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Truncated \(K\)-moment problems in several variables. (English) Zbl 1119.47304
Summary: 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 \(r:\equiv\;\text{rank}\;\mathcal M(n)\). 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 \(\mathcal Q\equiv\{q_i\}^m_{i=1}\subseteq\mathbb R[t_1,\dots,t_N]\) 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 \(\mathcal M(n)-\;\text{rank}\;\mathcal M_{q_i}(n+[\frac{1+\deg q_i}{2}])\) atoms in \(\mathcal Z(q_i)\equiv\{t\in\mathbb R^N:q_i(t)=0\},1\leq i\leq m.\)

47A57 Linear operator methods in interpolation, moment and extension problems
44A60 Moment problems
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
15B57 Hermitian, skew-Hermitian, and related matrices
15-04 Software, source code, etc. for problems pertaining to linear algebra
47N40 Applications of operator theory in numerical analysis
47A20 Dilations, extensions, compressions of linear operators