Rotationally invariant quadratures for the sphere

C. Ahrens, G. Beylkin
2009 Proceedings of the Royal Society A  
We construct near-optimal quadratures for the sphere that are invariant under the icosahedral rotation group. These quadratures integrate all (N + 1) 2 linearly independent functions in a rotationally 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
more » ... 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 centred and mostly concentrated at nodes of the quadrature, 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(N 3 ) for using these grids and representations. Finally, we note that our approach is also applicable to other discrete rotation groups.
doi:10.1098/rspa.2009.0104 fatcat:f2asuzf6mfcslfpdomnqr5lefm