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Journal of Operator Theory

Volume 60, Issue 2, Fall 2008  pp. 343-377.

Truncated multivariable moment problems with finite variety

Authors Lawrence A. Fialkow
Author institution: Department of Computer Science, State University of New York, New Paltz, NY 12561, USA

Summary:  Let $\beta \equiv \{\beta_{i}\}_{i\in\mathbb{Z}_{+}^{d},|i| \leqslant 2n}$ be a real $d$-dimensional multisequence of degree $2n$, with moment matrix $ {\mathcal{M}}(n)$, and let $ {\mathcal{V}} \equiv V( {\mathcal{M}}(n))$ denote the associated algebraic variety. For the case $v\equiv \mathrm{card} {\mathcal{V}} < +\infty$, we prove that $\beta$ has a representing measure if and only if $r\equiv \mathrm{rank} {\mathcal{M}}(n) \leqslant v$ and there exists a positive moment matrix extension $ {\mathcal{M}} \equiv {\mathcal{M}}(n+v-r+1)$ satisfying $\mathrm{rank} {\mathcal{M}}\leqslant \mathrm{card} V( {\mathcal{M}})$. For the class of {\it recursively determinate} moment matrices $ {\mathcal{M}}(n)$, we present a computational algorithm for establishing the existence (or nonexistence) of an extension $ {\mathcal{M}}$ as above and, in the positive case, for computing a minimal representing measure for $\beta$. We also show that for the case $r < v < +\infty$, it is possible for $\beta$ to admit a representing measure $\mu$ with $\mathrm{card}\, \mathrm{supp}\,\mu < v$; equivalently, in this case $\mathrm{supp}\,\mu$ may be a proper subset of $V( {\mathcal{M}}(n))$.

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