Pairs of Frequencybased Nonhomogeneous Dual Wavelet Frames in the Distribution Space
Abstract.
In this paper, we study nonhomogeneous wavelet systems which have close relations to the fast wavelet transform and homogeneous wavelet systems. We introduce and characterize a pair of frequencybased nonhomogeneous dual wavelet frames in the distribution space; the proposed notion enables us to completely separate the perfect reconstruction property of a wavelet system from its stability property in function spaces. The results in this paper lead to a natural explanation for the oblique extension principle, which has been widely used to construct dual wavelet frames from refinable functions, without any a priori condition on the generating wavelet functions and refinable functions. A nonhomogeneous wavelet system, which is not necessarily derived from refinable functions via a multiresolution analysis, not only has a natural multiresolutionlike structure that is closely linked to the fast wavelet transform, but also plays a basic role in understanding many aspects of wavelet theory. To illustrate the flexibility and generality of the approach in this paper, we further extend our results to nonstationary wavelets with real dilation factors and to nonstationary wavelet filter banks having the perfect reconstruction property.
Key words and phrases:
Nonhomogeneous wavelet systems, dual wavelet frames, distribution space, homogeneous wavelet systems, nonstationary dual wavelet frames, oblique extension principle, real dilation factors2000 Mathematics Subject Classification:
42C40, 42C151. Introduction and Motivations
In wavelet analysis, we often use translation, dilation, and modulation of functions. For a function , throughout the paper we shall use the following notation
(1.1) 
where denotes the imaginary unit. In this paper we shall use as a dilation factor. In applications, is often taken to be a positive integer greater than one, in particular, the simplest case is often used.
Classical wavelets are often defined and studied in the time/space domain with the generating wavelet functions belonging to the square integrable function space . For and for a subset of square integrable functions in , linked to discretization of a continuous wavelet transform (see [2, 9, 29, 30]), the following homogeneous wavelet system
(1.2) 
is generated by the translation and dilation of the wavelet functions in and has been extensively studied in the function space in the literature of wavelet analysis. To mention only a few references here, see [1]–[32]. In this paper, however, we shall see that it is more natural to study a nonhomogeneous wavelet system in the frequency domain. It is important to point out here that the elements in a set of this paper are not necessarily distinct and in a summation means that visits every element (with multiplicity) in once and only once. For example, for , all the functions are not necessarily distinct and in (1.2) means .
Most known classical homogeneous wavelet systems in the literature are often derived from scalar refinable functions or from refinable function vectors in ([2, 9, 29, 30]). Let us recall the definition of a scalar refinable function here. A function or distribution on is said to be refinable (or refinable) if there exists a sequence of complex numbers, called the refinement mask or the lowpass filter for the scalar refinable function , such that
(1.3) 
with the above series converging in a proper sense, e.g., in . Wavelet functions in the generating set of a homogeneous wavelet system are often derived from the refinable function by
(1.4) 
where are sequences on , called wavelet masks or highpass filters. For the infinite series in (1.3) and (1.4) to make sense, one often imposes some decay condition on the refinable function and wavelet filters so that all the infinite series in (1.3) and (1.4) are welldefined in a proper sense. Nevertheless, even for the simplest case of a compactly supported scalar refinable function (or distribution) with a finitely supported mask , the associated refinable function with mask does not always belong to . In fact, it is far from trivial to check whether in terms of its mask , see [20, 21] and references therein for detail. One of the motivations of this paper is to study wavelets and framelets without such stringent conditions on either the generating wavelet functions or their wavelet filters for .
For , the Fourier transform used in this paper is defined to be , and can be naturally extended to square integrable functions and tempered distributions. Under certain assumptions, taking Fourier transform on both sides of (1.3) and (1.4), one can easily rewrite (1.3) and (1.4) in the frequency domain as follows:
(1.5) 
and
(1.6) 
provided that all the periodic (Lebesgue) measurable functions and similarly are properly defined. In the following, we shall see that it is often more convenient to work with (1.5) and (1.6) in the frequency domain rather than (1.3) and (1.4) in the time/space domain. If there exist positive real numbers and such that the periodic measurable function satisfies for almost every (this condition is automatically satisfied with if is a periodic trigonometric polynomial with ), for a dilation factor such that , then it is easy to see (also c.f. section 3) that one can define a measurable function such that
(1.7) 
Regardless of whether in (1.7) is a square integrable function or not, is a welldefined measurable function obviously satisfying the frequencybased refinement equation (1.5) with being replaced by . The function is called the (frequencybased) standard refinable function with mask and dilation . All the wavelet functions in (1.6) with are also welldefined measurable functions provided that all are measurable. This motivates us to study refinable functions and wavelets in the frequency domain using (1.5) and (1.6) so that we can avoid some technical issues such as the convergence of the infinite series in (1.3) and (1.4) as well as membership in of the generating refinable function and the generating wavelet functions .
For wavelets derived from refinable functions or refinable function vectors, one of the most important key features of wavelets and framelets is its associated fast wavelet transform, which is based on the following nonhomogeneous wavelet system:
(1.8) 
where is an integer, representing the coarsest decomposition (or scale) level of its fast wavelet transform. In fact, a onelevel fast wavelet transform is just a transform between two sets of wavelet coefficients of a given function represented under two nonhomogeneous wavelet systems at two consecutive scale levels. Naturally, for a multilevel wavelet transform, there is a underlying sequence of nonhomogeneous wavelet systems at all scale levels, instead of just one single wavelet system. For a given at some scale , we shall see in this paper that via the dilation operation it naturally produces a sequence of nonhomogeneous wavelet systems for all integers with almost all properties preserved. Consequently, it often suffices to study only one nonhomogeneous wavelet system instead of a sequence of them. This desirable property of nonhomogeneous wavelet systems is not shared by homogeneous wavelet systems. Furthermore, as , the limit of the sequence will naturally lead to a homogeneous wavelet system . Hence, in certain sense, a homogeneous wavelet system could be regarded as the limit system of a sequence of nonhomogeneous wavelet systems. See section 3 for more detail.
For a homogeneous wavelet system that is derived from a refinable function or a refinable function vector, due to the absence of a refinable function in the system, the homogeneous wavelet system does not automatically correspond to a fast wavelet transform without ambiguity. In fact, the wavelet functions in of could be derived from many other (equivalent) refinable functions, which correspond to different fast wavelet transforms with different sets of wavelet filters. More precisely, for a periodic measurable function such that for almost every , define , then it is evident that is also refinable and satisfies
Such a change of generators from a refinable function to another equivalent refinable function is in fact the key idea in the oblique extension principle (OEP) in [3, 11, 12, 23] (also see [13, 14, 19, 24, 25, 26, 27]) to construct compactly supported homogeneous wavelet frames in with high vanishing moments derived from refinable functions and refinable function vectors. See [12, 23] for a detailed discussion on a fast wavelet transform based on a homogeneous wavelet system obtained via OEP from refinable function vectors. The effect of the change of generators on its fast wavelet transform is addressed in [23]. As we shall see in sections 3 and 4, nonhomogeneous wavelet systems are closely related to nonstationary wavelets (see [4, 7, 26]) and are naturally employed in a pair of nonhomogeneous dual wavelet frames in a pair of dual Sobolev spaces introduced in [27]. Due to these and other considerations, it seems more natural and more important for us to study nonhomogeneous wavelet systems rather than the extensively studied homogeneous wavelet systems . This allows us to understand better many aspects of wavelet theory such as a wavelet filter bank induced by OEP and its associated wavelets in the function setting without a priori condition on the generating wavelet functions.
Following the standard notation, we denote by the linear space of all compactly supported (test) functions with the usual topology, and denotes the linear space of all distributions, that is, is the dual space of . By duality, the definition in (1.1) for translation, dilation and modulation can be easily generalized from functions to distributions. Moreover, when in (1.1) is a square integrable function or more generally a tempered distribution, for , we have
(1.9) 
In this paper, we shall use boldface letters to denote functions/distributions (e.g., ) or sets of functions/distributions (e.g., ) in the frequency domain.
Let and be two sets of distributions on . For an integer and , we define a frequencybased nonhomogeneous wavelet system to be
(1.10) 
Similarly, a frequencybased homogeneous wavelet system is defined as follows:
(1.11) 
By (1.9), it is straightforward to see that under the Fourier transform, the images of and with are simply and , respectively, where and .
For , by we denote the linear space of all measurable functions such that for every compact subset of , with the usual modification for saying that is essentially bounded over . Note that is just the set of all measurable functions that can be globally identified as distributions. So, is the most natural space for us to study wavelets and framelets in the distribution space. For , it is evident that . However, a distribution may not be a tempered distribution and therefore, Fourier transform may not be applied so that holds for some tempered distribution . Under the setting of tempered distributions on which the Fourier transform can apply, although all the definitions and results of this paper in the frequency domain could be equivalently translated into the time/space domain by the inverse Fourier transform on tempered distributions, to avoid notational complexity and to avoid the a priori underlying assumption that is a tempered distribution if the notion is used, it seems very natural and convenient for us to work in the frequency domain in this paper.
For and , we shall use the following paring
(1.12) 
When and , the duality pairings and are understood similarly as . Now we are ready to introduce the key notion in this paper. Let
(1.13) 
be subsets of . Let and , we say that the pair , where is defined in (1.10), forms a pair of frequencybased nonhomogeneous dual wavelet frames in the distribution space if the following identity holds
(1.14) 
where the infinite series in (1.14) converge in the following sense:

For every , the following series
(1.15) converge absolutely for all integers , , and .

For every , the following limit exists and
(1.16)
As we shall discuss in section 3, the above introduced notion enables us to completely separate the perfect reconstruction property in (1.14) from its stability property in function spaces. Since the test function space is dense in many function spaces, one could extend the perfect reconstruction property (or “wavelet expansion”) in (1.14) to other function spaces, provided that the involved wavelet systems have stability in these function spaces. Let us give a simple example here to illustrate this connection for the particular function space . Let and in (1.13) be two subsets of distributions in . We shall see in section 3 that forms a pair of frequencybased nonhomogeneous dual wavelet frames in the distribution space, if and only if, are subsets of and
(1.17) 
Moreover, when , as a direct consequence of (1.17), one automatically has
Nonhomogeneous wavelet systems also have a close relation to refinable functions and refinable function vectors. Suppose that (that is, multiply every element in by the factor ) is an orthonormal basis of . Denote . Without assuming in advance that is a refinable function vector and all are derived from , we can deduce that must be a refinable function vector and all must be derived from via similar relations as in (1.5) and (1.6). See section 3 for more detail.
To have some rough ideas about our results on nonhomogeneous wavelet systems, here we present two typical results. The following result is a special case of Theorem 6.
Theorem 1.
Let be an integer such that . Let in (1.13) be subsets of . Then forms a pair of frequencybased nonhomogeneous dual wavelet frames in the distribution space for some integer (or for all integers ), if and only if, the following three statements hold

For all integers ,
(1.18) 
For all integers ,
(1.19) 
The following identity holds in the sense of distributions:
(1.20) more precisely, for all .
In the following, we make some remarks about Theorem 1. We assumed in Theorem 1 that all the generating functions in are from the space . Note that includes the Fourier transforms of all compactly supported distributions and of all elements in all Sobolev spaces. This assumption on membership in can be weakened and is only used to guarantee the absolute convergence of the infinite series in (1.15). See the remark after Lemma 3 in section 2 for more detail on this natural assumption.
If we assume additionally that for all and if (1.20) holds for almost every , by Lebesgue dominated convergence theorem, then (1.20) holds in the sense of distributions. If all elements in are essentially nonnegative measurable functions, then it is not difficult to verify that the conditions in items (i) and (ii) of Theorem 1 are equivalent to the following simple conditions:
(1.21) 
and
(1.22) 
As we shall see in section 2, items (i) and (ii) of Theorem 1 correspond to a natural multiresolutionlike structure, which is closely linked to a fast wavelet transform. The condition in item (iii) of Theorem 1 is a natural normalization condition which is related to (1.16).
Comparing with the characterization of a pair of homogeneous dual wavelet frames in the space or a homogeneous orthonormal wavelet basis in (e.g., see [10, 17, 18, 28, 31, 32]), Theorem 1 has several interesting features. Firstly, the characterization in items (i)–(iii) of Theorem 1 does not involve any infinite series or infinite sums; this is in sharp contrast to the homogeneous setting in . Secondly, as we shall see in section 2, all the involved infinite sums in the proof of Theorem 1 are in fact finite sums. This allows us to easily generalize Theorem 1 to any real dilation factors and to nonstationary wavelets, see sections 2 and 4 for detail. Thirdly, we do not require any stability (Bessel) property of the wavelet systems, while the homogeneous setting in needs the stability property to guarantee the convergence of the involved infinite series. Fourthly, we do not require in Theorem 1 that the generating wavelet functions possess any order of vanishing moments or smoothness, while all the generating wavelet functions in the homogeneous setting require at least one vanishing moment. Lastly, from a pair of nonhomogeneous dual wavelet frames in , we shall see in section 3 that one can always derive an associated pair of homogeneous dual wavelet frames in . In fact, most homogeneous wavelet systems in the literature are derived in such a way. We mention that weak convergence of wavelet expansions has been characterized in [16] for homogeneous wavelet systems. Similar weak convergence of wavelet expansions that are related to (1.16) also appeared in the study of homogeneous dual wavelet frames in and their frame approximation properties, for example, see [12, 17, 18]. We also point out that the approach in this paper can be extended to frequencybased homogeneous wavelet systems in the distribution space .
The following result generalizes the Oblique Extension Principle (OEP) and naturally connects a wavelet filter bank with a pair of frequencybased nonhomogeneous dual wavelet frames in .
Theorem 2.
Let be an integer such that . Let and , be periodic measurable functions on . Suppose that there are measurable functions satisfying
(1.23) 
Define as in (1.13) with
(1.24) 
and
(1.25) 
Assume that all the elements in belong to . Then forms a pair of frequencybased nonhomogeneous dual wavelet frames in the distribution space for some integer (or for all integers ), if and only if,
(1.26) 
with
(1.27) 
and the following fundamental identities are satisfied:
(1.28) 
and
(1.29) 
for all , where and
(1.30) 
In particular, if all are periodic measurable functions in and if there exist positive real numbers and such that
(1.31) 
then the frequencybased standard refinable measurable functions with masks and the dilation factor , which are defined by
(1.32) 
are welldefined for almost every and in fact . Then all elements in belong to . Moreover, forms a pair of frequencybased nonhomogeneous dual wavelet frames in the distribution space for some integer (or for all integers ), if and only if, the identities (1.28) and (1.29) are satisfied for all , and in the sense of distributions.
Note that (1.31) is automatically satisfied with if and are periodic trigonometric polynomials with . A similar result to Theorem 2 also holds when and are refinable measurable function vectors. The identities in (1.28) and (1.29) with and are called the oblique extension principle in [12], provided that all elements in belong to and satisfy some technical conditions to guarantee the Bessel (stability) property of the homogeneous wavelet systems and in the space (see [3, 11, 12, 23, 31, 32]). In contrast, our results here generally do not require any a priori condition on the generating wavelet functions and provide a natural explanation for the connection between the perfect reconstruction property induced by OEP in (1.28) and (1.29) in the discrete filter bank setting to wavelets and framelets in the function setting.
The structure of the paper is as follows. In order to prove Theorems 1 and 2, we shall introduce some auxiliary results in section 2. In particular, we shall provide sufficient conditions in section 2 for the absolute convergence of the infinite series in (1.15). Then we shall prove Theorems 1 and 2 in section 2. To explain in more detail about our motivation and importance for studying frequencybased nonhomogeneous wavelet systems, we shall discuss in section 3 nonhomogeneous wavelet systems in various function spaces such as and Sobolev spaces, as initiated in [27]. We shall see in section 3 that under the stability property, a pair of frequencybased nonhomogeneous dual wavelet frames can be naturally extended from the distribution space to a pair of dual function spaces. In section 3, we shall also explore the connections between nonhomogeneous and homogeneous wavelet systems in the space . To illustrate the flexibility and generality of the approach in this paper, we further study nonstationary wavelets which are useful in many applications, since nonstationary wavelet filter banks can be implemented in almost the same way and efficiency as a traditional fast wavelet transform. However, except a few special cases as discussed in [4, 7, 26], only few theoretical results on nonstationary wavelets are available in the literature, probably partially due to the difficulty in guaranteeing the membership of the associated refinable functions in and in establishing the stability property of the nonstationary wavelet systems in . In section 4, we present a complete characterization of a pair of frequencybased nonstationary dual wavelet frames in the distribution space. Though the statements and notation in section 4 on nonstationary wavelets seem a little bit more complicated comparing with the stationary case in sections 1–3, it is worth our effort to provide a better picture to understand nonstationary wavelets, since there are few theoretical results on this topic in the literature.
To understand and study wavelet systems in various function spaces, it is our opinion that there are two key fundamental ingredients to be considered. One ingredient is the notion investigated in this paper of a pair of frequencybased nonhomogeneous dual wavelet frames in the distribution space which enables us to completely separate its perfect reconstruction property from its stability property in function spaces. The other ingredient is the stability issue of nonhomogeneous wavelet systems in function spaces which we didn’t discuss in this paper but shall be addressed elsewhere.
2. Frequencybased Nonhomogeneous Wavelet Systems in the Distribution Space
In this section, we study pairs of frequencybased nonhomogeneous dual wavelet frames in the distribution space. To prove Theorems 1 and 2, we first present some sufficient conditions for the absolute convergence of the infinite series in (1.15).
For , by we denote the set of all periodic measurable functions such that (with the usual modification for ).
By the following result, we always have the absolute convergence of the infinite series in (1.15) provided that all the frequencybased wavelet functions are from the space .
Lemma 3.
Let be a nonzero real number and let . Then for all ,
(2.1) 
with the series on the lefthand side converging absolutely. Note that the infinite sum on the righthand side of (2.1) is in fact finite.
Proof.
By we denote the linear space of all compactly supported measurable functions in . Note that . More generally, we prove (2.1) for . Denote
Now we show that are welldefined functions in . In fact, since , has compact support and therefore, is essentially supported inside for some with depending on . Now it is easy to see that
(2.2) 
Since and , we see that for every . Therefore, is a welldefined periodic function in . Similarly, we have . Note that
and . Since , by the Parseval identity, we have
and
with the series on the lefthand side converging absolutely. By the finite sum in (2.2), we have
which completes the proof. ∎
The condition in Lemma 3 is only used in this paper to guarantee the identity (2.1). As long as (2.1) holds for and the frequencybased wavelet functions belong to , all the claims in this paper still hold. Note that is just the set of all measurable functions that can be globally identified as distributions. So, is the most natural and weakest space for us to study wavelets and framelets in the distribution space. The condition in Lemma 3 could be replaced by other conditions. For example, for every positive integer , if there exist positive numbers and such that and for all , then it is not difficult to check by (2.2) that with is a periodic Lipschitz function with some Lipschitz exponent . By Bernstein Theorem, has an absolutely convergent Fourier series. Now for any , it is easy to prove that (2.1) indeed holds for all . Other assumptions could be used to guarantee (2.1). But is a large space containing the Fourier transforms of all compactly supported distributions and of all elements in all Sobolev spaces. For simplicity of presentation, we shall stick to the space for our discussion of frequencybased wavelets and framelets.
Lemma 4.
Let be a sequence of nonzero real numbers such that . Let and be elements in with and . Then
(2.3) 
if and only if,
(2.4) 
Proof.
By Lemma 3, we have