Abstract
Balanced multiwavelet transform can process the vector-valued data sparsely while preserving a polynomial signal. Yang et al. (2006) constructed balanced multiwavelets from the existing nonbalanced ones. It will be proved, however, in this paper that if the nonbalanced multiwavelets have antisymmetric component, it is impossible for the balanced multiwavelets by the method mentioned above to have symmetry. In this paper, we give an algorithm for constructing a pair of biorthogonal symmetric refinable function vectors from any orthogonal refinable function vector, which has symmetric and antisymmetric components. Then, a general scheme is given for high balanced biorthogonal multiwavelets with symmetry from the constructed pair of biorthogonal refinable function vectors. Moreover, we discuss the approximation orders of the biorthogonal symmetric refinable function vectors. An example is given to illustrate our results.
1. Introduction
Multiwavelets have been studied extensively in the literature; for example, to mention only a few here, see [1–7] and the references therein. Implementing multiwavelet transform, one will face a serious problem related to the discrepancy of their approximation properties between the function setting and the discrete vector data setting [1, 8–10]. As such, a multiwavelet must be prefiltered or balanced in advance [1, 2, 8–10]. Balanced multiwavelet transform can sparsely process the vector-valued data efficiently. Moreover, they can preserve the polynomial structure of a signal. As such, constructing balanced multiwavelets is of interest for researchers of wavelet analysis. In this paper, we will construct pairs of biorthogonal symmetric multiwavelets with high balanced orders from any orthogonal refinable function vector, which simultaneously has symmetric and antisymmetric components. Before introducing our motivation, let us give some conceptions and notations.
Let denote the set of nonnegative integers while denotes the th derivative of , . Suppose that a compactly supported -refinable function vector satisfies the refinement equation with the integer (2) being regarded to as a dilation factor. We get from (1) that , with being called the mask symbol of and Fourier transform of being defined to be , which can be naturally extended to square integrable functions and temper distributions. We say that has sum rules if there exists a matrix of -periodic trigonometric polynomials with such that We say that has approximation orders if there exists a sequence of vectors such that, for any polynomial , , it can be represented as It is well known that has approximation orders if and only if its mask symbol has sum rules.
Next, let us introduce the relationship between the mask symbol of a refinable function vector and its symmetry. Suppose that in (1) satisfies Then, it is not difficult to check that with , . In the words of element, (5) is equivalent to where is the -element of . Suppose is an matrix of -periodic trigonometric polynomials, which satisfies with , , , and then it is not difficult to check that where . We will use (7) in the proof of Theorem 3.
Let and be two -refinable function vectors in . Their mask symbols are and , respectively. They are a pair of biorthogonal refinable function vectors, that is, if and only if where denotes the transpose conjugate. When , we call an orthogonal -refinable function vector.
Let be an orthogonal -refinable function vector. We say that it has balanced orders if There are a few equivalent definitions of balanced multiwavelets in the literature; see [11]. In [10, 11], the conception of balanced multiwavelets is extended to the case of biorthogonal multiwavelets. Next, we will introduce our motivation. The following remark is necessary.
Remark 1 (a balanced refinable function vector does not has antisymmetric component). Suppose that satisfies (2) and has balanced orders. By (11), we get that satisfies Condition ; that is, is the simple eigenvalue of while the other eigenvalues are strictly smaller than in modulus. It follows from (2) that , which leads to the fact that , , and consequently has no antisymmetric component.
In [8], Yang and Peng constructed balanced multiwavelets via PTST method. Before introducing our motivation, let us briefly introduce the main idea of PTST. Let be an orthogonal -refinable function vector. Construct via with being a paraunitary matrix; that is, . It is not difficult to check that is still orthogonal. Some conditions are imposed on such that has desired balanced orders [8, Theorem 2, Theorem 4]. However, the balanced can not preserve the symmetry of if it has antisymmetric component. Next, we will explain the reason, which is the motivation of this paper. Without losing generality, assume that , . Suppose that there exists a paraunitary matrix such that is balanced and all the components of have symmetry. By Remark 1, all the components of are symmetric. Then, it is easy to see that , , are all antisymmetric polynomials, which contradicts the fact that is invertible. In conclusion, there does not exist invertible matrix , such that is simultaneously balanced and symmetric if has antisymmetric component.
Since many famous refinable function vectors satisfy (13), for example, CL multiwavelets [12], we naturally face the following problem.
How can we construct symmetric multiwavelets with high balanced orders from such ones that have antisymmetric component?
In this paper, we are interested in the case of (13) with . That is, In fact, many classical multiwavelets satisfy (14). Readers are referred to [13] for many examples.
We will need the following results related to the canonical form of mask symbols of refinable function vectors.
Lemma 2 (see [2]). Let be a compactly supported -refinable function vectors satisfying (1), (2), and (13). Then, for any , there exists a strongly invertible matrix ; that is, and are both matrices of -periodic trigonometric polynomials, such that the mask symbol takes the form where , , , and are some , , , and matrices of -periodic trigonometric polynomials, respectively. Moreover, constructed via also satisfies (13) with being replaced by , , and , , .
Note. (I) The algorithm for constructing in Lemma 2 can be seen in the proof of [2, Theorem 2.5]. (II) There exist a number of orthogonal multiwavelets in the literature, whose mask symbols take the form of (15); just see [12, 13] for some examples.
2. Main Results
Theorem 3. Let be an orthogonal compactly supported -refinable function vector satisfying (1), (2), and (14). Then, for any positive integer , we can construct an matrix of -periodic trigonometric polynomials such that the compactly supported -refinable , which is defined via is symmetric and has balanced orders. Moreover, the -refinable function vector , which is defined via is also symmetric and a dual refinable function vector of . On the other hand, and have and approximation orders, respectively.
Theorem 4. Let be as in Theorem 3. Moreover, its mask symbol takes the form of (15) with . Then, defined by (18) is compactly supported.
3. Proof of Main Result and Algorithm for High Balanced Biorthogonal Multiwavelets
Proof of Theorem 3. has sum rules. Of course, it has sum rule. According to Lemma 2, there exists a strongly invertible matrix , such that the mask symbol has the property of (15); that is,
where , , , and are some , , , and matrices of -periodic trigonometric polynomials, respectively. Define via
Then, satisfies (13). Define and . Then, it is not difficult to check that
By (21) and (6), we have
Again by (6), we know (22) is equivalent to
where and is defined by . In other words, compared with , the components of are all symmetric.
From the aspect of two-scale similarity transforms (TST), is the singular TST of . Therefore, by [14, Theorem ], has sum rules. Define a strongly invertible matrix as follows:
with satisfying such that
where is as in (23).
Construct via
Then, by (7) and (8), we get
Moreover, (25) is equivalent to
That is, has balanced orders. Select and define . Then, .
Next, we will prove that , which is defined via (18), is a dual refinable function vector of . In fact,
which together with
leads to the fact that . Thus, is a dual refinable function vector of . By (7) and (8), we have
from which we deduce that
That is, is symmetric. Moreover, , , . From the aspect of two-scale similarity transforms (TST), is the singular inverse TST of . Therefore, by [14, Theorem ], has sum rules.
Next, we summarize the process of constructing in Theorem 3.
Algorithm 5. Let be as in Theorem 3. Then, based on [2], can be constructed through the following steps.
Step 1. By Theorem 3, construct a strongly invertible matrix of -periodic trigonometric polynomials such that the mask symbol takes the property of (19). Moreover, can generate a -refinable function vector, whose symmetry takes the form of (14).
Step 2. Define and . Then, takes the form of (21). Compared with , all the components of the -refinable function vector , which is defined by , satisfy (23).
Step 3. Define a strongly invertible matrix as in (24), with satisfying such that (25) holds.
Step 4. Construct via . Then, has balanced orders.
Step 5. Define via (18). Then, it is the dual symmetric refinable function vector of .
Proof of Theorem 4. If the mask symbol of in Theorem 3 takes the form of (15), then the dual mask symbol of isThe transform matrix in Step 3 of Algorithm 5 is strongly invertible. Therefore, the mask symbol is a matrix of -periodic trigonometric polynomials. That is, the dual refinable function vector , which satisfies , is compactly supported.
4. High Balanced Biorthogonal Multiwavelets from CL Multiwavelets
In [12], Chui and Lian constructed an orthogonal refinable function vector , which satisfies the following refinement equation: where with It is easy to see that takes the form of (15) with and . Specifically, straightforward observation gives us that , , and . Moreover, by [12], has sum rules. See Figure 1 for the graphs of and . Select . Compute where Then, by Theorem 3, the components of , satisfying , are all symmetric. has sum rules. By [3], we found out that the Sobolev exponent of is . According to Theorem 3, the dual refinable function vector of is given by . That is, has sum rules.
(a)
(b)
Next, we will construct balanced symmetric refinable function vectors and their dual refinable function vectors. We compute from that We will find a -periodic trigonometric polynomial and construct by where such that is balanced and symmetric.
By (25), if satisfies , then defined in (41) has balanced order. Take ; for example, its graph is illustrated in Figures 2(a) and 2(b).
(a)
(b)
(c)
(d)
(e)
(f)
By (25), if then defined in (41) has balanced orders. For this, construct such that (43) is satisfied. Now, the graph of , given by (41) with being as in (44), is shown in Figures 2(c) and 2(d).
By (25), select , such that and Construct such that (45) is satisfied. Then, has balanced orders and is symmetric. See Figures 2(e) and 2(f) for its graph.
According to Algorithm 5, the mask symbol of , the dual refinable function vector of , is . By [3], we found that the Sobolev exponent of is 0.4418.
Moreover, for the pair of dual refinable function vectors, one can still get the dual multiwavelets. More specifically, suppose that is the mask symbol of . Then, and defined by are the mask symbols of their corresponding multiwavelets, where That is,
5. Future work
In this paper, from an orthogonal refinable function vector , which has antisymmetric component and is not balanced, we give an algorithm for constructing a pair of symmetric biorthogonal refinable function vectors and such that has high balanced orders. When compared with , the sum rule of increases by one while that of decreases by one. Therefore, the Sobolev smoothness of may be smaller than that of . In future, based on , we will study how to construct a new symmetric and smooth dual refinable function vector of .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors thank the two reviewers for their valuable suggestions which improve the paper a lot. The paper is supported by the Natural Science Foundation of Guangxi Province (no. 2013GXNSFBA019010), Natural Science Foundation of China (nos. 11126343, 11461002), and The Midwest Universities Comprehensive Strength Promotion Project of Guangxi University.