Limits of positive flat bivariate moment matrices

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.