Sampling in Grassmannians

We shall provide an overview of recent results on cubature points in Grassmannians. First, we consider the covering radius, which measures how well a finite point set covers the underlying space and observe that low-cardinality cubature points cover the Grassmannian asymptotically optimal. Therefore, cubature points are well-suited for several approximation tasks on the Grassmannian. Then we outline results on the approximation of integrals and functions on the Grassmannian via cubature point samples. Last, we connect frames for polynomial spaces with the concept of cubatures enabling a direct construction of cubature points.