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

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 $\beta \equiv \beta ^{(2d)}=\{\beta _{i}\}_{ i\in \mathbb {Z}_{+}^{n},|i| \leq 2d }$, $\beta _{0}>0$, 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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