Characterization Of The Image Space In The Wavelets Transform

In this thesis, the main problem investigated is the characterization of the image space in the wavelets transform by making use of the relationship between wavelets transform and the theory of reproducing kernel. Firstly, the general characterization of the image space in the wavelets transform is discussed and its sampling theorem is given from the conclusion that the image space in the wavelets transform is a reproducing kernel Hilbert space. Secondly, it is given that the characterization of the image spaces in two classic wavelets transforms, Shannon wavelets transform and Marr wavelets transform. 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. When fixing the scale factor, concretely, for Shannon wavelets transform the reproducing kernel of its image space is the difference between two reproducing kernels of known reproducing kernel spaces; and for Marr wavelets transform that is the fourth order partial derivatives of a reproducing kernel of a known reproducing kernel space. According to the fine structures of reproducing kernels, the characterization of the image space in the wavelets transform and the isometric identities formulas are given by the perfect theory of reproducing kernel space. It shows that the theory of reproducing kernel will give a unified understanding of the wavelets transform, multiresolution analysis and sampling theorem. That provides not only the theoretic bases for discussing the image space of wavelets transform, but also a new method to investigate the wavelets analysis theory further.Moreover,the problem of multiresolution analysis in reproducing kernel space is investigated in this thesis. Firstly, a two-dimension tensor product space is constructed, in which reproducing kernel exists. Secondly a multiresolution analysis is constructed in the reproducing kernel space. Then an orthonormal basis in is obtained. Thus,the reproducing kernel space can be expressed by wavelets spaces. Wavelets approximation formula and sampling formula can also be obtained in the space. Besides, the wavelets approximation formula is simple and easy for calculation. That completes the theory of multiresolution analysis in finite interval fully and provides theoretical base and algorithm for application.