Wavelets and applications

In the first chapter, a pair of wavelets are constructed on the basis of Hermite cubic splines. These wavelets are in C 1 and supported on [-1, 1]. Moreover, one wavelet is symmetric, and the other is anti-symmetric. These spline wavelets are then adapted to the interval [0,1]. The construction of boundary wavelets is remarkably simple. Furthermore, global stability of the wavelet basis is established. The wavelet basis is used to solve the Sturm-Liouville equation with the Dirichlet boundary condition. Numerical examples are provided. The computational results demonstrate the advantage of the wavelet basis.; In the second chapter, we develop an efficient preconditioning method on the basis of the modified hierarchy basis (stable wavelet basis in Sobolev space) for solving the singular boundary value problem by the Galerkin method. After applying the preconditioning method, we show that the condition number of the linear system arising from the Galerkin method is uniformly bounded. In particular, the condition number of the preconditioned system will be bounded by 2 for the case q(x) = 0 (see equation (2.1) in the thesis). Some numerical results are presented.; The third chapter is devoted to the construction of C 1 continuous wavelets for Powell-Sabin elements on the three-direction meshes in two dimensional space. Discrete L2 inner products are introduced to help build the wavelet subspaces. We provide a simple estimate for a kind of projection operators, and thus propose a fairly general theory to show the stability of the wavelet system in the Sobolev space H2.; In Appendix A, we prove that BPX method, or so called additive Schwarz type preconditioner is applicable to the nested finite dimensional subspaces generated by Hermite cubic splines for the second order elliptic problem. It is shown that the preconditioned system is well conditioned.