A Double Exponential Formula for the Fourier Transforms

In this paper, we propose a new and efficient method that is applicable for the computation of the Fourier transform of a function which may possess a singular point or slowly converge at infinity. The proposed method is based on a generalization of the method of the double exponential (DE) formula; the DE formula is a powerful numerical quadrature proposed by H. Takahasi and M. Mori in 1974 [1]. Although it is a widely applicable formula, it is not effective in computing the Fourier transform of a slowly decreasing function. Actually it is not very efficient even if one wants to compute the value of a Fourier transform at a particular point, i.e., a Fourier-type integral. To conquer this weakness at least for a Fourier-type integral, M. Mori and the author proposed a new DE formula in 1991 [2]. See also [3] for a further improvement. The method proposed there is effective for Fourier-type integrals, but it is still weak in the computation of the Fourier transform. Here we propose another DE formula which is applicable to the computation of the Fourier transform. One point in the new method proposed here is that it makes use of fixed sampling points even if we change the point where the Fourier integral is evaluated. In this paper we propose the new method and illustrate the efficiency of the new method in several concrete examples through the comparison with the older methods.