The Multivariable moment problems and recursive relations

TL;DR

This paper explores multivariable moment problems, demonstrating that truncated sequences are subsequences of recursively generated sequences, providing alternative proofs of existing results, and establishing new findings in the characterization of representing measures.

Contribution

It introduces an approach linking truncated moment sequences to recursively generated sequences, offering new proofs and results in multivariable moment problem theory.

Findings

Every truncated moment sequence is a subsequence of an infinite recursively generated multisequence.

Provides an alternative proof of Curto-Fialkow's results on moment matrices.

Derives new results on the existence and structure of representing measures.

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 $rank M(\beta)$-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 $rank M(n)=r$, 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 $M_{q_j}(n +[\frac{\deg q_j +1}{2}])$ are positive semidefinite; moreover, $\mu$ has precisely $rank M(n) - rank M_{q_j}(n +[\frac{\deg q_j +1}{2}])$ 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.