Location
Cupples I Room 113
Start Date
7-22-2016 2:30 PM
End Date
22-7-2016 2:50 PM
Description
Let $K$ denote a nonempty closed subset of $\mathbb{R}^{n}$ and let $\beta\equiv \beta^{(m)} = \{\beta_{i}\}_{i\in \mathbb{Z}_{+}^{n}, |i|\le m}$, $\beta_{0}>0$, 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's Theorem and its generalizations and applications to TKMP; representing measures arising from a nonempty \textit{core variety}.
Positivity and representing measures in the truncated moment problem
Cupples I Room 113
Let $K$ denote a nonempty closed subset of $\mathbb{R}^{n}$ and let $\beta\equiv \beta^{(m)} = \{\beta_{i}\}_{i\in \mathbb{Z}_{+}^{n}, |i|\le m}$, $\beta_{0}>0$, 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's Theorem and its generalizations and applications to TKMP; representing measures arising from a nonempty \textit{core variety}.