Truncated K-moment problems in several variables
By R. Curto, L. Fialkow. 2005
180 citations recorded by the source corpus.
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 $\mathcal{Z}(q_{i})\equiv {t\in\mathbb{R}^{N}:q_{i}(t)=0}$, $1\leq i\leq m$.
Subfields: Computational Theory and Mathematics, Numerical Analysis
View Paper: https://www.semanticscholar.org/paper/1f1cba1620a2ca32de6797d8ce4affee6cef1ace
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Citations in the corpus
- Positivity of Riesz Functionals and Solutions of Quadratic and Quartic Moment Problems (2009, background): “A fundamental result of [4, 8] shows that y admits a rank Md(y)-atomic representing measure if and only Md(y) is positive semi-definite and Md(y) admits a flat (i.”
- Moment/Sum-of-Squares Hierarchy for Complex Polynomial Optimization (2015, background): “1) different from existing literature [21]; the Lasserre hierarchy [41] can thus be viewed as a special case of the proposed complex hierarchy (Figure 2); 2.”
- An analogue of the Riesz–Haviland theorem for the truncated moment problem (2008, background): “(The result in [11] is given in terms of the truncated complex moment problem for measures on C, but the complete equivalence of this problem to the real truncated moment problem for measures on R2 is established in [15].”