Truncated K-moment problems in several variables

Let $\beta\equiv\beta^{(2n)}$ be an N-dimensional real multi-sequence of degree 2n, with associated moment matrix $\mathcal{M}(n)\equiv \mathcal{M}(n)(\beta)$, and let $r:=rank \mathcal{M}(n)$. We prove that if $\mathcal{M}(n)$ is positive semidefinite and admits a rank-preserving moment matrix extension $\mathcal{M}(n+1)$, then $\mathcal{M}(n+1)$ has a unique representing measure \mu, which is r-atomic, with supp \mu$equal to$\mathcal{V}(\mathcal{M}(n+1))$, the algebraic variety of$\mathcal{M}(n+1)$. Further, \beta has an r-atomic (minimal) representing measure supported in a semi-algebraic set$K_{\mathcal{Q}}$subordinate to a family$\mathcal{Q}% \equiv\{q_{i}\}_{i=1}^{m}\subseteq\mathbb{R}[t_{1},...,t_{N}]$if and only if$\mathcal{M}(n)$is positive semidefinite and admits a rank-preserving extension$\mathcal{M}(n+1)$for which the associated localizing matrices$\mathcal{M}_{q_{i}}(n+[\frac{1+\deg q_{i}}{2}])$are positive semidefinite$(1\leq i\leq m)$; in this case, \mu (as above) satisfies supp \mu\subseteq K_{\mathcal{Q}}$, and \mu has precisely rank \mathcal{M}(n)-rank \mathcal{M}_{q_{i}}(n+[\frac{1+\deg q_{i}}{2}])$atoms in$\mathcal{Z}(q_{i})\equiv {t\in\mathbb{R}^{N}:q_{i}(t)=0}$,$1\leq i\leq m$.Comment: 33 pages; to appear in J. Operator Theor

Flat extensions of positive moment matrices: recursively generated relations

We develop new computational tests for existence and uniqueness of representing measures in the Truncated Complex Moment Problem: γij = Z zizj d (0 i + j 2n):(TCMP) We characterize the existence of nitely atomic representing measures in terms of positivity and extension properties of the moment matrix M(n)(γ) associated with γ γ(2n): γ00; : : : ; γ0;2n; : : : ; γ2n;0, γ00> 0 (Theorem 1.5). We study conditions for flat (i.e., rank-preserving) extensions M(n + 1) of M(n) 0; each such extension corresponds to a distinct rank M(n)-atomic representing measure, and each such measure is minimal among representing measures in terms of the cardinality of its support. For a natural class of moment matrices satisfying the tests of recursive generation, recursive consis-tency, and normal consistency, we reduce the existence problem for minimal representing measures to the solubility of small systems of multivariable alge