Wavelet Analysis Method And Its Applications In Reproducing Kernel Spaces

Wavelet analysis is a newly arisen subject that develops gradually on the basis of the wavelet transform. It is not only of the profound theoretical significance, but also of the extensive applications. Because wavelet transform is the basis of the wavelet analysis, discussing wavelet transform deeply is of theoretical significance and practical value.In this paper, the multiresolution analysis (MRA) method in reproducing kernel space H~1 [0,1] is acquired and the properties of nesting are verified by establishing the isomorphism mapping between L~2[0,1] and H~1[0,1], at the same time the wavelet approximation formula and the corresponding sampling formula in reproducing kernel space H~1[0,1] is given. Consequently, these provide a new frame for solving practical problems in the reproducing kernel space by the wavelet theory. Moreover, a method which characterizes the image space of wavelet transform is put forward by combining the reproducing kernel theory with wavelet transform, and this method is applied to two typical wavelet transforms. The content and the main results studied in the paper are as follows:Firstly, the analytic expression of the reproducing kernel in H~1[0,1] is given, and the MRA in the reproducing kernel space H~1[0,1] is established by means of the MRA in L~2 [0,1]. The wavelet approximation method and the sampling theorem are given in H~1[0,1].Secondly, the two-dimensional tensor product space H~1(Ω) is constructed, and the space H~1(Ω) with a reproducing kernel is verified; in the reproducing kernel space H~1[0,1] multiresolution analysis is established and criterion orthogonal basis is acquired so that the reproducing kernel space H~1[0,1] can be characterized by wavelet space, thus the wavelet series and the corresponding sampling formula are given, and the given wavelet series is very simple and can be easily applied to numerical analysis. These perfect MRA on finite interval.Thirdly, the analytic expressions of the reproducing kernels of the image spaces of two typical wavelet transforms are given by making use of the...