Large Structures Seminar: Cordian Riener

Let $d$ be a positive integer and let $\mu$ be a positive Borel measure on $\mathbf{R}^2$ possessing moments up to degree $2d-1$. Every such measure defines a linear from $L\in\mathbf{R}[X,Y]_{2d-1}^*$, which can be decomposed into a positive combination of point evaluations, i.e.,

$$L(f)=\sum_{v\in S\subset\mathbf{R}^2} \lambda_v f(v),$$

where $S$ is a finte set of nodes and $\lambda_v\geq 0$ are the corresponding weights. Such a decomposition is called a cubature rule for the given measure. We study the question of bounding the minimal number of nodes necessary and give generalizations of the well known Gauß-quadrature on the real line to the two-dimensional case.

In particular, we show that for every $\mu$ there is always a cubature rule with at most $\frac{3}{2}d(d-1) + 1$ many nodes, which is chosen such that its value on a fixed positive definite form of degree $2d$ is minimized.

(joint work with Markus Schweighofer)