關於週期性波包近似值的理論與應用

Abstract

在本篇論文裏，我們將使用所謂的週期化(periodization)的裝置作用於Daubechies` compactly supported wavelets上而得到一族構成L²([0,1])和H^s-periodic (the space of periodic function locally in H^s)基底的正交的週期性波包(orthonormal periodic wavelets)。然後我們給出了對於一函數的波包近似值的誤差估計（參閱定理6）以及對於週期性邊界值的常微分方程問題的解的波包近似值的誤差估計（參閱定理7）。對於Burger equation的數值解也當作一個應用來討論。

In this thesis,we shall construct a family of orthonormal periodic wavelets which form a basis of L²([0,l]) and H^s-periodic (the space of periodic functions locally in H^s) by using a device called periodization ([10,7]) on Daubechies` compactly supported wavelets.We then give the error estimates for the wavelet approximation to a given function (see theorem 6) and to a solution of periodic boundary value problem for ordinary differential equation(see theorem 7). Numerical solution for Burger equation is also discussed as an application.

摘要\r\nContents-----1\r\nAbstract-----2\r\n1 Introduction-----3\r\n2 Multiresolution analysis-----5\r\n 2.1 Multiresolution analysis-----5\r\n 2.2 Examples of orthogonal wavelets-----10\r\n3 Periodic wavelets-----14\r\n4 The fast periodic wavelet transform-----18\r\n 4.1 Wavelets with finitely many non-zero filter coefficients-----18\r\n 4.2 Decomposition algorithm-----19\r\n 4.3 Reconstruction algorithm-----22\r\n 4.4 The pyramid scheme-----24\r\n 4.5 Two-dimensional periodic wavelets-----29\r\n5 Approximation and error estimates-----33\r\n6 Applications-----37\r\n 6.1 Application to ordinary differential equation with periodic boundary conditions-----37\r\n 6.2 Application to Burgers` equation with periodic boundary conditions-----40\r\n7 Conclusions-----44\r\nReferences-----45\r\nAppendix-----46