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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 $\mathcal{CV}(L)$.

Keywords: truncated moment problems, representing measure, positive Riesz functional, moment matrix

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