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.

Contents Full-Text PDF