Chunji Li, Hongkai Liang Northeastern University; Dalian University of Technology

Abstract : Let $\gamma ^{\left( n\right) }\equiv \{\gamma _{ij}\}\,(0\leq i+j\leq 2n,\,|i-j|\leq n)$ be a sequence in the complex number set $\mathbb{C}$ and let $E\left( n\right) $ be the Embry truncated moment matrices corresponding from $\gamma ^{\left( n\right) }$. For an odd number $n$, it is known that $ \gamma ^{\left( n\right) }$ has a rank $E\left( n\right) $\textit{-}atomic representing measure if and only if $E(n)\geq 0$ and $E(n)$ admits a flat extension $E(n+1)$. In this paper we suggest a related problem: if $E(n)$ is positive and nonsingular, does $E(n)$ have a flat extension $E(n+1)$? and give a negative answer in the case of $E(3)$. And we obtain some necessary conditions for positive and nonsingular matrix $E\left( 3\right)$, and also its sufficient conditions.