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Limits of positive flat bivariate moment matrices


Author: Lawrence A. Fialkow
Journal: Trans. Amer. Math. Soc. 367 (2015), 2665-2702
MSC (2010): Primary 47A57, 44A60, 42A70, 30E05; Secondary 15A57, 15-04, 47A20
DOI: https://doi.org/10.1090/S0002-9947-2014-06393-9
Published electronically: December 3, 2014
MathSciNet review: 3301877
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Abstract: The bivariate moment problem for a sequence $ \beta \equiv \beta ^{(6)}$ of degree $ 6$ remains unsolved, but we prove that if the associated $ 10 \times 10$ moment matrix $ M_{3}(\beta )$ satisfies $ M_{3}\succeq 0$ and $ rank~M_{3}\le 6$, then $ \beta $ admits a sequence of approximate representing measures, and $ \beta ^{(5)}$ has a representing measure. More generally, let $ \overline {\mathcal {F}_{d}}$ denote the closure of the positive flat moment matrices of degree $ 2d$ in $ n$ variables. Each matrix in $ \overline {\mathcal {F}_{d}}$ admits computable approximate representing measures, and in 2013, Jiawang Nie and the author began to study concrete conditions for membership in this class. Let $ \beta \equiv \beta ^{(2d)}=\{\beta _{i}\}_{ i\in \mathbb{Z}_{+}^{n},\vert i\vert \leq 2d }$, $ \beta _{0}>0$, denote a real $ n$-dimensional sequence of degree $ 2d$. If the corresponding moment matrix $ M_{d}\equiv M_{d}(\beta )$ is the limit of a sequence of positive flat moment matrices $ \{M_{d}^{(k)}\}$, i.e., $ M_{d}^{(k)}\succeq 0$ and $ rank~M_{d}^{(k)} = rank~M_{d-1}^{(k)}$, then i) $ M_{d}\succeq 0$, ii) $ rank~M_{d} \le \rho _{d-1} \equiv dim~\mathbb{R}[x_{1},\ldots ,x_{n}]_{d-1}$, and iii) $ \beta ^{(2d-1)}$ admits a representing measure. We extend our earlier results by proving, conversely, that for $ n=2$, if $ M_{d}$ satisfies certain positivity and rank conditions related to i)-iii), then $ M_{d}$ is the limit of positive flat moment matrices.


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Additional Information

Lawrence A. Fialkow
Affiliation: Department of Computer Science, State University of New York, New Paltz, New York 12561
Email: fialkowl@newpaltz.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-06393-9
Keywords: Truncated moment problem, Riesz functional, moment matrix extension, flat extensions of positive matrices, positive functional
Received by editor(s): February 10, 2013
Published electronically: December 3, 2014
Article copyright: © Copyright 2014 American Mathematical Society