Concrete Solution to the Nonsingular Quartic Binary Moment Problem

Abstract

Given real numbers $\beta \equiv {\beta }^{\left(4\right)}:{\beta }_{00}$, ${\beta }_{10}$, ${\beta }_{01}$, ${\beta }_{20}$, ${\beta }_{11}$, ${\beta }_{02}$, ${\beta }_{30}$, ${\beta }_{21}$, ${\beta }_{12}$, ${\beta }_{03}$, ${\beta }_{40}$, ${\beta }_{31}$, ${\beta }_{22}$, ${\beta }_{13}$, ${\beta }_{04}$, with ${\beta }_{00}>0$, the quartic real moment problem for $\beta$ entails finding conditions for the existence of a positive Borel measure $\mu$, supported in ${\mathbb{R}}^2$, such that ${\beta }_{ij}=\int s^i{t}^{j}d\mu (0\le i+j\le 4)$. Let $\mathcal{M}(2)$ be the 6 x 6 moment matrix for ${\beta }^{(4)}$, given by $\mathcal{M}(2{)}_{\mathbf{i},\mathbf{j}}:={\beta }_{\mathbf{i}+\mathbf{j}}$, where $\mathbf{i},\mathbf{j}\in {\mathbb{Z}}_{+}^2$ and $\left|\mathbf{i}\right|,\left|\mathbf{j}\right|\le 2$. In this note we find concrete representing measures for ${\beta }^{(4)}$ when $\mathcal{M}(2)$ is nonsingular; moreover, we prove that it is possible to ensure that one such representing measure is 6-atomic.

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• The truncated moment problem on the union of parallel lines
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