H~2(I)空间上的多分辨分析

Sobolev空间在偏微分方程理论中占有重要的地位 ,且关于 Sobolev空间 H20 (I)的多分辨分析 (MRA)也已经被建立起来。然而在处理偏微分方程的边值问题时就显得有些不足 ,尤其是捕捉靠近边界层附近的激波。在本文中给出了 Sobolev空间 H2 (I)的尺度函数以及它们之间的双尺度关系。在其基础上可以直接建立 Sobolev空间 H2 (I)的多分辨分析 (V0 V1 … ) ,其中 Vj 是由尺度函数通过平移与伸缩得到的。最后分析了关于Sobolev空间 H20 (I)与 H2 (I)的多分辨分析关系并给出了 Sobolev空间 H2 (I)的小波分解算法。

A Multiresolution Analysis of H~2(I)

Sobolev space plays an important role on partial diff er ential equations (PDEs) theories. Multiresolution analysis (MRA) of Sobolev spac e H~2_0(I) is constructed. But it is not enough to deal with the bound value problems of PDEs, especially to capture the shocks ne ar the bound. In this paper some scaling functions of Sobolev space H~2(I ) and their two-scale relationships are given. Based on them we can directly construct a MRA (V_0 V_1 ...) for Sobolev space H~2(I) where V_j is generated by the scaling functions through dilations and translati ons. Finally, the algorithm of wavelet decomposition is given for H~2(I) through the relationship between Sobolev spaces H~2_0(I) and H~2(I).