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Anisotropic curl-free wavelets with boundary conditions
Journal of Inequalities and Applications volume 2012, Article number: 205 (2012)
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 . Next, the characterization of Sobolev spaces is studied. Finally, we give the concrete construction of wavelets satisfying the initial assumptions.
Due to their potential use in many physical problems, like the simulation of incompressible fluids or electromagnetism, curl-free wavelet bases have been advocated in several papers and all results focus on the cases of and [1–4]. Moreover, it is questionable whether they are appropriately called bases and whether they can be used to characterize Sobolev spaces. However, it is reasonable to study the corresponding wavelet bases on bounded domains because of some practical use. At the same time, the boundary conditions, the stability and the characterization of Sobolev spaces are also necessary in some applications such as adaptive wavelet methods. In references [5, 6], anisotropic divergence-free wavelets which satisfy the specific boundary conditions on the hypercube are studied. 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. In Section 2, based on some assumption, the desired curl-free wavelets are constructed through the orthogonal decomposition of vector-valued . Section 3 is devoted to studying the characterization of Sobolev spaces. We give the concrete construction of wavelets satisfying the initial assumption in the final section.
For two 2D vectors and , is defined as
Then for , we define the 2D curl-operator by
and for , the 3D curl-operator is defined by
2 Decomposition of
In this part, we will construct curl-free wavelets that satisfy tangent boundary conditions by the orthogonal decomposition of vector-valued .
Let . For , we firstly define the following spaces:
For a scalar function , define . Then integration by parts shows
Let , . Furthermore, set
For , we define and
Finally, let , .
The following result will be proved in Section 4:
Assumption 2.1 There exist bi-orthogonal Riesz bases and for (of wavelet type) such that
Proposition 2.1 It holds that , and are Riesz bases for (), and , respectively.
Proof For any (), we know
then with . Finally, it is easy to verify by the definition of and curl that
the remaining results can be proved similarly. □
Proposition 2.2 The following decompositions hold:
Proof We only prove the case of , the others can be proved similarly. Since
for any , then . On the other hand, since , then . Therefore, . □
Now, we consider the orthogonal decomposition of . Let . Then there are the following orthogonal decompositions:
Therefore, we obtain the following decomposition:
By Proposition 2.2, , . Moreover,
Finally, we obtain and .
Now, we will construct Riesz bases for (), and . For , we define the embedding by
For , define by
It is obvious that , . Moreover, the image satisfies
Furthermore, we know from (2.1) and (2.2) that ,
Since , and
we obtain and .
In view of Proposition 2.1, we obtain
Theorem 2.1 In the situation of Assumption 2.1, the collections () are Riesz bases for (). and are Riesz bases for and , respectively.
Note 2.1 In fact, for . , which is defined in Section 4.
3 Characterization of
This part will show that the curl-free wavelets constructed above can be used to characterize Sobolev spaces. For and , define the following Sobolev spaces:
The following result will be verified in Section 4:
Assumption 3.1 The collection from Assumption 2.1 can be constructed so that, normalized in , it is a Riesz basis for
Based on this assumption, we obtain:
Theorem 3.1 In the situation of Assumptions 2.1 and 3.1, the collection , normalized in , is a Riesz basis for .
Proof Since , then for any , we know and by Assumption 3.1,
with , where denotes the primal wavelets corresponding to . Furthermore, since , then
with . □
4 Construction of wavelets
In this section, we will give the construction of wavelets satisfying Assumptions 2.1 and 3.1.
Lemma 4.1 ([, Corollary 3.3])
Suppose that the collections and are bi-orthogonal in . In addition, for some , ,
Define the collections and by
Then it holds that
Moreover, and are bi-orthogonal.
Note 4.1 It has been pointed out in  that such wavelet bases can be obtained by taking standard bi-orthogonal wavelet bases for that satisfy the corresponding Jackson and Bernstein assumptions of d, , γ and with and reading as and (see ), and then removing those scaling functions without a vanishing moment.
The following result can be proved by the same method as Corollary 3.7 of .
Corollary 4.1 For and , the sets
are Riesz bases for
respectively. For , the corresponding collections are bi-orthogonal in .
For , we define the vector-valued wavelets
From Corollary 4.1 and the definition of , we obtain
Proposition 4.1 For and , the sets
are Riesz bases for the vector spaces
and , respectively. For , the collections are bi-orthogonal Riesz bases for .
Now, we are in the position to apply the basis transform. Let be an orthogonal matrix with its 1st row given by
Such an example is known as the Householder transform
in the case and . Defining
Applying the property of an orthogonal transform, we infer the following result.
Proposition 4.2 For and , the sets
are Riesz bases for the vector spaces
and , respectively. In particular, for , the collections and are bi-orthogonal Riesz bases for .
In the following, we are mainly concerned with the cases and because of the complicated form of curl operators in .
Theorem 4.1 Let and . Then
(), or .
is a Riesz basis for the vector valued space .
Proof (i) It is easy to see that for , then
Furthermore, and . Therefore, (). In addition,
Therefore, we obtain . Finally, suppose that a, b and c are the solutions of
whose existence can be guaranteed by the orthogonality of . Then
Similarly, if a, b and c are the solutions of the equation
then we can also obtain
Since , taking in Proposition 4.2, we know the set is a Riesz basis for . Furthermore, it is easy to verify
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The project is supported by the National Natural Science Foundation of China (No. 11201094, 11161014), the 863 Project of China (No. 2012AA011005), the project of Guangxi Innovative Team (No. 2012jjGAG0001), the fund of Education Department of Guangxi Province (No. 201012M9094, 201102ZD015, 201106LX172).
The author declares that they have no competing interests.
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Jiang, Y. Anisotropic curl-free wavelets with boundary conditions. J Inequal Appl 2012, 205 (2012). https://doi.org/10.1186/1029-242X-2012-205
- bounded domains
- boundary conditions