| In this thesis, the characterization about the image space of wavelet transform is investigated by making use of the relationship between the image space of wavelet transform and the reproducing kernel Hilbert space. Firstly, the general characterization about the image space of wavelet transform is discussed from the conclusion that the image space of wavelet transform is a reproducing kernel Hilbert space. Secondly, for two classic wavelets transforms, DOG(Difference of Gaussian)wavelet transform and the modified Morlet wavelet transform, the characterizations of their image spaces are given concretely. 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 A is fixed. When fixing the scale factor A , concretely, for DOG wavelet transform the reproducing kernel of its image space is the difference between the sum of the first two reproducing kernels of two known reproducing kernel spaces and the third reproducing kernel of a known reproducing kernel space; and for the modified Morlet wavelet transform the reproducing kernel of its image space is a wavelet from variable replacement. According to the fine structures of reproducing kernels, the characterizations about the image spaces of the two wavelets transforms and the isometric identities formulas are given concretely by the perfect theory of reproducing kernel space. That provides not only the theoretic bases for discussing the image spaces of other wavelets transforms, but also a new method to investigate the wavelet analysis theory further.Moreover,the reproducing kernel Krein spaces are introduced in this thesis. And two different Krein spaces with the same reproducing kernel are constructed.
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