Journal of Operator Theory

Volume 84, Issue 1, Summer 2020 pp. 185-209.

The core variety and representing measures in the truncated moment problem

Authors: Grigoriy Blekherman

$1$ , Lawrence Fialkow

$2$
Author institution:

$1$ School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, U.S.A.

$2$ Department of Computer Science, State University of New York, New Paltz, NY 12561, U.S.A.

Summary: The truncated moment problem asks for conditions so that a linear functional

$L$ on the vector space of real

$n$ -variable polynomials of degree at most

$d$ can be written as integration with respect to a positive Borel measure

$\mu$ on

$\mathbb{R}^n$ . More generally, let

$L$ act on a finite dimensional space of Borel-measurable functions defined on a

$T_{1}$ topological space

$S$ . Using an iterative geometric construction, we associate to

$L$ a subset of

$S$ called the \textit{core variety}

$\mathcal{CV}(L)$ . Our main result is that

$L$ has a representing measure

$\mu$ if and only if

$\mathcal{CV}(L)$ is nonempty. In this case,

$L$ has a finitely atomic representing measure, and the union of the supports of such measures is precisely

$\mathcal{CV}(L)$ .

DOI: http://dx.doi.org/10.7900/jot.2019mar15.2239
Keywords: truncated moment problems, representing measure, positive Riesz functional, moment matrix

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