Cubic column relations in truncated moment problems

Abstract

For the truncated moment problem associated to a complex sequence ${\gamma }^{(2n)}=\{{\gamma }_{ij}{\}}_{i,j\in Z_{+},i+j\le 2n}$ to have a representing measure $\mu$, it is necessary for the moment matrix $M(n)$ to be positive semidefinite, and for the algebraic variety ${\mathcal{V}}_{\gamma }$ to satisfy $\mathrm{rank}M(n)\le$ card${\mathcal{V}}_{\gamma }$ as well as a consistency condition: the Riesz functional vanishes on every polynomial of degree at most $2n$ that vanishes on ${\mathcal{V}}_{\gamma }$. In previous work with L. Fialkow and M. Möller, the first-named author proved that for the extremal case (rank$M(n)=$ card${\mathcal{V}}_{\gamma }$), positivity and consistency are sufficient for the existence of a representing measure. In this paper we solve the truncated moment problem for cubic column relations in $M(3)$ of the form $Z^3=itZ+u\bar{Z}$ ($u,t\in \mathbb{R}$); we do this by checking consistency. For $(u,t)$ in the open cone determined by $0<\left|u\right|<t<2\left|u\right|$, we first prove that the algebraic variety has exactly $7$ points and $\mathrm{rank}M(3)=7$; we then apply the above mentioned result to obtain a concrete, computable, necessary and sufficient condition for the existence of a representing measure.

18 pages, 2 figures

References in corpus (1)

• Truncated K-moment problems in several variables