Numerical calculation of Fourier transforms based on hyperfunction theory

In this paper, we propose a numerical method for calculating Fourier transforms based on hyperfunction theory, a theory of generalized functions based on complex function theory. In the proposed method, we first obtain analytic functions giving the desired Fourier transform as a hyperfunction, and, then, we compute the Fourier transform by using continued fractions to determine the analytic continuation of these analytic functions onto the real axis. Using the proposed method, we can evaluate Fourier transforms that are difficult to evaluate using the conventional numerical integration rules. In addition, we can use the proposed method to evaluate Fourier transforms of hyperfunctions. Numerical examples show the efficiency of the presented method compared to previous methods. (C) 2020 The Author(s). Published by Elsevier B.V.