The truncated moment problem via homogenization and flat extensions (Q452366)

The paper studies the truncated moment problem in \(n\) variables, namely, to find a positive measure on \(\mathbb{R}^n\) such that matches the prescribed moments \(y_\alpha\) in NEWLINE\[NEWLINE\int x^\alpha d\mu=\int x_1^{\alpha_1}\dots x_n^{\alpha_n}\,d\mu=y_\alpha,\quad \alpha\in\mathbb{N}^n,\quad |\alpha|=\alpha_1+\dots+\alpha_n\leq d.NEWLINE\]NEWLINE The problem is reformulated in homogeneous form, which means to rewrite \(x^\alpha=x_1^{\alpha_1}\dots x_n^{\alpha_n}\) as \(x_0^{d-|\alpha|}x^\alpha\). The original problem admits a solution supported in \(\mathbb{R}^n\) if and only if the homogeneous problem admits a solution supported on the unit sphere of \(\mathbb{R}^{n+1}\). Furthermore, by associating with this truncated moment problem also a moment matrix and a positive definite linear functional (the Riesz functional) acting on the space of polynomials, it is shown that the existence of a solution and, in fact, a solution itself (if it exists) can be obtained by solving a sequence of semidefinite programming problems (a sequence depending on the degree of the polynomials involved). This increase of degree corresponds to extensions of the moment problem. If a flat extension is reached, i.e., if it does not increase the rank of the moment matrix, a solution is found. Some examples illustrate the ideas.