# 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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