Abstract:
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 degree $m$. The Truncated Moment Problem for $\beta$ (TMP) concerns the existence of a positive Borel measure $\mu$, supported in $\mathbb{R}^{n}$, such that $\beta_{i} = \int_{\mathbb{R}^{n}} x^{i}d\mu ~~~~( i\in \mathbb{Z}_{+}^{n},~~|i|\le m).$ (Here, for $x\equiv (x_{1},\ldots,x_{n})\in \mathbb{R}^{n}$  and $i\equiv (i_{1},\ldots,i_{n})\in \mathbb{Z}_{+}^{n}$, we set $|i| = i_{1}+\cdots + i_{n}$ and $x^{i} = x_{1}^{i_{1}}\cdots x_{n}^{i_{n}}$.) A measure $\mu$ as above is a representing measure for $\beta$. We discuss a solution of TMP based on methods of convex analysis. In work with G. Blekherman [J. Operator Theory, to appear] we use an iterative procedure to associate to $\beta^{m}$ an algebraic subset of $\mathbb{R}^{n}$ called the core variety of $\beta$, which evidently contains the support of every representing measure. We discuss the main result, which shows that  $\beta$ has a representing measure if and only if the core variety is nonempty, in which case the core variety is the union of supports of all finitely atomic representing measures. The result can also be adapted to representing measures for a linear functional defined on a finite dimensional subspace of Borel measurable functions on a $T_{1}$ topological space. Applications include a version for the Full Moment Problem and a new proof of the Richter-Tchakaloff Theorem.