Using NFFT 3---A Software Library for Various Nonequispaced Fast Fourier Transforms
ACM Transactions on Mathematical Software (TOMS)
The uselessness of the Fast Gauss Transform for summing Gaussian radial basis function series
Journal of Computational Physics
Fast Approximation of the Discrete Gauss Transform in Higher Dimensions
Journal of Scientific Computing
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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.