The general perfect translation invariance theorem and its application to an orthogonal complex wavelet basis on the classical hardy space

In this paper, the general perfect translation invariance theorem is proved, which ensures the condition of perfect translation invariance for complex discrete wavelet transforms of an arbitrary complex square integrable function. Next, by using this theorem, an orthogonal complex wavelet basis on the classical Hardy space is defined and its calculation method is designed. Finally, by extending the general perfect translation invariance theorem to the case of using the discrete Fourier transform, the fast calculation algorithm for this wavelet basis is proposed.