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

DOI:
https://doi.org/10.1090/S0025-5718-1982-0669645-5

Corrigendum:
Math. Comp. **43** (1984), 347.

Corrigendum:
Math. Comp. **43** (1984), 347.

MathSciNet review:
669645

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Abstract | References | Similar Articles | Additional Information

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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Additional Information

Keywords:
Fourier integral,
<I>B</I>-splines,
fast Fourier transform,
continuous Fourier transform

Article copyright:
© Copyright 1982
American Mathematical Society