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Fast convolution with radial kernels at nonequispaced knots

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Summary.

We develop a new algorithm for the fast evaluation of linear combinations of radial functions based on the recently developed fast Fourier transform at nonequispaced knots. For smooth kernels, e.g. the Gaussian, our algorithm requires arithmetic operations. In case of singular kernels an additional regularization procedure must be incorporated and the algorithm has the arithmetic complexity if either the points y j or the points x k are “reasonably uniformly distributed”. We prove error estimates to obtain clues about the choice of the involved parameters and present numerical examples for various singular and smooth kernels in two dimensions.

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Authors and Affiliations

  1. University of Lübeck, Institute of Mathematics, 23560, Lübeck, Germany

    Daniel Potts

  2. University of Mannheim, Faculty of Mathematics and Computer Science, 68131, Mannheim, Germany

    Gabriele Steidl & Arthur Nieslony

Authors
  1. Daniel Potts
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  2. Gabriele Steidl
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  3. Arthur Nieslony
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Corresponding author

Correspondence to Gabriele Steidl.

Additional information

Mathematics Subject Classification (2000): 65T40, 65T50, 65F30

Revised version received December 3, 2003

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Potts, D., Steidl, G. & Nieslony, A. Fast convolution with radial kernels at nonequispaced knots. Numer. Math. 98, 329–351 (2004). https://doi.org/10.1007/s00211-004-0538-5

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  • DOI: https://doi.org/10.1007/s00211-004-0538-5

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