Algorithm 418: calculation of Fourier integrals [D1]

Abstract

The most commonly used formula for calculating Fourier integrals is Filon's formula, which is based on the approximation of the function by a quadratic in each double interval. In order to obtain a better approximation the cubic spline fit is used in [1]. The obtained formulas do not need the explicit calculation of the spline fit, but in addition to the function values at all intermediate points, the values of the first and second derivatives at the boundary points are required. However, these values are often obtained from symmetry conditions. If the derivatives at the end-points are unknown, they may be calculated from a cubic spline fit, for example by using some exterior points or by using two extra interior conditions for the spline fit. It can also be noted that in certain periodic cases the terms containing the derivatives will cancel, and their values will be superfluous. The use of Algorithm 353 [2] is recommended if the frequency ω / π is a positive integer and the interval is [0,1]. Test computations reported in [1] indicate that the spline formula is more accurate than Filon's formula. Both are of the fourth order. The expansion of the error term in powers of the step length contains only even powers, and therefore the use of Richardson extrapolation is very efficient.