Construction Of A Class Of Multi-wavelets

Construction of wavelet plays an important role in wavelet analysis. Inparticular, multi-wavelets not only have good properties of single wavelet, butalso overcome the defect which single wavelet can't simultaneously possesssymmetry, orthogonality and compact support. Multi-wavelets can simultaneouslyprovide perfect reconstruction, while preserving length, good performanceat the boundaries, and a high order of approximation. Hence, multi-waveletsrecently were widely used.In this paper, wavelet function and multi-wavelets functions are constructedby using some related properties of the theory of wavelet analysis and the theoryof the reproducing kernel. The wavelet function and the multi-wavelets functionshave many good properties. The wavelet function can be used to reconstruct thebounded functions ofL² (R) and the multi-wavelets functions can be used todecompose the Hilbert space H₀¹ (0,1 ), which gives a new method ofdecomposition of general space. Concretely, two aspects are finished in thisthesis as follows:Firstly, on the base of the reproducing kernel of Sobolev HilbertspaceH¹(R;a,b), the reproducing kernel of Sobolev Hilbert space Hⁿ(R;a,b)is derived from computing the convolution of the reproducing kernel.Furthermore, we discuss related properties of the reproducing kernel of SobolevHilbert space Hⁿ(R;a,b)and give a class of wavelet function on the base of thereproducing kernels. The reproducing kernel of Sobolev Hilbert spaceHⁿ(R;a,b)is symmetric and has odd order vanishing moment; the waveletfunction is anti- symmetric and has even order vanishing moment. Finally, we thebounded functions ofL² (R) is reconstruct by using the class of wavelet.Secondly, two scaling functions are constructed on bases of Beta functions.One function is symmetric, the other is anti-symmetric. The vector generated bythe two scaling functions satisfies refinement equation. So a class of multiwaveletsfunctions can be constructed on the base of the two scaling functions. The multi-wavelets functions are supported on [-1, 1], and one wavelet function is symmetric, the other is anti-symmetric. The supported multi-wavelets functions are then adapted to the interval [0, 1]. Finally, we use the multi-wavelets functions to decompose the Hilbert space H₀¹(0,1)...