Abstract
Radial functions are a powerful tool in many areas of multi-dimensional approximation, especially when dealing with scattered data. We present a fast approximate algorithm for the evaluation of linear combinations of radial functions on the sphere 
. The approach is based on a particular rank approximation of the corresponding Gram matrix and fast algorithms for spherical Fourier transforms. The proposed method takes 
(L) arithmetic operations for L arbitrarily distributed nodes on the sphere. In contrast to other methods, we do not require the nodes to be sorted or pre-processed in any way, thus the pre-computation effort only depends on the particular radial function and the desired accuracy. We establish explicit error bounds for a range of radial functions and provide numerical examples covering approximation quality, speed measurements, and a comparison of our particular matrix approximation with a truncated singular value decomposition.
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Keiner, J., Kunis, S. & Potts, D. Fast Summation of Radial Functions on the Sphere. Computing 78, 1–15 (2006). https://doi.org/10.1007/s00607-006-0169-z
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DOI: https://doi.org/10.1007/s00607-006-0169-z