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MRA Parseval Frame Wavelets and Their Multipliers in
We characterize all generalized lowpass filters and multiresolution analysis( MRA) Parseval frame wavelets in with matrix dilations of the form , where A is an arbitrary expanding matrix with integer coefficients, such that . At first, we study the pseudo-scaling functions, generalized lowpass filters, and multiresolution analysis (MRA) Parseval frame wavelets and give some important characterizations about them. Then, we describe the multiplier classes associated with Parseval frame wavelets in and give an example to prove our theory.
Wavelet theory has been studied extensively in both theory and applications since 1980’s (see [1–3]). One of the basic advantage of wavelets is that an even can be simultaneously described in the frequency domain as well as in the time domain. This feature permits a multiresolution analysis of data with different behaviors on different scales. The main advantage of wavelets is their time-frequency localization property. Many signals can be efficiently represented by wavelets.
The classical MRA wavelets are probably the most important class of orthonormal wavelets. Because they guarantee the existence of fast implementation algorithm, many of the famous examples often used in applications belong to this class. However, there are useful filters, such as , that do not produce orthonormal basis; nevertheless, they do 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 can produce systems having these properties.
A tight wavelet frame is a generalization of an orthonormal wavelet basis by introducing redundancy into a wavelet system . By allowing redundancy in a wavelet system, one has much more freedom in the choice of wavelets. Tight wavelet frames have some desirable features, such as near translation invariant wavelet frame transforms, and it may be easier to recognize patterns in a redundant transform. For advantages and applications of tight wavelet frames, the reader is referred to [4–15] and many references therein. Recently, the theory of high dimensional wavelet is widely studied by the people, such as [16, 17].
In , authors discussed wavelet multipliers, scaling function multipliers, and lowpass filter multipliers in . In , authors introduced a class of generalized lowpass filter that allowed them to define and construct the MRA Parseval frame wavelets. This led them to an associated class of generalized scaling functions that were not necessarily obtained from a multiresolution analysis. Also, they generalized notions of the wavelet multipliers in  to the case of wavelet frame and got several properties of the multipliers of Parseval frame wavelets.
In this paper, we characterize all generalized lowpass filter and MRA Parseval frame wavelets (PFWs) in with matrix dilations of the form , where is an arbitrary expanding matrix with integer coefficients, such that . Firstly, we study some properties of the generalized wavelets, scaling functions, and filters in . Our result is a generalization of the construction of PFWs from generalized lowpass filters that is introduced in . Though we follow  as a blueprint, it is well known that the situation in higher dimension is so complex that we have to recur to some special matrices to solve problem. Thus, our ways are different from original ones. And we borrow some thoughts and technique in . Then, we give some characterizations of the multiplier classes associated with Parseval frame wavelets in .
Let us now describe the organization of the material that follows. Section 2 is of a preliminary character: it contains various results on matrices belonging to the class and some facts about a Parseval frame wavelet. In Section 3, we study the pseudoscaling functions, the generalized lowpass filters, and the MRA PFWs and give some important characterizations about them. In Section 4, we describe the multiplier classes associated with Parseval frame wavelets in . At last, we give an example to prove our theory.
Let us now establish some basic notations.
We denote by the -dimensional torus. By we denote the space of all -periodic functions (i.e., is 1-periodic in each variable) such that . The standard unit cube will be denoted by C. The subsets of invariant under translations and the subsets of are often identified.
We use the Fourier transform in the form where denotes the standard inner product in .
For we denote the function as follows: In particular, for , we will write , which is named as the bracket function of . For , we let be the -translation invariant subset of defined, modulo a null set, by
The Lebesgue measure of a set will be denoted by . When measurable sets and are equal up to a set of measure zero, we write .
Then we introduce some notations and the existing results about expanding matrices.
Let denote the set of all expanding matrices such that . The expanding matrices mean that all eigenvalues have magnitude greater than 1. For , we denote by the transpose of . It is obvious that .
Lemma 2.1. Let be any integer matrix such that . Then the group is isomorphic to and the order of is equal to 2. In particular, if and , then and .
Our standard example that will be frequently used is the quincunx matrix Observe that acts on as rotation by composed with dilation by . In the quincunx case, our standard choice will be
In this paper, we will work with two families of unitary operators on . The first one consists of all translation operators defined by . The second one consists of all integer powers of the dilation operator defined by with .
Let us now fix an arbitrary matrix . For a function , we will consider the affine system defined by
Let us recall the definition of a Parseval frame and a Parseval frame wavelet.
Definition 2.2. We say that a countable family , in a separable Hilbert space H, is a Parseval frame (PF) for H if the equality is satisfied for all .
Definition 2.3. We say that is a Parseval frame wavelet (briefly: PFW) if the system (2.3) is a Parseval frame for .
Then we recall a result from  that characterizes Parseval frame wavelets associated with more general matrix dilations. We state the special case of that theorem appropriate to the discussion in this paper.
In the following, we will give some definitions which will be used in this paper. In fact, they are some generalizations of the notations in .
Definition 2.5. A measurable -periodic function on is called a generalized filter if it satisfies where is defined in Lemma 2.1.
We will denote by the set of generalized filters and put . Observe that .
Definition 2.6. A function is called a pseudoscaling function if there exists a filter such that
Notice that is not uniquely determined by the pseudo-scaling function . Therefore, we shall denote by the set of all such that satisfies (2.7) for . For example, if , then, If is a scaling function of an orthonormal MRA wavelet, then is a singleton; its only element is the lowpass filter associated with . Notice that for a pseudo-scaling function , the function is also a pseudo-scaling function, and if , then .
Suppose that . Since , the function is well defined a.e.; moreover, we have Following , the function defined by (2.9) belongs to and the function is a pseudo-scaling function such that .
Consequently, if is an arbitrary generalized filter, then the function is a pseudo-scaling function and .
Definition 2.7. For , define We say that is a generalized low-pass filter if . The set of all generalized low-pass filters is denoted by .
Then, we will give the definition of MRA PFW.
Definition 2.8. A PFW is an MRA PFW if there exists a pseudo-scaling function and and a unimodular function such that
Let us conclude this introductory section by noting that many of the results that follow can be proved for dilations by expanding integer matrices with arbitrary determinant. Some of these extensions are obtained easily with essentially the same proofs; others require subtler and more involved arguments. But, for the sake of simplicity, we restrict ourselves to the class .
3. MRA Parseval Frame Wavelets
The main purpose of this section is to study the pseudo-scaling functions, the generalized filters, and the MRA PFWs in . We give some important characterizations about them.
In the following we firstly give several lemmas in order to prove our main results.
Lemma 3.1. Suppose that is a pseudo-scaling function and . If then, and .
Lemma 3.2. If , then, for , .
Proof. Assuming that and applying the monotone convergence theorem we obtain
It follows that for a.e. , is finite. Therefore, for , .
Then, we will give a characterization of the generalized lowpass filter.
Proof. Since is an MRA PFW, from (2.4), (2.6), and (2.11), 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 lowpass filter.
The following theorem provides a way of constructing MRA PFWs from generalized lowpass filters.
In order to obtain main result in this part, we firstly introduce a result in [16,Lemma 2.8].
Lemma 3.4. Let be an expanding matrix. Let be a -periodic, unimodular function. Then there exists a unimodular function that satisfies
Theorem 3.5. Suppose that is a generalized lowpass filter. Then, there exist a pseudo-scaling function and an MRA PFW such that they satisfy (2.11).
Proof. Suppose that is a generalized lowpass filter. Consider first the signum function for :
Clearly, is a measurable unimodular function such that, for all ,
By Lemma 3.4, there exists a unimodular measurable function such that
Now take the function constructed from by (2.9), and put
Obviously, . Using (2.9), (3.6), (3.10), and the definition of the signum function , we find
Thus, is a pseudo-scaling function.
For any unimodular function , we define
We affirm that is an MRA PFW. Lemma 2.4 shows that it is enough to prove that satisfies (2.4) and (2.5).
From the fact that is a generalized lowpass filter, we can deduce that (3.1) holds. Using Lemma 3.2, we obtain
This computation shows that the first characterizing condition (2.4) in Lemma 2.4 is satisfied precisely when (3.1) holds.
Let us now prove that the function given by above satisfies (2.5).
Let us fix and , and write
To compute the first term on the right-hand side of (3.14), by the fact ,and (2.6), we have
To compute the second term on the right-hand side of (3.14), by (2.6), (2.7), and (2.11), we have
This shows that the expression in (3.14) is equal to 0. Hence, satisfies (2.5), and is a PFW.
4. The Multiplier Classes Associated with PFWs
In this section, we will describe the multiplier classes associated with PFWs in . We firstly give their definitions.
Definition 4.1. (1)A PFW multiplier is a function such that is a PFW whenever is a PFW.(2)An MRA PFW multiplier is a function such that is an MRA PFW whenever is an MRA PFW.(3)A pseudo-scaling function multiplier is a function such that is a pseudo-scaling function associated with an MRA PFW whenever has the same property.(4)A generalized lowpass filter multiplier is a function such that is a generalized lowpass filter whenever is a generalized lowpass filter.
At first, we will obtain a property of PFW multiplier.
Property 4.2. If a measurable function is a PFW multiplier, then, is unimodular.
Proof. Let be a PFW multiplier; that is, there exist PFWs and such that .
Let be any wavelet satisfying for a.e. . By assumption, for every , we easily deduce that is a PFW and, thus, satisfies (2.4):
In particular, for a.e. and every
This is only possible if a.e. since almost never vanishes. Using (2.4) for PFWs and such that , we easily obtain which is only possible if all terms vanish. Thus, a.e.
The next theorem gives a characterization of PFW multiplier in .
Theorem 4.3. If a measurable function is unimodular and is -periodic, then, this measurable function is a PFW multiplier.
Proof. PFW is characterized as an element of satisfying (2.4) and (2.5) in Lemma 2.4.
Let be a PFW, so that (2.4) and (2.5) hold. Let us assume that is unimodular and is a -periodic function, necessarily unimodular. Let For the function , (2.4) obviously holds from the fact that is a unimodular function and . In the following, for the function , let us consider (2.5). Let , and let . Then
If , according to Lemma 3.4, there exists a unimodular function that satisfies
By -periodicity and unimodularity of , we have
Then, we can repeat the above argument until we obtain
By the equalities (4.4), (4.7), and summing over , we obtain
Since the above equation of the right-hand side is 0 by (2.5) in Lemma 2.4, we conclude that also satisfies (2.5).
Hence, from Lemma 2.4, is a PFW, thus, is a PFW multiplier.
Remark 4.4. It is proved in [19, Theorem 3.2] that if a measurable function is a PFW multiplier, is Z-periodic. However, it still is a unsolved question whether this conclusion holds in . We will discuss it in the future. The next result give a characterization of the MRA PFW multipliers. In order to obtain this result, we firstly introduce a result in [16,Lemma 2.8].
Lemma 4.5. Let be an expanding matrix such that Let and . Suppose that is a - periodic, unimodular function on . Then there exists a -periodic, unimodular function on such that
Theorem 4.6. A measurable function is an MRA PFW multiplier if and only if is unimodular, and is -periodic.
Proof. (if) Let be unimodular, and let
be -periodic, necessarily unimodular. We now use Lemma 4.5 to obtain a unimodular, -periodic function such that
Then is unimodular, and is a -periodic function. In the above computation, we have used (4.9) and (4.10) as well as the unimodularity and the periodicity of .
Let be an MRA PFW, and let be an associated pseudo-scaling function with such that (2.11) holds. Let
Then, we have where the second equation is deduced by (4.13) and the definition of pseudo-scaling function.
It is obvious that
Thus, we obtain
Since is a PFW, we can apply Lemma 2.4 to deduce that is a PFW. It follows, that is an MRA PFW, since where we have used (4.13) and the definition of .
(only if) The Haar wavelet is, in particular, an MRA TFW; we can then proceed as in the proof of Property 4.2 to show the unimodularity of . The proof that is -periodic is similar to Theorem 2 in , and we omit it.
Thus, we complete the proof.
The next theorem characterizes the class of the generalized lowpass filter multipliers. Note that the generalized lowpass filters are the functions for some pseudo-scaling function satisfying Definition 2.5, that is, and . We will use this fact in the following result.
Theorem 4.7. A measurable function is a generalized lowpass filter multiplier if and only if is unimodular and -periodic.
Proof. (if) Suppose that is unimodular and -periodic, and is a generalized lowpass filter.
Then, from the definition of the generalized lowpass filter, we have and . Therefore, according to the function being unimodular and -periodic, it is obvious that these properties are also true for .
(only if) We proceed similarly in the proof of the two previous theorems.
Let be any wavelet satisfying for a.e. , and the corresponding lowpass filter. Then, it follows that a.e. .
By the assumption that the measurable function is a generalized lowpass filter multiplier, we have . In particular, is -periodic, and so is also -periodic.
Applying repetitively, we obtain
This implies that a.e. Unimodularity follows, since both and are in and, thus, satisfy
In a conclusion, is unimodular and -periodic.
The above results provide description of PFW, MRA PFW, and generalized lowpass filter multipliers. These classes are identical with the respective multiplier classes of wavelets. This fact is basically a consequence of the fact that all of these multiplier operations necessarily preserve the norm of the PFW .
Let us introduce a notation for the next theorem. For a measurable function , let and on E.
The following result shows that the situation for pseudo-scaling function multipliers is completely different from others.
Theorem 4.8. is a pseudo-scaling function multiplier if and only if (1) a.e. and a.e.;(2) extends to a -periodic function;(3)If , and is an odd multiple of , then .
Proof. (if) Let be a pseudo-scaling function satisfying (2.6), and suppose that satisfies (1)–(3).
Using condition (1), we see that satisfies
Let us now examine the scale equation from (4.15).
We claim that there exists such that
If , then, by (1), and, thus, (4.25) is satisfied automatically.
If , then (4.25) is equivalent to
The condition (2) implies that defined on by (4.26) is -periodic on E. We will now define a -periodic extension of to satisfying
Let us consider . Define on (part of) by the following: if for some , then
The definition is consistent with the periodicity of on . Let , and . Then one of the following conditions must hold:(a), such that and ;(b) such that and for any , ;(c) For any , such that and .
In (a) case, is an odd multiple of , so by (3), . Thus, we have
Similarly, we can deduce Since , this implies We have either or , and so (4.27) holds.
In (b) case we extend the definition of to the set of all such by
and so we get (4.27).
In (c) case, we let if , and if , whenever or .
We have thus extended to the entire interval so that (4.27) holds if and are in . We now extend to by -periodicity.
Clearly, , and (4.15) is satisfied.
(only if) Let be the Shannon scaling function in ; that is, Then By (2.6) for , we have
This establishes (2), since the right-hand side is -periodic, The remain proof is similar to ones in , and so we omit them. Then, we will provide an example to prove our results.
Example 4.9. Let be the quincunx matrix then we get Furthermore, we have and where is the standard unit square in .
Let us define
Now we extend to such that the equality is satisfied for all , where we take and extend to by -periodicity.
From the definitions of the functions and , we easily deduce that is a pseudo-scaling function, and is a generalized lowpass filter.
Finally, we define
Therefore, by Theorem 3.5, we know that we get an MRA Parseval frame wavelet.
However, from , we know that this MRA Parseval frame wavelet is not a Parseval frame wavelet associated to MRA, which does not permit fast algorithm.
In this paper, we characterize all generalized lowpass filter and MRA Parseval frame wavelets in with matrix dilations of the form , where is an arbitrary expanding matrix with integer coefficients, such that . We study some properties of the multipliers of generalized wavelets, scaling functions, and filters in . Our result is a generalization of the construction of PFWs from generalized low-pass filters that is introduced in . However, our ways are different from original ones. Then, we describe the multiplier classes associated with Parseval frame wavelets in and give an example to prove our theory.
The authors would like to express their gratitude to the referee for his (or her) valuable comments and suggestions that lead to a significant improvement of their manuscript. This work was supported by the National Natural Science Foundation of China (no. 60802061) and the Natural Science Foundation for the Education Department of Henan Province of China (no. 2008B510001).
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