YINGCHUN JIANG and YOUMING LIU∗ Department of Applied Mathematics Beijing University of Technology Pingle Yuan 100, Beijing 100022, P. R. China ∗[email protected] Received 8 October 2006 Revised 9 March 2007 Divergence-free and curl-free vector wavelets have many applications for the analysis and numerical simulation of incompressible flows and certain electromagnetic phenom- ena. Because special domains are required in those applications, Bittner and Urban constructed interpolating divergence-free multi-wavelets based on cubic Hermite splines in 2005. In this paper, we construct interpolating curl-free multi-wavelets and give two wavelet estimates for a class of vector Besov spaces. Keywords: Interpolatory wavelet; curl-free; Besov spaces. Mathematics Subject Classification 2000: 42C15, 42C401. Introduction and Preliminaries Wavelets are by now a widely accepted tool in signal and image processing as well as in numerical simulation. Divergence-free wavelets have been used for the analysis of the Navier–Stokes equations and also for the analysis of experimental data of incompressible turbulent flows. 1,4,6,10,14 The construction of divergence-free wavelets can be found in Refs. 2, 12, 13, and 15. However, the application of divergence-free wavelets in numerical simulations is so far restricted to rather academic simple domains like rectangles. In general cases, interpolatory wavelets are required, which can easily be adapted to compli- cated domains. 11 Bittner and Urban constructed interpolating divergence-free and compactly supported wavelets based on cubic Hermite splines. 3 On the other hand, note that the following two facts: 1. the curl operator is involved in Maxwell equations, which describe certain elec- tromagnetic phenomena;844 Y. C. Jiang & Y. M. Liu 2. curl-free and divergence-free vector fields give a Hodge decomposition, which has been successfully applied to compute the nonlinear term of 2D and 3D incompressible flows. 6 Then we believe that interpolating curl-free vector wavelets should have some potential applications. In fact, the construction of curl-free vector wavelets was discussed in 2001. 15 In this current paper, we shall construct interpolating curl-free vector wavelets, by using the knowledge from Refs. 3 and 15 Secondly, two wavelet estimates for a class of vector Besov spaces are given; Finally, the stability of our vector wavelets is shown in the sense of (L2(R3))3. Now, we begin with some notations and formulae, which will be used later on. All are taken from Refs. 3 and 15 if not pointed out particularly......
无分歧和无旋度向量小波在不可压缩流动和稳定电磁现象的分析和数值模拟上有很多的应用。在2005年，由于特殊领域所需要的那些应用，比特纳和城市构造插值无分歧多小波是基于三次Hermite样条的。在这篇文章，我们构造插值无旋度多小波和为这一类的向量Besov空间提供了两个小波估算。
1.简介和预备知识
2. 插值无旋度向量小波
3.特性
4.无旋度小波基的稳定性
