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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