Fourier变换在处理频谱问题中的一个应用

AN APPLICATION OF THE FOURIER TRANSFORM IN DEALING WITH A PROBLEM OF THE FREQUENCY SPECTRUM

本文利用Fourier变换理论处理了如下问题(T): 设函数X(t)∈L′_((-∞,∞))的Fourier变换为现给函数X(t)在[a,b]外为0,试求一函数X_1(t),使得并且X_1(t)的频谱的截止频率为f_c,其中f_c>0为常数。所得到的结论是X_1(t)不存在。进而,我们给出了在X(t)满足一定光滑条件下,求X_1(t)的方法以及它的频谱在|f|>f_c的估计式。

In this paper, we have resolved the following problem(T)by the theory of the Fourier transforms: Let the Fourier transform of a given function X(t) be Seek for a function X_1 (t), such that and the truncated frequency of the frequency spectrum of X_1(t) is f_e, that is, a positive constant. We conclude that the X_1(1) does not exist and we give solution of X_1(t) and an estimation of X_1^(F) for when X_1 (t) satisfies certain Smooth conditions.