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本文构造了一个新的分布空间■′_H,并且把经典的Hilbert变换的定义推广到这个空间上.证明了推广的Hilbert变换是把空间■′H映射到自身的同胚映射,且■′_H是满足该条件的■′的最大子空间.进一步,证明了L~P空间中经典的Hilbert变换和周期函数空间中周期Hilbert变换都是这一推广Hilbert变换的特款.对空间■_H的性质进行了刻画,并给出了空间■′_H中的一类非常有用的非线性相位信号.最后给出了该推广Hilbert变换的两个简单的应用.
In this paper, a new distribution space is constructed, and the definition of classical Hilbert transform is generalized to this space. It is proved that the generalized Hilbert transform maps the space H ’to its homeomorphism, Is the largest subspace that satisfies this condition.Furthermore, it is proved that both the classical Hilbert transform in L-P space and the periodic Hilbert transform in periodic function space are the specializations of this extended Hilbert transform. And gives a very useful class of nonlinear phase signals in space ’H. Finally, two simple applications of this generalized Hilbert transform are given.