Reconstrction Of The Wavelet Transform In The Reproducing Kernal Space And Differential Operator

The main problem is discussed in this thesis is that the reconstruction of the wavelet transform in the reproducing kernel space by making use of the relationship between the wavelet transform and the theory of reproducing kernel. Firstly, on the base of the Mallat reconstruction algorithm of the wavelet transform modulus maxima, general differential equation in more general meaning is given, which is satisfied with the reconstruction of the wavelet transform. Secondly, the reproducing kernel space H¹ [a,b] is constructed for which reproducing kernel exists. The image space in the wavelet transform is a reproducing kernel Hilbert space. Combing reproducing kernel space with spline function space and reproducing kernel function with delta function in the reproducing kernel space, a method to solve general differential equation is given. Meanwhile, the reconstruction method of the wavelet transform is obtained, which makes up the deficiency of the Mallat reconstruction algorithm of the wavelet transform modulus maxima. Because the interpolation expression of reconstruction can be expressed by section, it is simple and easy for calculation. Furthermore, it shows that the theory of reproducing kernel will give a unified understanding of the wavelet transform, general differential equation. That provides not only the theoretic bases for discussing the image space of wavelet transform, but also a new method to investigate the wavelet analysis theory further. Moreover, the spline function acts as a widely applied mathematics tool, it is concerned by some areas works. On the base of the spline function, the problem of the B-spline function, which is defined by differential operator in the reproducing kernel space W₂¹[ a , b ], is investigated in this thesis. According to differential equation, firstly, different B-spline functions are constructed in the reproducing kernel space W2 1 [ a , b ]. Secondly, it is proved that the B-spline functions is close to reproducing kernel functions in the reproducing kernel space W2 1 [ a , b ]. The same a way, B-spline functions are base of spline function space Sp ( L,Ï€), the B-spline interpolation functions which act as operator is uniform with the best approximate operator.