Gruppentheoretische Aspekte der Quadratur und Kubatur

Abstract

The notion of t-design is well established in combinatorics. Replacing the “discrete” sphere by the compact Euclidean unit sphere S_n-1 of IRⁿ(n≧2) and the action of the symmetric group by the action of the special orthogonal group SO(n,IR), the notion of spherical t-design in IRⁿ emerges. Since spherical t-designs allow to measure certain regularity properties of finite subsets X of S_n-1, this notion has computational besides theoretical significance. In particular, spherical t-designs are useful for the explicit construction of cubature formulae for surface integrals over S_n-1 by averaging over X (Section 1). The purpose of the present note is to deal mainly with the one-dimensional case (n=1). It will be shown that in this case the action of the real Heisenberg group (Section 2) gives rise to a trapezoidal rule for improper integrals. The error of this quadrature formula will be represented by a complex contour integral with noncompact integration path (Section 3). Finally, a guide to various different applications of our geometric and analytic methods is provided (Section 4).