用Ooura的连续欧拉变换和分数FFT的数值公式误差控制
Error control of a numerical formula for the Fourier transform by Ooura's continuous Euler transform and fractional FFT
AI总结：
本文提出一种快速计算衰减缓慢函数傅里叶变换的方法，通过连续欧拉变换和分数FFT结合，实现给定频率范围内的高精度计算。
摘要：
本文考虑了一种快速计算衰减缓慢函数傅里叶变换的方法，该方法在给定频率范围内具有给定的精度。在这些领域中，提出了一些有用的傅里叶变换公式，以克服由于缓慢衰减和被积函数振荡导致的计算困难。特别是，Ooura提出了带有连续欧拉变换的公式，并展示了其有效性。然而，也报告了这些公式在某些频率范围外的误差会变得很大。因此，为了说明代表性公式，我们根据其误差分析选择参数，以在给定频率范围内计算具有给定精度的傅里叶变换。此外，结合该公式和分数FFT，一种快速傅里叶变换（FFT）的推广方法，可以在与FFT相同的时间顺序内执行计算。
英文摘要：
In this paper, we consider a method for fast numerical computation of the Fourier transform of a slowly decaying function with given accuracy in given ranges of the frequency. In these decades, some useful formulas for the Fourier transform are proposed to recover difficulty of the computation due to the slow decay and the oscillation of the integrand. In particular, Ooura proposed formulas with continuous Euler transformation and showed their effectiveness. It is, however, also reported that errors of them become large outside some ranges of the frequency. Then, for an illustrating representative of the formulas, we choose parameters in the formula based on its error analysis to compute the Fourier transform with given accuracy in given ranges of the frequency. Furthermore, 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 the FFT.
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