A Double Exponential Formula for the Fourier Transforms

Abstract

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.

A fast and accurate numerical method for symmetric Lévy processes based on the Fourier transform and sinc-Gauss sampling formula

“…Step 1 Computation of the Fourier transform (Ooura, 2005) with sampling points y = y j ( j = −M − , . .…”

“…Therefore, the discretization of (3.6) by the mid-point rule can yield accurate approximation (3.3) for some h independent of ζ , and sufficiently large M + and M − . In Ooura (2005), the error of approximation (3.3) is bounded by c…”

“…We begin with the review of the DE formula for the Fourier transforms (2.9) proposed by Ooura (2005). Let ζ 0 and h be positive constants and let the function ϕ : R → (0, ∞) be defined by…”

“…Therefore, we can compute approximate solutions of p(x,t) for any t with the same computational cost, and errors of numerical time integration do not occur. The second objective of the proposed method is to present new applications of Ooura's methods (Ooura, 2001(Ooura, , 2005 for the Fourier transforms to obtain the solution of PIDEs. The high accuracy of the proposed method is due to the very fast convergence of Ooura's methods, and speeding up of computations by Ooura's methods is realized by combining them with the nonuniform FFT (Dutt & Rokhlin, 1993, 1995Greengard & Lee, 2004;Potts et al, 2001;Steidl, 1998) or the fractional FFT (Bailey & Swarztrauber, 1991;Chourdakis, 2005;Tanaka, 2014a).…”

“…To propose the method, we use Ooura's accurate numerical formulas for the Fourier transform (Ooura, 2001(Ooura, , 2005, and propose a numerical indefinite integration formula based on the sinc-Gauss sampling formula (Tanaka et al, 2008) to compute the integrals with respect to the Lévy measures in the equation. Furthermore, we combine the fast Fourier transform (FFT) with these formulas to speed up the numerical computations.…”