Non-Polynomial Realization Of Sigmoid Function In Meyer Wavelet Construction

Wavelet analysis is a new mathematical theory and method of development in the mid-1980 s,it is considered a breakthrough development of Fourier analysis method and it is characterized by deep theoretical and extensive application.Wavelet transform is the basis of wavelet analysis.We take different wavelets to do the wavelet transform and its application field is different.The Meyer type wavelet with limited spectrum has certain characteristics of differentiability,good smoothness and fast attenuation,etc.And its properties are based on the sigmoid function selected in the scale function,so it plays an important role in the construction of sigmoid function in Meyer wavelet.Moreover,Meyer wavelet is widely used in signal processing and harmonic detection of power system.Therefore,this paper has studied the spectral finite wavelet and realized the structure of the non-polynomial sigmoid function in Meyer wavelet,and the main contents are as follows:Firstly,we've talked about some of spectrum limited wavelet functions that are represented by the Shannon scale function and their properties,have attenuation in time domain and have tight support in frequency domain,and these are either orthogonal or non-orthogonal wavelets.We can all use the wavelet transform and the reproducing kernel space theory to get the general expression of the reproducing kernel function of wavelet transform image space.When the scale factor is fixed,we can also obtain the analytic expression of the reproducing kernel function of the wavelet transform image space.This provides a theoretical basis for describing the image space of the spectrum limited wavelet transform.In addition,the Shannon wavelet has an ideal frequency domain,but its locality is very poor in the time domain.We can get the Meyer wavelet scale function by smoothing the Shannon scale function in the sharp edge value in the frequency domain,so we study the orthogonal spectrum limited Meyer type wavelet function.Because the properties of sigmoid function directly affect the differentiability,smoothness and attenuation speed of Meyer wavelets in the construction of Meyer type wavelets,the selection of sigmoid functions is very important.On the one hand,by constructing a fully smooth polynomial sigmoid function,we study its properties,and give the corresponding Meyer wavelets with sufficient smoothing,high order vanishing moments and infinitely differentiability.On the other hand,we use the multiresolution analysis method to construct a spectrum limited scale function,and then get a wider range of Meyer wavelet functions,making the common Meyer wavelet in engineering as its special case.It provides a theoretical basis for the application of Meyer wavelet function to signal detection and image processing.Finally,the fully smooth sigmoid function of the non-polynomial type presented in this paper is used as the activation function in BP neural network function approximation,which can achieve better approximation results.