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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Limits of positive flat bivariate moment matrices
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by Lawrence A. Fialkow PDF
Trans. Amer. Math. Soc. 367 (2015), 2665-2702 Request permission


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 $\beta \equiv \beta ^{(2d)}=\{\beta _{i}\}_{ i\in \mathbb {Z}_{+}^{n},|i| \leq 2d }$, $\beta _{0}>0$, 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:
  • Received by editor(s): February 10, 2013
  • Published electronically: December 3, 2014
  • © Copyright 2014 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 367 (2015), 2665-2702
  • MSC (2010): Primary 47A57, 44A60, 42A70, 30E05; Secondary 15A57, 15-04, 47A20
  • DOI:
  • MathSciNet review: 3301877