A unified approach to scattered data approximation on $\mathbb{S}^{\bf 3}$ and SO(3)

Abstract

In this paper we use the connection between the rotation group SO(3) and the three-dimensional Euclidean sphere $\mathbb{S}^{3}$ in order to carry over results on the sphere $\mathbb{S}^{3}$ directly to the rotation group SO(3) and vice versa. More precisely, these results connect properties of sampling sets and quadrature formulae on SO(3) and $\mathbb{S}^{3}$ respectively. Furthermore we relate Marcinkiewicz---Zygmund inequalities and conditions for the existence of positive quadrature formulae on the rotation group SO(3) to those on the sphere $\mathbb{S}^{3}$ , respectively.