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 $y$, we show that $y$ 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 $L_y$ is $K$-positive. For a determining set $K$, we prove that if $L_y$ is strictly $K$-positive, then $y$ admits a representing measure supported in $K$. As a consequence, we are able to solve the truncated $K$-moment problem of degree $k$ in the cases: (i) $(n,k)=(2,4)$ and K=\re^2; (ii) $n\geq 1$, $k=2$, and $K$ 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 $\beta\equiv\beta^{(2n)}$ 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

Positivity and representing measures in the truncated moment problem

Let $K$ denote a nonempty closed subset of $\mathbb{R}^{n}$ and let $\beta\equiv \beta^{(m)} = \{\beta_{i}\}_{i\in \mathbb{Z}_{+}^{n}, |i|\le m}$, $\beta_{0}\u3e0$, denote a real $n$-dimensional multisequence of finite degree $m$. \textit{The Truncated $K$-Moment Problem (TKMP)} concerns the existence of a positive Borel measure $\mu$, supported in $K$, such that $$\beta_{i} = \int_{\mathbb{R}^{n}} x^{i}d\mu ~~~~~~~~( i\in \mathbb{Z}_{+}^{n},~~|i|\le m).$$ We describe a number of interrelated techniques for establishing the existence of such \textit{$K$-representing measures}. We discuss $K$-representing measures arising from \textit{$K$-positivity} or \textit{strict K-positivity} of the Riesz functional $L_{\beta}$ associated with $\beta$; 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 $K$ denote a nonempty closed subset of $\mathbb{R}^{n}$, let $m=2d$, and let $\beta\equiv \beta^{(m)} = \{\beta_{i}\}_{i\in \mathbb{Z}_{+}^{n}, |i|\le m}$, $\beta_{0}\u3e0$, denote a real $n$-dimensional multisequence of finite degree $m$. %and let $K$ denote a closed subset of $\mathbb{R}^{n}$. \textit{ The Truncated $K$-Moment Problem} concerns the existence of a positive Borel measure $\mu$, supported in $K$, such that $$\beta_{i} = \int_{\mathbb{R}^{n}} x^{i}d\mu ~~~~~~~~( i\in \mathbb{Z}_{+}^{n},~~|i|\le m).$$ The \textit{core variety} of $\beta$, $\mathcal{V} \equiv \mathcal{V}(\beta)$, is an algebraic variety in $\mathbb{R}^{n}$ that contains the support of any such \textit{$K$-representing measure}. In previous work we showed, conversely, that if $\mathcal{V}$ is a nonempty compact set, or $\mathcal{V}$ is nonempty and is a determining set for polynomials of degree at most $m$ (in particular, if $\mathcal{V}= \mathbb{R}^{n}$), then $\beta$ admits a $\mathcal{V}$-representing measure. We describe some additional cases where a nonempty core variety implies the existence of a representing measure