How well does the finite Fourier transform approximate the Fourier transform?
How well does the finite Fourier transform approximate the Fourier transform?
复制标题
DOI：
10.1002/cpa.20064
复制
发表时间：
2005-10-01
影响因子：
3
通讯作者：
Epstein, CL
中科院分区：
文献类型：
--
作者：
Epstein, CL
我们证明了标题中问题的答案是“确实很好。“特别是，我们证明，在整个最大可能范围内，有限傅立叶系数提供了一个很好的近似分段连续函数的傅立叶系数。对于连续的周期函数，误差的大小是根据函数的连续模来估计的。随着函数变得更平滑，估计值得到了显著改善。我们还表明，有限傅里叶变换的部分和基本上与普通傅里叶级数的部分和一样好地逼近函数及其导数。沿着的方式，我们建立类似的Riemann-Lebesgue引理和本地化原则。(c)2004 Wiley Periodicals，Inc.
We show that the answer to the question in the title is "very well indeed." In particular, we prove that, throughout the maximum possible range, the finite Fourier coefficients provide a good approximation to the Fourier coefficients of a piecewise continuous function. For a continuous periodic function, the size of the error is estimated in terms of the modulus of continuity of the function. The estimates improve commensurately as the functions become smoother. We also show that the partial sums of the finite Fourier transform provide essentially as good an approximation to the function and its derivatives as the partial sums of the ordinary Fourier series. Along the way we establish analogues of the Riemann-Lebesgue lemma and the localization principle. (c) 2004 Wiley Periodicals, Inc.