TRUNCATED K-MOMENT PROBLEMS IN SEVERAL VARIABLES

Let β ≡ β(2n) be an N-dimensional real multi-sequence of degree 2n, with associated moment matrix M(n) ≡ M(n)(β), and let r := rank M(n). We prove that if M(n) is positive semidefinite and admits a rank-preserving moment matrix extension M(n + 1), then M(n + 1) has a unique representing measure μ, which is r-atomic, with supp μ equal to V(M(n + 1)), the algebraic variety of M(n + 1). Further, β has an r-atomic (minimal) representing measure supported in a semi-algebraic set KQ subordinate to a family $\mathcal{Q}\equiv \{{\mathrm{q}}_{\mathrm{i}}{\}}_{\mathrm{i}=1}^{\mathrm{m}}\subseteq \mathrm{\mathbb{R}}\left[{\mathrm{t}}_{1,\mathrm{\ldots},}{\mathrm{t}}_{\mathrm{N}}\right]$ if and only if M(n) is positive semidefinite and admits a rank-preserving extension M(n + 1) for which the associated localizing matrices ${\mathcal{M}}_{{\mathrm{q}}_{\mathrm{i}}}(\mathrm{n}+\left[\frac{1+\mathrm{deg}{\mathrm{q}}_{\mathrm{i}}}{2}\right])$ are positive semidefinite, 1 ≤ i ≤ m; in this case, μ (as above) satisfies supp μ ⊆ KQ, and μ has precisely rank M(n) – rank ${\mathcal{M}}_{{\mathrm{q}}_{\mathrm{i}}}(\mathrm{n}+\left[\frac{1+\mathrm{deg}{\mathrm{q}}_{\mathrm{i}}}{2}\right])$ atoms in Z(qi) ≡ {t ∈ RN : qi(t) = 0}, 1 ≤ i ≤ m.