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Mathematics of Computation

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Evaluation of Fourier integrals using $B$-splines

Authors: M. Lax and G. P. Agrawal
Journal: Math. Comp. 39 (1982), 535-548
MSC: Primary 65D30; Secondary 42A15, 65R10
Corrigendum: Math. Comp. 43 (1984), 347.
Corrigendum: Math. Comp. 43 (1984), 347.
MathSciNet review: 669645
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Abstract: Finite Fourier integrals of functions possessing jumps in value, in the first or in the second derivative, are shown to be evaluated more efficiently, and more accurately, using a continuous Fourier transform (CFT) method than the discrete transform method used by the fast Fourier transform (FFT) algorithm. A B-spline fit is made to the input function, and the Fourier transform of the set of B-splines is performed analytically for a possibly nonuniform mesh. Several applications of the CFT method are made to compare its performance with the FFT method. The use of a 256-point FFT yields errors of order ${10^{ - 2}}$, whereas the same information used by the CFT algorithm yields errors of order ${10^{ - 7}}$—the machine accuracy available in single precision. Comparable accuracy is obtainable from the FFT over the limited original domain if more than 20,000 points are used.

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Keywords: Fourier integral, <I>B</I>-splines, fast Fourier transform, continuous Fourier transform
Article copyright: © Copyright 1982 American Mathematical Society