Rotationally Invariant Quadratures for the Sphere

Abstract. We construct nearly optimal quadratures for the sphere that are invariant under the icosahedral rotation group. These quadra-tures integrate all (N +1)2 linearly independent functions in a rotation-ally invariant subspace of maximal order and degree N. The nodes of these quadratures are nearly uniformly distributed and the number of nodes is only marginally more than the optimal (N +1)2/3 nodes. Using these quadratures, we discretize the reproducing kernel on a rotationally invariant subspace to construct an analogue of Lagrange interpolation on the sphere. This representation uses function values at the quadrature nodes. In addition, the representation yields an expansion that uses a single function centered and mostly concentrated at nodes of the quadra-ture, thus providing a much better localization than spherical harmonic expansions. We show that this representation may be localized even further. We also describe two algorithms of complexity O(N3) for using these grids and representations. Finally, we note that our approach is also applicable to other discrete rotation groups