The Multivariable moment problems and recursive relations

Abstract

Let $\beta \equiv \{{\beta }_{\mathbf{i}}{\}}_{\mathbf{i}\in {\mathbb{Z}}_{+}^d}$ be a $d$-dimensional multisequence. Curto and Fialkow, have shown that if the infinite moment matrix $M(\beta )$ is finite-rank positive semidefinite, then $\beta$ has a unique representing measure, which is $rankM(\beta )$-atomic. Further, let ${\beta }^{(2n)}\equiv \{{\beta }_{\mathbf{i}}{\}}_{\mathbf{i}\in {\mathbb{Z}}_{+}^d,\mid \mathbf{i}\mid \le 2n}$ be a given truncated multisequence, with associated moment matrix $M(n)$ and $rankM(n)=r$, then ${\beta }^{(2n)}$ has an $r$-atomic representing measure $\mu$ supported in the semi-algebraic set $K=\{(t_1,\dots ,t_d)\in {\mathbb{R}}^d:q_j(t_1,\dots ,t_d)\ge 0,1\le j\le m\}$, where $q_j\in \mathbb{R}[t_1,\dots ,t_d]$, if $M(n)$ admits a positive rank-preserving extension $M(n+1)$ and the localizing matrices $M_{q_j}(n+[\frac{°q_j+1}{2}])$ are positive semidefinite; moreover, $\mu$ has precisely $rankM(n)-rank{M}_{q_j}(n+[\frac{°q_j+1}{2}])$ atoms in $\mathcal{Z}(q_j)\equiv \{t\in {\mathbb{R}}^d:q_j(t)=0\}$. In this paper, we show that every truncated moment sequence ${\beta }^{(2n)}$ is a subsequence of an infinite recursively generated multisequence, we investigate such sequences to give an alternative proof of Curto-Fialkow's results and also to obtain a new interesting results.

13 pages

References in corpus (1)

• Truncated K-moment problems in several variables