Anisotropic curl-free wavelets with boundary conditions

Yingchun Jiang

doi:10.1186/1029-242x-2012-205

Abstract

This paper deals with the construction of anisotropic curl-free wavelets that satisfy the tangent boundary conditions on bounded domains. Based on some assumptions, we first obtain the desired curl-free Riesz wavelet bases through the orthogonal decomposition of vector-valued L 2 . Next, the characterization of Sobolev spaces is studied. Finally, we give the concrete construction of wavelets satisfying the initial assumptions. MSC: 42C20

Highlights

This paper deals with the construction of anisotropic curl-free wavelets that satisfy the tangent boundary conditions on bounded domains
Inspired by the fact that a div-free space and a curl-free space form the orthogonal Helmholtz decomposition, we mainly study the anisotropic curl-free wavelet bases satisfying the tangent boundary conditions on bounded domains in this paper, which is organized as follows
The following result will be proved in Section : Assumption

Summary

Then integration by parts shows

The following result will be proved in Section : Assumption. ), Proof For any u ∈ H(In) (n = , ), we know and H I ⊥ curl H I × H I × H I , u = u,. H m In =: u ∈ Hm In n : u × n = or on , V In =: H m In ∩ H In. The following result will be verified in Section : Assumption. Proof Since H(In) = H(In) = H(In) ∩ L (In)n, for any u ∈ V (In), we know u ∈ H m(In) and by Assumption .

Then it holds that

Such an example is known as the Householder transform

It is easy to see that