David P. Kimsey : A moment matrix method for computing cubature rules of odd degree

Let $\sigma$ be a positive Borel measure on $\mathbb{R}^d$ having convergent power moments up to at least degree $m$, i.e.
$$s_{j_1, \ldots, j_d} = \int \cdots \int_{\mathbb{R}^d} x_1^{j_1} \cdots x_d^{j_d} d\sigma(x_1, \ldots, x_d) \quad {\rm for}\quad{\rm all} \quad 0 \leq |j| \leq m,$$
where $|j| = j_1+ \ldots + j_d$, converge absolutely. A cubature rule for $\sigma$ of degree $m$ and size $t$ consists of distinct points $w_1, \ldots, w_t \in \RR^d$ and positive real numbers $\varrho_1, \ldots, \varrho_t$ such that
$$s_{j} = \sum_{q=1}^t \varrho_q w_q^j \quad\quad {\rm for}\quad {\rm all} \quad 0 \leq |j| \leq m$$
where $w_q^j = (w_1^{(q)})^{j_1} \cdots (w_d^{(q)})^{j_d}$ for $w_q = (w_1^{(q)}, \ldots, w_d^{(q)})$ and $j = (j_1, \ldots, j_d)$.
In this talk, we will outline a method for constructing cubature rules when $m$ is odd and $t$ is prescribed. The method is based on constructing matrices built from the finite multisequence $\{ s_{j} \}_{0 \leq |j| \leq m}$ and checking commutativity conditions.