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Mathematical Problems in Engineering
Volume 2010, Article ID 128294, 26 pages
http://dx.doi.org/10.1155/2010/128294
Research Article

Characterizations of Tight Frame Wavelets with Special Dilation Matrices

1Institute of Information and System Science, Beifang University of Nationalities, Yinchuan 750021, China
2Department of Foundation, Beifang University of Nationalities, Yinchuan 750021, China

Received 22 May 2010; Revised 29 October 2010; Accepted 25 November 2010

Academic Editor: Angelo Luongo

Copyright © 2010 Huang Yongdong and Zhu Fengjuan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study all generalized low-pass filters and tight frame wavelets with special dilation matrix (M-TFW), where M satisfies and generates the checkerboard lattice. Firstly, we study the pseudoscaling function, generalized low-pass filters and multiresolution analysis tight frame wavelets with dilation matrix (MRA M-TFW), and also give some important characterizations about them. Then, we characterize all M-TFW by showing precisely their corresponding dimension functions which are nonnegative integer valued. Finally, we also show that an M-TFW arises from our MRA construction if and only if the dimension of a particular linear space is either zero or one.

1. Introduction

Wavelet analysis with its fast algorithms is used in many fields of applied mathematics, such as image or signal analysis and numerical treatment of operator equations (see [14]). Moreover, wavelet bases, recently also wavelet frames, are applied to the characterization of the function space [5]. The classical MRA wavelets are probably the most important class of orthonormal wavelets. Many of the better known examples as well as those often used in applications belong to this class. However, Journe's wavelet is not an MRA wavelet. Thus, it was a natural question to find necessary and sufficient conditions for an orthonormal wavelet to be an MRA wavelet. An interesting approach to this involves the dimension function. On the other hand, there are useful filters, such as , that do not produce orthonormal basis; nevertheless, they produce systems that have the reconstruction property, as well as many other useful features.

It is natural, therefore, to develop a theory involving more general filters that do, indeed, produce systems having these properties. A natural setting for such a theory is provided by frames [1]. Several authors have considered this problem and have shown how more general filters produce such frames. A successful development of these ideas is provided by the papers [6, 7]. These results, however, do involve certain restrictions and technical assumptions such as semiorthogonality. In particular, they exclude the use of the filter we described above. A related approach can be found in [8]. Note that the design of tight wavelet frames is still a challenging problem and a number of references have dealt with this subject (see [325]).

In [20, 21], authors successfully developed a theory of univariate tight frame wavelets and MRA tight frame wavelets. In [24], authors revealed the deep and rich structure of the set of dyadic tight frame wavelets. In [9, 10, 25], authors extended the one-dimensional case to the multidimensional case with the expending dilation matrix of determinant 2, but it is a pity that authors only studied the pseudoscaling function, generalized low-pass filters, and the multiplier classes associated with M-TFWs, but they did neither study the relationship between MRA M-TFW and M-TFW, nor study the relationship between MRA M-TFW and dimension function. In [18], authors constructed multivariate compactly supported tight wavelet frames with dilation matrix , unfortunately, they obtained that at least tight frame generators can generate a tight wavelet frame. Therefore, with the dimension of space increasing, the computational complexity increases. In [8], the author gave the characterizations of the abstract tight frames with arbitrary dilation matrix for ; due to arbitrariness of dilation matrix, the author did not give the expression of tight frame generators (in the time domain or the frequency domain). In fact, even in one-dimensional case, it is very difficult to give the explicit expression of tight frame generators with dilation factor by means of scaling function (see [16]). With the dimension increasing, the computational complexity increases, with the absolute value of determinant of the dilation matrix increasing, the computational complexity increases, that is, at least generators can generate tight wavelet frames for . Therefore, we have to recur some special dilation matrices to solve the problem in . In the paper, we study all the generalized low-pass filters and tight frame wavelets with special dilation matrix (M-TFW), where satisfies and generates the checkerboard lattice [14], that is, In this case, we only need one function such that the system is a tight frame (with ) for (in other word, is a M-TFW), then we can give an explicit expression of by means of pseudoscaling function in the frequency domain; this is due to . Firstly, we study the pseudoscaling function, generalized low-pass filters, MRA M-TFW, and also give some important characterizations about them. Secondly, we characterize all M-TFW by showing precisely their corresponding dimension functions which are nonnegative integer valued. Finally, we also show that a M-TFW arises from our MRA construction if and only if the dimension of a particular linear space is either zero or one. Our result is a generalization of the construction of TFW from generalized low-pass filters that is introduced in [20, 21]. But, it is well known that the situation in higher dimension is more complicated than the situation in one dimension.

Let us now describe the organization of the material as follows. Section 2 presents preliminaries and basic definitions. In Section 3, we study the pseudoscaling function, generalized low-pass filters, and MRA M-TFW, and give some important characterizations about them. In Section 4, we characterize all M-TFW in by dimension function.

2. Preliminaries and Basic Definitions

In this paper, we denote by the transpose of , and , is a constant vector and . For , and , we denote

Let us recall the concept of frame.

Given a countable index set , a collection in a Hilbert space is called a frame in if there exist two constants such that If we can choose in (2.2), then is called a tight frame.

Now, we give some basic definitions which will be used in this paper. In fact, they are some generalizations of the notations in [20, 21].

Definition 2.1. A function is a tight frame wavelet with dilation matrix (briefly: M-TFW) if and only if the system is a tight frame (with ) for .

Definition 2.1 implies that It is clearly that (2.3) is equivalent to for all , where the sum converges unconditionally in .

Bownik gave a deeper result as follows.

Proposition 2.2 (see [8, Theorem  4.2]). is a M-TFW if and only if

A M-TFW is said to be semi-orthogonal if and only if is orthogonal to whenever (for all ). This is equivalent to the orthogonality of the subspaces and if , where as ranges throughout .

Proposition 2.3 (see [3, Chapter  7]). Suppose is a semi-orthogonal M-TFW, then, for each ,

Definition 2.4. A measurable periodic function on is a generalized filter with dilation matrix (briefly: M-GF) if it satisfies
As what was done in [20, 21], we will denote by the set of generalized filters with dilation matrix and let . Observe that .

Definition 2.5. A function is called a pseudoscaling function with dilation matrix (briefly: M-PSF) if there exists a generalized filter such that

Remark 2.6. Notice that is not uniquely determined by the M-PSF . Therefore, we will denote by the set of all such that satisfies (2.10) for . For example, if , then . If is a scaling function of MRA wavelet, then is a singleton. On the other hand, if is an M-PSF, then is also an M-PSF, and if , then .

Definition 2.7. For , let We say that is a generalized low-pass filter with dilation matrix (briefly: M-GLPF) if .

Now, we give the definition of MRA M-TFW.

Definition 2.8. A M-TFW is an MRA M-TFW if there exists a M-PSF and such that where is a measurable periodic function and .

In Section 3, we will prove that if is an MRA M-TFW, then has to be more than just a generalized filter; has to be a M-GLPF.

3. The Characterizations of M-PSF, M-GF, and MRA M-TFW

The main purpose of this section is to study the M-PSF, the M-GF, and the MRA M-TFW in . We will give some important characterizations about them.

Lemma 3.1. Suppose is a M-PSF and . If then, and .

Proof. By (2.10), we have Using (3.1), we obtain that and, therefore (3.2) and are clearly satisfied.

Lemma 3.2 (see [25, Lemma  3.2]). If , then, for a.e. , .

Proof. As the proof is simple, we omit it.

Theorem 3.3. Suppose that is an MRA M-TFW and is a M-PSF satisfying (2.12). Then defined by (2.12) is a generalized low-pass filter.

Proof. Suppose that is an MRA M-TFW, by (2.5), (2.9) and (2.12), we can obtain
Since , Lemma 3.2 implies for a.e. . This shows that for a.e. , , thus, by Lemma 3.1, is a generalized low-pass filter.

Suppose that is a M-TFW, let We say that is the dimension function of M-TFW.

Lemma 3.4. Suppose that is an MRA M-TFW, the corresponding M-PSF of the is , then

Proof. By Definition 2.8, there exists a M-GF such that we obtain that Thus Notice that , hence is an increased sequence and bounded with 1. Moreover, Therefore, by Fatou lemma, So , consequently, .

Lemma 3.5. Suppose that is an MRA M-TFW, then Furthermore, if is a M-TFW, then (3.12) is also valid.

Proof. We only prove the case of is an MRA M-TFW. By Definition 2.8, there exists an M-GF such that By Lemma 3.4, we can obtain Thus, Analogously, we have Thus,

Theorem 3.6. Suppose that is an MRA M-TFW, the corresponding M-PSF of the is , then

Proof. Let , and it is the corresponding M-GLPF of the . Let where denotes all the vertices of cube , . It is clearly that . Now, we need to prove that is an orthonormal system for . Denote So, if , then However, Thus we deduce from (3.19) and the above equation that if , then Similarly, by (3.19), (3.23) also holds if , or , or . Therefore, for a.e. By Fatou lemma,

Recall that the MRA TFW are precisely those that can be constructed from a generalized low-pass filter as described in [20, 21]. What are the properties of when is a M-TFW? We will see that the answers to this question are important for determining the properties of .

For (not necessarily a M-TFW), let us consider the principal shift-invariant space . When is an M-TFW, then the space we defined in (2.7) is the space . We will be interested in examining what type of spanning set for is . If is an orthonormal wavelet, then is an orthonormal basis for . What will happen if is a M-TFW?

It is not hard to see that we can find such that is a tight frame (with constant 1) for . Indeed, let if and if , where . Straight forward calculations show that defined by provides us with the desired function. Moreover, and . This result can be extended to the following known result.

Lemma 3.7 (see [5,Theorem  7.2.3]). , is a tight frame with constant for if and only if

When is a M-TFW, for the space , one has the following results.

Lemma 3.8 (see [21,Theorem  2.7]). Suppose that is a M-TFW. The followings are equivalent:(a) is a tight frame (of constant 1) for (i.e., is a ),(b),(c) is semi-orthogonal,(d) a.e. on .

Proof. It is clear that (a) implies (b), this is (2.3) with . Since and is a tight frame, (b) implies whenever . This clearly implies the semiorthogonality. Thus, (b) implies (c). If we assume (c), so that whenever and , an application of (2.3) gives us and we see that (c) implies (a).
Lemma 3.7 with tells us that (a) and (d) are equivalent.

Although the proof of Lemma 3.8 is rather simple, the result is not obvious. Some aspects of this lemma are counter intuitive: the properties (a), (b), and (d) are tied to the inner structure of . On the other hand, (c) provides information about the relationship between and the other space . The assumption that is an M-TFW is very important. If we do not assume this to be the case, (a) and (d) are still equivalent. However, (b) is not equivalent to (d) (see [21]).

Now, we need to make some facts clear. For Theorem 3.3, Lemma 3.4, and Lemma 3.8, the assumption of can be weakened into , and the assumption of integer-valued dilation matrix can be weakened into noninteger-valued dilation matrix, Theorem 3.3, Lemma 3.4, and Lemma 3.8 still hold; their proofs work more or less in an unchanged form from the original ones. On the other hand, for Lemma 3.5 and Theorem 3.6, the assumption of integer-valued dilation matrix cannot be weakened into noninteger-valued dilation matrix, because we can obtain and if and only if dilation matrix is integer valued, in fact, during the course of these proofs we always use .

4. Dimension Function of M-TFW

The dimension function of a multivariate orthonormal wavelet is integer value; moreover, unless is an MRA wavelet, it attains each of the integer values in the interval , where is the supremum of , on sets of positive measure (see [8, 21]). We will investigate the properties of the dimension function for M-TFW.

Theorem 4.1. Suppose that is a M-TFW. Then is semi-orthogonal if and only if is integer valued a.e.

Proof. First, let us prove the sufficiency. Suppose that is integer valued a.e. From Lemma 3.5, we see that must also be integer valued a.e. By (3.18) and a.e., we have a.e. By Lemma 3.8, part (c) and (d), we conclude that is semi-orthogonal.
Now, we prove the necessity. Let us assume that is semi-orthogonal, denote
Apply Schwart's inequality to the sum in (4.1) with respect to , we obtain Using the fact that a.e. and appling Schwart's inequality to the sum over , we obtain that Therefore, the sum in (4.1) is absolutely convergent. The fact allows us to interchange the sums in the expression . By Proposition 2.3, we have where is defined as in (2.6). Since , a.e. , , we obtain On the other hand, Therefore, we have shown that , and consequently, for . By Propositions 2.2 and 2.3, we have
This shows that
Finally, since is a semi-orthogonal M-TFW, is either 0 or 1 a.e.; this and the last fact, which implies that when , give us
Now, let Using the fact that once again, we see that for a.e. . But, from (2.6), (2.7), and (2.8), we see that for .
By the definition of , we have Hence, we conclude that where ; this is a well-defined subspace of . Obviously, (4.13) implies that is integer valued a.e. The proof of Theorem 4.1 is completed.

Theorem 4.1 provides us with the following interesting statement.

Corollary 4.2. Suppose that is a M-TFW. Then , for a.e. , if and only if is semi-orthogonal.

The dimension function of an orthonormal wavelet which attains value on a set of positive measure, must also attain value on a set of positive measure (for details, see [14]). Equation (3.12) is used there to prove that Fortunately, (4.14) is also valid for semi-orthogonal M-TFWs. More precisely, one has the following result.

Proposition 4.3. Suppose that is a semi-orthogonal M-TFW. Let be an integer. If there exits a set of positive Lebesgue measure, such that for all , then there exits a set of positive Lebesgue measure such that for all .

There is another interesting consequence of Theorem 4.1. Notice that a consequence of (2.5) is that for every M-TFW , one has An interesting class consists of those M-TFW for which attains only values 0 and 1; in accordance with the orthonormal wavelet terminology one will call such M-TFW, MSF M-TFW. One may expect that MSF M-TFW may or may not be semi-orthogonal. Theorem 4.1, however, implies that they have to lie within the realm of semi-orthogonal M-TFW since the corresponding dimension function must be integer-valued. Let us state this in the form of a corollary (see [21]).

Corollary 4.4. If is an MSF M-TFW, then is semi-orthogonal.

Theorem 4.5. Suppose that is a M-TFW, then, is an MRA M-TFW if and only if , for a.e. .

Proof of Theorem 4.5. First, we prove the necessity. Suppose that is an MRA M-TFW, thus, it satisfies (2.12) for an appropriate and . If , for some , then there exist such that and are linearly independent in . It is then impossible to have a vector such that both and are in the one-dimensional subspace . However, by (2.10) and (2.12), for every , we have Since is in , for a.e. , by Lemma 3.5 and Theorem 3.6, we conclude that is at most 1, for a.e. .
Now, we will prove the sufficiency. In other words, we have to prove that is an MRA M-TFW, assuming that is an M-TFW such that is either 0 or 1, for a.e. . We noticed that is always defined for a M-TFW and is a closed subspace of . Theorem 4.1 implies that, for , for a.e. .
Therefore, we need to prove that there exist a generalized filter and a corresponding pseudoscaling function such that (2.12) is satisfied, that is, a.e.
Thus, we begin to prove that it is enough to show that there exist a generalized filter , a corresponding pseudoscaling function , and a periodic, unimodular function such that
Let, and, for , we define the set to be
It is clearly that is the set where all vectors , are zero (or, equivalently, where ). Moreover, the set and the , are periodic, measurable and they form a partition of .
We define as follows:
Observe that (4.21) makes sense and defines a measurable function . Furthermore, the function clearly satisfies Using (4.22), we obtain that . It follows that . Thus, .
Next, we divide the argument into six steps (six lemmas).
Lemma 4.6. For a.e. , one has Proof. If , then, (4.23) is trivially true. We only need to consider . Thus, and, by assumption, . This implies that for every there exists a periodic, measurable function , such that, for a.e. , Coordinatewise, this means that for every , a.e.
It follows that the right hand side of (4.23) satisfies for a.e. . Keeping in mind formula (4.21) for , (4.26) implies that for a.e. . On the other hand, we obtain
Using (4.26), (4.27), and the above equation, we obtain Combine (4.29) with definition of , for a.e. , we obtain This completes the proof of Lemma 4.6 since and cover all of .
Lemma 4.7. There exists a periodic, measurable function such that Proof. Notice that the periodicity of is consistent with the periodicity of the measurable set
Define as the following formula
Since all the functions and sets involved in (4.34) are measurable and periodic, it is obvious that (4.34) defines a periodic, measurable function . Furthermore, implies that (4.31) is valid. For , (4.32) is the direct consequence of (4.21) and (4.34). For . In particular, ; thus (4.32) is trivially satisfied. The proof of Lemma 4.7 is completed.
Lemma 4.8. There exists a periodic, measurable function such that Proof. Again, by (4.33), the periodicity will be clear from the argument, and we need only to prove (4.36) and (4.37) on .
First, we consider . For a.e. , we obtain This and the definition (4.20) of imply that for a.e. , there exists such that . It is easy to check, using (4.20), that is periodic and measurable (defined on ). Since, according to our assumption, for a.e. , there exists a periodic, measurable function such that, for a.e. , Notice that , since otherwise we would have which would imply ; this is impossible for . Hence, (4.39) implies that, for a.e. , Notice that (4.41) shows that there exists a periodic, measurable function such that, for a.e. , At the same time it follows directly from (4.21), that there exists a periodic, measurable function such that, for a.e. , and
We define on by By (4.42) and (4.43), it is clear that is a periodic, measurable function, and satisfies (4.37) on . Therefore, it remains to define on . Observe that for , either , or, otherwise, there is a periodic, measurable function , such that . If , then we define to be 0. Otherwise, we define, by (4.21), that there exists a periodic, measurable function such that Finally, we consider , we know that there exists a periodic, measurable function , such that and a periodic, measurable function , such that Hence, for and , we define by It is now clear that is a periodic, measurable function on and satisfies (4.37) on . It follows that we have a periodic, measurable function , such that (4.37) is satisfied.
Let us prove (4.36). For , we have, by (4.22), that there exists such that . Since is a periodic we obtain, by (4.37), Hence However, Lemma 4.6 implies that for a.e. . Thus, , and this completes the proof of Lemma 4.8.

Now, let us extend the definition of and to , as well. Since is already defined on , the following definition of on makes sense: Using Lemmas 4.7 and 4.8 and (4.51), we obtain that is a periodic, measurable function, such that Moreover, we conclude that (4.32) is now satisfied for a.e. . By Lemma 4.7, we need to check (4.32) only on . But, for , we have that , by (4.21), and , since . Hence, on (4.32) is satisfied irrespective of the value on . We have established, that
Let us turn our attention to . Now that is defined on all of , we define on by Obviously, is a periodic, measurable function, such that Again, we obtain that such satisfies (4.37) on . And, again, because of Lemma 4.8, it is enough to check (4.37) on . Since , for , it is enough to show that , for . But this is an immediate consequence of Lemma 4.6. We conclude that, for a.e. ,
The following lemma connects and .
Lemma 4.9. For a.e. Proof. (1) If and , then, by (4.51) and (4.37), , and, since , so Obviously, (4.57) is satisfied.
(2) If , and , then by (4.51) and (4.54), Hence, (4.57) is going to be satisfied if we can prove it on . We present this argument.
Lemma 4.6 implies that, for a.e. From (4.53) and (4.56) we obtain that, for a.e. Using the fact that and are periodic, we can periodize (4.61). This periodization and (4.22) provides us with Recall that on ; thus (4.62) provides us with (4.57) on .

The following lemma establishes that and are generalized filters, and provides the crucial step to find such that (4.18) is satisfied. It is interesting to observe that the proof of this lemma relies on the fact that .
Lemma 4.10. For a.e. Proof. We only need to prove (4.63). Indeed, (4.63) and Lemma 4.9 imply that