then $y$ has a representing measure (and $L_{y}$ is positive). Let $\mathcal{F}_{d} := \{y\equiv y^{(2d)}: M_{d}(y)\succeq 0 \mbox{ is flat}\}$. If $y\in \overline{\mathcal{F}}_{d}$ (the closure), then $y$ does not necessarily have a representing measure, but $L_{y}$ is positive, so $M_{d}(y)\succeq 0$ and, moreover, $\mathrm{rank}M_{d}(y)\leqslant \mathrm{dim}\mathbb{R}_{d-1}[x_{1},\ldots,x_{n}]$. We prove, conversely, that these positivity and rank conditions for $M_{d}(y)$ are sufficient for membership in $\overline{\mathcal{F}}_{d}$ in two basic cases: when $n=1$, $d\geqslant 1$, and when $n=d=2$.

Keywords:  truncated moment sequence, Riesz functional, $K$-positivity, flat moment matrix, representing measure, polynomial optimization

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