David Kimsey: Multidimensional moment problems, the subnormal completion problem in several variables and cubature formulas

Abstract. Given a positive integer \(t\), a set \(K \subseteq \mathbb{R}^d\) and a real multisequence \(s = \{ s_{\gamma_1, \ldots, \gamma_d} \}_{0 \leq \gamma_1+\ldots+\gamma_d \leq m}\) we will formulate new moment matrix conditions for \(s\) to have a \(K\)-representing measure \(\sigma= \sum_{q=1}^t \varrho_q \delta_{w_q}\) with \(t\) atoms, i.e.,
\(\quad\displaystyle s_{\gamma_1, \ldots, \gamma_d} = \int_{\mathbb{R}^d} x_1^{\gamma_1} \cdots x_d^{\gamma_d} d\sigma(x_1, \ldots, x_d) \quad {\rm for} \quad 0 \leq \gamma_1+\ldots + \gamma_d \leq m\)
and
\(\quad w_1, \ldots, w_t \in K.\)
Using these conditions, we will establish new minimal inside cubature rules for planar measures in \(\mathbb{R}^2\) and also pose a solution to the subnormal completion problem in \(d\) variables, i.e., given a collection of positive numbers \(\mathcal{C} = \{ \alpha_{\gamma}^{(1)}, \ldots, \alpha_{\gamma}^{(d)}) \}_{0 \leq |\gamma| \leq m}\) we wish to determine whether or not \(\mathcal{C}\) gives rise to a \(d\)-variable subnormal weighted shift operator whose initial weights are given by \(\mathcal{C}\).

We will also study the following moment problem in a countably infinite number of real variables. Given a real-valued sequence \(s = \{ s_{n_1, n_2, \ldots} \},\) and a closed set \(K \subseteq \mathbb{R}^{\mathbb{N}}\) (with the Tychonoff topology), where all but finitely many \(n_j\) are positive integers, we wish to a find a measure \(\sigma\) on \(\mathbb{R}^{\mathbb{N}}\) so that
\(\quad\displaystyle s_{n_1, n_2, \ldots} = \int_{\mathbb{R}^{\mathbb{N}}} \prod_{j=1}^{\infty} x_j^{n_j} \,d\sigma(x_1, x_2, \ldots)\)
and
\(\quad {\rm supp}\, \sigma \subseteq K.\)
We will see that a natural analogue of Haviland's theorem holds in this setting.

This talk is partially based on joint work with Daniel Alpay and Palle Jorgensen.