1,857 research outputs found
Positivity of Riesz Functionals and Solutions of Quadratic and Quartic Moment Problems
We employ positivity of Riesz functionals to establish representing measures
(or approximate representing measures) for truncated multivariate moment
sequences. For a truncated moment sequence , we show that lies in the
closure of truncated moment sequences admitting representing measures supported
in a prescribed closed set K \subseteq \re^n if and only if the associated
Riesz functional is -positive. For a determining set , we prove
that if is strictly -positive, then admits a representing measure
supported in . As a consequence, we are able to solve the truncated
-moment problem of degree in the cases: (i) and K=\re^2;
(ii) , , and is defined by one quadratic equality or
inequality. In particular, these results solve the truncated moment problem in
the remaining open cases of Hilbert's theorem on sums of squares.Comment: 27 page
Truncated K-moment problems in several variables
Let be an N-dimensional real multi-sequence of
degree 2n, with associated moment matrix , and let . We prove that if
is positive semidefinite and admits a rank-preserving moment
matrix extension , then has a unique
representing measure \mu, which is r-atomic, with supp \mu\mathcal{V}(\mathcal{M}(n+1))\mathcal{M}(n+1)K_{\mathcal{Q}}\mathcal{Q}%
\equiv\{q_{i}\}_{i=1}^{m}\subseteq\mathbb{R}[t_{1},...,t_{N}]\mathcal{M}(n)\mathcal{M}(n+1)\mathcal{M}_{q_{i}}(n+[\frac{1+\deg q_{i}}{2}])(1\leq i\leq m), and \mu has precisely rank \mathcal{M}(n)-rank
\mathcal{M}_{q_{i}}(n+[\frac{1+\deg q_{i}}{2}])\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
Positivity and representing measures in the truncated moment problem
Let denote a nonempty closed subset of and let , , denote a real -dimensional multisequence of finite degree . \textit{The Truncated -Moment Problem (TKMP)} concerns the existence of a positive Borel measure , supported in , such that We describe a number of interrelated techniques for establishing the existence of such \textit{-representing measures}. We discuss -representing measures arising from \textit{-positivity} or \textit{strict K-positivity} of the Riesz functional associated with ; representing measures arising from extensions of moment matrices; Tchakaloff\u27s Theorem and its generalizations and applications to TKMP; representing measures arising from a nonempty \textit{core variety}
The core variety of a multisequence in the truncated moment problem
Let denote a nonempty closed subset of , let , and let , , denote a real -dimensional multisequence of finite degree . %and let denote a closed subset of . \textit{ The Truncated -Moment Problem} concerns the existence of a positive Borel measure , supported in , such that The \textit{core variety} of , , is an algebraic variety in that contains the support of any such \textit{-representing measure}. In previous work we showed, conversely, that if is a nonempty compact set, or is nonempty and is a determining set for polynomials of degree at most (in particular, if ), then admits a -representing measure. We describe some additional cases where a nonempty core variety implies the existence of a representing measure
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