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Journal of Operator Theory

Volume 42, Issue 2, Fall 1999  pp. 425-436.

Minimal representing measures arising from rank-increasing moment matrix extensions

Authors Lawrence A. Fialkow
Author institution: Department of Mathematics and Computer Science, State University of New York at New Paltz, New Paltz, NY 12561, U.S.A.

Summary:  If $\mu$ is a representing measure for $\gamma\equiv \gamma^{(2n)}$ in the Truncated Complex Moment Problem $\gamma_{ij}= \int \overline{z}^{i}z^{j}\,{\rm d}\mu$ ($0\leq i+j\leq 2n$), then $\card\supp\m u$ $\geq \rank M(n)$, where $M(n)\equiv M(n)(\gamma)$ is the associated moment matrix. We present a concrete example of $\gamma$ illustrating the case when $\card\supp \mu > \rank M(n)(\gamma)$ for every representing measure $\mu$. This example is based on an analysis of moment problems in which some analytic column $Z^{k}$ of $M(n)$ can be expressed as a linear combination of columns ${\overline{Z}^{i}Z^{j}}$ of strictly lower degree.

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