Wavelet Coorbit Spaces viewed as Decomposition Spaces
In this paper we show that the Fourier transform induces an isomorphism between the coorbit spaces defined by Feichtinger and Gröchenig of the mixed, weighted Lebesgue spaces with respect to the quasi-regular representation of a semi-direct product with suitably chosen dilation group , and certain decomposition spaces (essentially as introduced by Feichtinger and Gröbner) where the localized ,,parts“ of a function are measured in the -norm.
This equivalence is useful in several ways: It provides access to a Fourier-analytic understanding of wavelet coorbit spaces, and it allows to discuss coorbit spaces associated to different dilation groups in a common framework. As an illustration of these points, we include a short discussion of dilation invariance properties of coorbit spaces associated to different types of dilation groups.
There exist several methods in the literature for the construction of higher-dimensional wavelet systems. A rather general class of constructions follows the initial inception of the continuous wavelet transform in  and uses the language of group representations [25, 1, 15, 23]: Picking a suitable matrix group , the so-called dilation group, one defines the associated semidirect product . This group acts on via the (unitary) quasi-regular representation defined by
The associated continuous wavelet transform of a signal is then obtained by picking a suitable mother wavelet , and letting
A wavelet is called admissible if the operator is (a multiple of) an isometry as a map into , where denotes a left Haar measure on . By definition we thus have for admissible vectors that
alternatively expressed in the weak-sense inversion formula
An alternative, with somewhat less structure but higher design flexibility, is the semi-discrete approach described as follows: Pick a discretely labelled quadratic partition of unity in frequency domain, i.e. a family of functions satisfying
and consider the system of all translates of the inverse Fourier transforms . This system is a (continuously labelled) tight frame, i.e.
where denotes translation by . This norm equality can also be expressed in the weak-sense inversion formula
with . For compactly supported , the translation variable can be discretized as well, yielding a tight frame, and an associated unconditionally converging frame expansion for all .
First and second generation curvelets [28, 3] are special examples of this type of generalized wavelets, as well as discrete shearlet systems (see [21, Chapter 1] for an overview). In all these constructions, the desired degree of isotropy, directional selectivity, etc. in the generalized wavelet system is achieved by suitably prescribing the supports of the functions .
The similarity between the two approaches is best realized by noticing that the admissible functions in the sense of the group-theoretic wavelet transforms are characterized by the condition
for almost all , showing that the wavelet inversion formula associated to the continuous wavelet transform is also closely related to a quadratic partition of unity on the Fourier transform side, this time indexed by the dilation group .
For applications of these transforms, mathematical or otherwise, it is important to realize that each class of generalized wavelet transforms comes with a natural scale of related smoothness spaces, which are defined by norms measuring wavelet coefficient decay. In the group-related case, these are the so-called coorbit spaces introduced by Feichtinger and Gröchenig [9, 10, 11]. In the semi-discrete case, it has been realized recently that the decomposition spaces and their associated norms, as introduced by Feichtinger and Gröbner [8, 7], provide a similarly convenient framework for the treatment of approximation-theoretic properties of anisotropic (mostly shearlet-like) wavelet systems, see e.g. [2, 22].
For a long time, the prime examples of coorbit theory were provided by the modulation spaces, arising as coorbit spaces associated to the Schrödinger representation of the Heisenberg group, and the Besov spaces, which are coorbit spaces associated to the quasi-regular representation of the group (and their isotropic counterparts in higher dimensions). More recently, the introduction of shearlets (at least the group-theoretic version) triggered the systematic study of the associated coorbit spaces [4, 5]; coorbit spaces over the Blaschke group and their connection to complex analysis are discussed in . The recent papers [17, 18] pointed out that the study of wavelet coorbit spaces could be considerably extended to cover a multitude of group-theoretically defined wavelet systems in a unified approach that allows to prove the existence of easily constructed, nice wavelet systems and atomic decompositions in a large variety of settings.
However, with the introduction of ever larger classes of function spaces comes the necessity of developing conceptual tools helping to understand these spaces and the relationships between them. It is the chief aim of this paper to provide a bridge between the two types of generalized wavelet systems, by clarifying how wavelet coorbit spaces arising from a group action can be understood as decomposition spaces. There are several motives for this question. The first one is provided by pre-existing results in the literature pointing in this direction: In [9, Section 7.2] it was shown that (homogenous) Besov spaces arise as certain coorbit spaces of weighted, (mixed) Lebesgue spaces with respect to the quasi-regular representation of the group. On the other hand, these spaces can be defined by localizing the Fourier transform of using a dyadic partition of the frequency space and summing the -norms of the localized “pieces” in a certain weighted -space (cf. [19, Definition 6.5.1]).
In this paper we will show that this phenomenon is no coincidence, but merely a manifestation of the general principle that every coorbit space of a (suitably) weighted mixed Lebesgue space with respect to the quasi-regular representation of the semidirect product (with a closed subgroup ) arises as (the inverse image under the Fourier transform of) a certain decomposition space. This means that membership of in the coorbit space can be decided by localizing the Fourier transform with respect to a certain covering (called the covering induced by ) of the dual orbit and summing the -norms of the individual pieces in a suitable weighted -space.
Thus, wavelet coorbit theory becomes a branch of decomposition space theory. To some extent this was to be expected, because the structures underlying decomposition spaces – i.e., certain coverings of (subsets of) and subordinate partitions of unity – are much more flexible than the group structure of the dilation group associated to coorbit spaces. In some sense, passing from the dilation group and its associated scale of coorbit spaces to a suitable covering and its associated scale of decomposition spaces amounts to a loss of structure, as the group is replaced by a suitably chosen index set of a discrete covering. This passage is important from a technical point of view, because by (largely) discarding the dilation group, we become free to discuss coorbit spaces associated to different dilation groups in a common framework. This observation provides a second reason for studying the connection to decomposition spaces.
Possibly the most fundamental motivation for studying this connection is that it allows to discuss the approximation-theoretic properties of a wavelet system in terms of the frequency content of the different wavelets. To elaborate on this point, let us recall the well-understood case of wavelet ONB’s in dimension one: The typical vanishing moment and smoothness conditions on the wavelets can be understood as a measure of frequency concentration. Conceptually speaking, different scales of the wavelet system correspond to different frequency bands, and increasing the degrees of smoothness and vanishing moments amounts to improving the separation between the different frequency bands, which in turn allows larger classes of homogeneous Besov spaces to be characterized in terms of the wavelet coefficients with respect to a single wavelet ONB. The papers [17, 18] extend this type of reasoning to (possibly anisotropic) higher dimensional wavelet systems and their associated coorbit spaces; here the key concept was provided by the dual action and in particular the so-called “blind spot” of the wavelet transform.
However, in the study of wavelet systems in higher dimensions, the description of frequency content poses an increasingly difficult challenge: Different wavelet systems can be understood as prescribing different ways of partitioning the frequency space into (possibly oriented) “frequency bands”; we argue that their approximation-theoretic properties are describable in terms of this behaviour. It is important to note that precisely this intuition was also used in the inception of curvelets by Candés and Donoho , and the results of that paper provide further evidence that the frequency partition determines the approximation-theoretic properties. However, what is needed to systematically turn this intuition into provable theorems is a suitable language describing these partitions, and allowing to assess which properties of a partition are relevant for the approximation-theoretic properties of the corresponding wavelet systems. Our paper makes a strong case that this language is provided by the decomposition spaces introduced by Feichtinger and Gröbner in , and studied more recently in [2, 22].
To illustrate these points, we have included a discussion of dilation invariance properties of certain coorbit spaces in Section 9. Given a suitable coorbit space associated to a dilation group , we would like to identify those invertible matrices such that is invariant under dilation by . It is clear that the set of these matrices contains ; this follows from the fact that the wavelet transform intertwines (a suitably normalized) dilation by with left translation by , and from left invariance of the Banach function space entering the definition of the coorbit space. It is much less clear whether there are further invertible matrices which leave invariant. It will be seen in Section 9 that this property depends on : If is the similitude group in dimension two and the associated coorbit spaces are the isotropic Besov spaces, they are in fact invariant under arbitrary dilations. By contrast, there are shearlet coorbit spaces that are not invariant under dilation by a ninety degree rotation.
While these observations are of some independent interest (for example, the lack of rotation invariance for shearlet coorbit spaces seems to be a new observation), we have included this discussion mostly because of the way it highlights the role of the decomposition space viewpoint in understanding the different coorbit spaces. It also illustrates the importance of being able to compare coorbit spaces associated to different dilation groups: One quickly realizes that dilation by is an isomorphism , where is a suitably chosen Banach function space over the group . Thus invariance of under dilation by is equivalent to an embedding statement .
2. Notation and Preliminaries
In this paper we will always be working in the following setting: We assume that is a closed subgroup for some and we consider the semidirect product with multiplication . For the convenience of the reader we recall that a (left) Haar integral on the locally compact group is then given by
where denotes integration against left Haar measure on . The modular function on is given by
We then consider the so-called quasi-regular representation of acting unitarily on by
where we use the operators (and later on also) defined by
for , and .
We use the following version of the Fourier transform:
for the (inverse) Fourier transform of .
Using this convention, we note that on the Fourier side the quasi-regular representation is given by
There is a such that the dual orbit is an open set of full measure (i.e. , where denotes Lebesgue measure on ) and
the isotropy group of with respect to the dual action of is compact. In this case, the isotropy group is a compact subgroup of for every .
In the following, we will always assume that these conditions are met. We will then see below (cf. Theorem 9) that is indeed an integrable representation, i.e. there exists with , where the Wavelet transform of with respect to is defined by
for . It should be noted that this definition of the Wavelet transform coincides with the voice transform as defined in , because
as Feichtinger uses a scalar-product that is antilinear in the first component, i.e.
whereas we adopt the convention that the scalar-product is antilinear in the second component.
The paper is organized as follows: In section 3 we clarify the exact definitions of the mixed Lebesgue spaces and the requirements on the weight for which we will later prove the isomorphism of to a suitable decomposition space. Furthermore, we show that the coorbit theory is indeed applicable in this setting. The only point for the applicability of coorbit theory that we do not cover in section 3 is the existence of analyzing vectors.
This gap is closed in the ensuing section 4 in which we recall the most important definitions from coorbit theory and show that any Schwartz function whose Fourier transform is compactly supported in the dual orbit is admissible as a so-called analyzing vector (even as a “better vector”). Furthermore, we show that the “reservoir” from which the elements of the coorbit space are taken can naturally be identified with a subspace of the space of distributions on the dual orbit determined by , as well as with a subspace of , where we use the notation for the space of smooth functions with compact support in the open set . This notation will also be used in the remainder of the paper.
In section 5 we define the concept of an “induced covering” of the dual orbit and we give a precise definition of the decomposition spaces for which we will later show that the Fourier transform induces an isomorphism . Furthermore, we recall the essential definitions for the theory of decomposition spaces (i.e. the concepts of admissible coverings, BAPUs, etc.) and show that the induced covering is a structured admissible covering of (cf. Definition 13).
In section 6 we construct a specific partition of unity subordinate to the induced covering . In principle, one could use any partition of unity subordinate to for which is uniformly bounded, but our construction has the advantage that the localizations can be explicitly expressed in terms of the Wavelet transform of .
This explicit formula will be prominently exploited in section 7 where we prove that the Fourier transform extends to a bounded linear map for a suitable weight .
In section 8 we show that the inverse Fourier transform is continuous. In this section we also show that instead of the reservoir for the elements of the coorbit space, one can use the more invariant reservoir .
In section 9 we apply the established isomorphism between the coorbit space and the decomposition space to investigate the invariance of certain specific coorbit spaces under conjugation of the group. Using the decomposition space view, we show that all coorbit spaces with respect to the similitude group are invariant under conjugation, whereas the same is in general not true for the coorbit spaces with respect to the shearlet group.
We close the technical preliminaries by noting that while the most important results of the present paper are Theorems 37 and 43, which establish the continuity of the (inverse) Fourier transform as a map from the coorbit space into the associated decomposition space (and vice versa), the most important parts of the paper in terms of ideas for the proof are the definition of the specific partition of unity (cf. equation (6.1)) and the calculation of the localization in terms of the wavelet transform (cf. Lemma 34), as well as Lemma 41, where we show for .
3. Applicability of Coorbit-theory for the spaces
As the quasi-regular representation is irreducible and square-integrable by our standing assumptions, the main requirement for the coorbit-theory as developed by Feichtinger and Gröchenig in [10, 11] is fulfilled. In the remainder of this paper, we will routinely abuse notation by identifying weights with their trivial extension , . We will exclusively consider the coorbit spaces where only depends on the second factor.
Nevertheless, we will sometimes have occasion to consider the Banach function space , where is arbitrary measurable (but we will not consider the coorbit space in this case). This space is defined by
for and and with
The weight need not be submultiplicative itself, but we will assume that is -moderate for a (measurable, locally bounded) submultiplicative weight , i.e. we assume
In this case, is invariant under left- and right translations. More precisely, we have the following:
is invariant under left- and right translations and we have the estimates
for all .
Let and . For we have
We first consider the left translation. For we get, using the above formula
as can be seen using the change-of-variables formula for and for using the fact that and its inverse map both map null-sets to null-sets.
Thus, we arrive at
Using the (isometric) invariance of under left-translations, we obtain
Applying this to instead of , we finally see
We now turn to the right translations. Using the translation-invariance of , we derive
Now, for and (measurable) , formula (2.26) of  implies
The same is true for , as right-translations map (left) null-sets to (left) null-sets.
Therefore, we arrive at
Using the -moderateness of and boundedness of on compact sets (from below and above) one can easily establish the same properties for . These properties (as stated in the next lemma) will be frequently used in the rest of the paper.
Let be compact. There is a constant such that
Furthermore, there are (only dependent on and ) with
Additionally, we note some easy closure-properties of submultiplicative functions that will be used below:
Let be submultiplicative. Then the same holds for and as well as for .
With these preparations, we can now show that the space satisfies all requirements of coorbit theory (with the exception of the requirement which we will establish in Theorem 9 below).
For we set
Then is a solid Banach function space, is a (locally bounded, measurable) submultiplicative weight that satisfies
for all , where we have written .
Interpreting as a submultiplicative weight on , we have
for all and , where we used the notation . Here, the integral defining the convolution converges (absolutely) for almost every .
We first note that implies that the constant map is submultiplicative. Now Lemma 3 easily shows (with the multiplicativity of and ) that is submultiplicative. As and are continuous and is locally bounded and measurable, the same is true of .
We first prove inequality (3.1). To this end, we notice that for and the inequality yields the estimate
In the following we will apply this for or .
Lemma 1 together with shows
where we used .
The Banach function space properties of are routinely checked. Finally, we establish the convolution relation (3.2). Here we first observe the identity
which is valid by left invariance. Now Minkowski’s inequality for integrals (cf. [14, Theorem 6.19]) with
yields (together with the solidity of ) the estimate
In particular, we conclude for almost every (depending on ) for -almost every . Sine we have for all and because is measurable (which is implied by Fubini’s theorem), we see for almost every . Thus, the convolution-defining integral
converges absolutely for almost every with . By solidity of , this implies and
4. Admissibility of as analyzing vectors and identification of with a subspace of
In this section we show that any Schwartz function whose Fourier transform is compactly supported in the dual orbit is admissible as an analyzing vector. This will also allow us to identify the “reservoir” that is used in the definition of coorbit spaces with (a subspace of) the space of distributions on the dual orbit as well as with a subspace of .
Before we go into the details of the proof, we recall some important definitions related to coorbit theory. First of all we recall the definition of the set of analyzing vectors
and of the set of “better vectors”
from [10, pages 317 and 321].
Here, we use the notion of the so-called Wiener amalgam space for a solid Banach function space (cf. [10, pages 312 and 315]). For the definition of this space, let be an open, precompact unit-neighborhood. For we define the (right sided) control function of with respect to by
The (right sided) Wiener amalgam space with local component and global component is then defined by
with norm . Here one should note that (as long as is first countable) is a lower semicontinuous (and hence measurable) function. Then is a Banach space that is independent of the actual choice of and is continuously embedded in . These properties are shown in [26, Lemma 2.1 and Lemma 2.2 together with Theorem 2.3] for the left sided Wiener amalgam space
where the (left sided) control function of with respect to is defined by
Note that we have
Thus, one can easily derive the analogous properties for the right sided amalgam spaces.
We mention that in , Feichtinger uses continuous functions as a cut-off for localization instead of the simple characteristic function that we use. As we use as the local component, this makes no difference.
Below, we will show that any Schwartz function with Fourier transform already satisfies for every (locally bounded, submultiplicative) weight that only depends on the second component, i.e. which satisfies for . This shows in particular that is nontrivial, which closes the gap for the applicability of coorbit theory that was left open in section 3 (cf. Lemma 4).
Moreover, we show that the map
is well-defined and continuous, where is defined by
for some fixed analyzing vector with norm <