An optimization view on the quadrature problem

Let d and k be positive integers and $\mu$ be a positive Borel measure on ${\mathbb{R}}^2$ possessing finite moments up to degree $2d-1$. Using methods from convex optimization, we study the question of the minimal $m\in \mathbb{N}$ such that there exists a quadrature rule for $\mu$ with m nodes which is exact for all polynomials of degree at most $2d-1$ We show that if the support of $\mu$ is contained in an algebraic curve of degree $k$, then there exists a quadrature rule for $\mu$ with at most $dk$ many nodes all placed on the curve (and positive weights). This generalizes Gauss quadrature where the curve is a line and (the odd case of) Szegö quadrature where the curve is a circle to arbitrary plane algebraic curves. In the even case, i.e., $2d$ instead of $2d-1$ this result generalizes to compact curves. We use this result to show that, any plane measure $\mu$ has a quadrature rule with at most $3/2d(d-1)$ many nodes, which is exact up to degree $2d-1$.

All our results are obtained by minimizing a certain linear functional on the polynomials of degree $2d$ and our proof uses both results from convex optimisation and from real algebraic geometry.

(Joint work with Markus Schweighofer)