The Multivariable moment problems and recursive relations

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 \leq 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, \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 $M_{q_j}(n+[\frac{\mathrm{deg}{q}_j+1}{2}])$ are positive semidefinite; moreover, $\mu$ has precisely $rank M(n) - rank M_{q_j}(n +[\frac{\deg q_j +1}{2}])$atomsin$$\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.