The Multivariable moment problems and recursive relations
Kaissar Idrissi, El Hassan Zerouali · arXiv (Cornell University) · 2016
Let $β\equiv \{ β_\mathbf{i} \}_{\mathbf{i} \in \mathbb{Z}_+^d}$ be a $d$-dimensional multisequence. Curto and Fialkow, have shown that if the infinite moment matrix $M(β)$ is finite-rank positive semidefinite, then $β$ has a unique representing measure, which is $rank M(β)$-atomic. Further, let $β^{(2n)} \equiv \{ β_\mathbf{i} \}_{\mathbf{i} \in \mathbb{Z}_+^d, \mid \mathbf{i} \mid \leq 2n}$ be a given truncated multisequence, with associated moment matrix $M(n)$ and $rank M(n)=r$, then $β^{(2n)}$ has an $r$-atomic representing measure $μ$ supported in the semi-algebraic set $K=\{ (t_1, \ldots, t_d) \in \mathbb{R}^d : q_j(t_1, \ldots, t_d) \geq 0, 1\leq j\leq m \}$, where $q_j \in \mathbb{R}[t_1, \ldots, t_d]$, 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, $μ$ has precisely $rank M(n) - rank M_{q_j}(n +[\frac{°q_j +1}{2}])$ 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 $β^{(2n)}$ 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.