与傅里叶变换有关的线性空间和代数

Linear Space and Algebra related to Fourier Transform

与三维欧氏空间一样,信号时域和频域都是希尔伯特空间。采用类比方法,将信号看作矢量,信号基函数看作基矢量,能建立时域空间和频域空间的几何直观性。可通过三维欧氏空间性质来直观理解时域空间和频域空间,而傅里叶系数本质上是信号在其基函数上的正交投影分量。信号的时域空间和频域空间是环同构,同构映射是傅里叶变换。根据希尔伯特空间性质和环同构,还可得出傅里叶变换的一些性质。

3D Euclidean space,signal time domain and frequency domain are Hilbert Space.By the analogy method,the signal is regarded as a vector,and the signal base function is regarded as a base vector,which can establish the geometric intuition of the time domain space and frequency domain space.The time domain space and frequency domain space can be intuitively understood using the properties of 3D Euclidean space,while the Fourier coefficient is essentially the orthogonal projection component of the signal on its base function.The time domain space and frequency domain space of the signal are ring isomorphism and the isomorphism map is Fourier transform.According to the properties of Hilbert space and ring isomorphism,some properties of Fourier transform are obtained.