Error control of a numerical formula for the Fourier transform by Ooura’s continuous Euler transform and fractional FFT

摘要

In this paper, we consider a method for fast numerical computation of the Fourier transform of a slowly decaying function with given accuracy in a given range of the frequency. Recently, some useful formulas for the Fourier transform have been proposed to resolve the difficulty of the computation caused by the slow decay and the oscillation of the integrand. In particular, Ooura proposed formulas with continuous Euler transformation and showed their effectiveness. It has, however, also been reported that their errors become large outside some ranges of the frequency. Then, for an illustrative representative of the formulas, in order to compute the Fourier transform with given accuracy in a given frequency range, we choose the parameters in the formula based on its error analysis. Furthermore, by combining the formula and fractional FFT, a generalization of the fast Fourier transform (FFT), we execute the computation in the same order of computation time as that of the FFT.