Evaluation of Fourier Integrals Using ß-Splines

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 ß-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.