The Fourier approximation of smooth but nonperiod

Authors: M Lyon J Picard

Publish Date: 2014/03/18

Volume: 40, Issue: 5-6, Pages: 1073-1092

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Abstract

We develop an algorithm to extend to the nonequispaced case a recentlyintroduced fast algorithm for constructing spectrallyaccurate Fourier approximations of smooth but nonperiodic data Fast Fourier continuation algorithms which allow for the Fourier approximation to be periodic in an extended domain are combined with the underlying ideas behind nonequispaced fast Fourier transform NFFT algorithms The result is a method which allows for the fast and accurate approximation of unevenly sampled nonperiodic multivariate data by Fourier series A particular contribution of the proposed method is that its formulation avoids the difficulties related to the conditioning of the linear systems that must be solved in order to construct a Fourier continuation The efficiency essentially equivalent to that of an NFFT and accuracy of the algorithm is shown through a number of numerical examples Numerical results demonstrate the spectral rate of convergence of this method for sufficiently smooth functions The accuracy for sufficiently large data sets is shown to be improved by several orders of magnitudes over previously published techniques for scattered data interpolation

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