Journal of the
Korean Mathematical Society JKMS

In this paper, we consider the quartic moment problem suggested by Curto-Fialkow [6]. Given complex numbers $\gamma \equiv \gamma ^{(4)}:\gamma _{00},\gamma _{01},\gamma _{10},$ $\gamma _{02},\gamma _{11},\gamma _{20},\gamma _{03}$, $\gamma _{12},\gamma _{21},\gamma _{30},\gamma _{04},\gamma _{13},\gamma _{22},\gamma _{31},\gamma _{40},$ with $\gamma _{00}>0$ and $\gamma _{ji}=\bar{\gamma}_{ij},$ we discuss the conditions for the existence of a positive Borel measure $\mu ,$ supported in the complex plane $\mathbb{C}$ such that $\gamma _{ij}=\int \bar{z}% ^{i}z^{j}d\mu \ (0\leq i+j\leq 4).$ We obtain sufficient conditions for flat extension of the quartic moment matrix $M(2).$ Moreover, we examine the existence of flat extensions for nonsingular positive quartic moment matrices $M(2).$

Keywords: the quartic moment problem, representing measure, flat extension

MSC numbers: 44A60, 47A57