Abstract

Abstract We develop a new algorithm for the fast evaluation of linear combinations of radial func-tions f(yj): = PNk=1 ffkK(||yj-xk||) (j = 1,..., M) for xk, yj 2 R2 based on the recentlydeveloped fast Fourier transform at nonequispaced knots. For smooth kernels, e.g. the Gaussian, our algorithm requires O(N + M) arithmetic operations. In case of singularkernels an additional regularization procedure must be incorporated and the algorithm has the arithmetic complexity O(N log pN + M) or O(M log pM + N) if either the points yj or the points xk are "reasonably uniformly distributed". We prove error estimates toobtain clues about the choice of the involved parameters and present numerical examples for various singular and smooth kernels in two dimensions. 2000 AMS Subject Classification: 65T40, 65T50, 65F30 Keywords: fast discrete summation, radial basis functions, fast Fourier transform at nonequispaced knots, fast Gauss transform, fast multipole method 1 Introduction In this paper, we develop a new algorithm for the fast computation of sums of the form

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