The Property Of The Image Space Of Wavelet Transform Based On Gauss Function

The researches of theory of wavelet analysis have acquired plentiful and substantial results after decades of rapid development. And the theory and application of wavelet analysis have been widely applied to many fields and provided a strong mathematical tool for many scientists. Because of the fine structures of reproducing kernels, the theory of reproducing kernel can solve many practice problems. With the rise of wavelet the theory of the reproducing kernel attracts more and more scholars'attention in some fields because reproducing kernel Hilbert space is the basis of continue wavelet transform and it is very important for us to reconstruct the continue wavelet transform.In this thesis, the main problem, which is investigated, is the characterization of the image space in the wavelets transform. It is given that two more types general wavelet function constructed by Gauss function and the characterization of the image spaces. The first type continuous wavelet function is constructed by the linear combination with Gauss and its derivative function, and the most typical example of it is the result of DOG wavelet function. The second type continuous wavelet function is constructed by convolution with the derivative function of Gauss function, and the most typical example of it is the result of Gauss wavelet function. Furthermore, it can be found that the number of partial derivatives of the reproducing kernel function in the image space is relevant to the number of partial derivatives in constructing the wavelet function. For the two classic wavelets transforms, the general characterizations of their image spaces are obtained by means of analytic extension. That is their images are extended analytically onto complex spaces to discuss. Using the structures and properties of the reproducing kernels, the properties of the functions in their image spaces can be studied when the scale factor is fixed. According to the fine structures of reproducing kernels, the characterization of the image space in the wavelets transform is given by the perfect theory of reproducing kernel space. That plays a guiding role and provides a new method for discussing the properties of the image space of more general wavelet transform.