Research Article | Open Access
A Note on Directional Wavelet Transform: Distributional Boundary Values and Analytic Wavefront Sets
By using a particular class of directional wavelets (namely, the conical wavelets, which are wavelets strictly supported in a proper convex cone in the -space of frequencies), in this paper, it is shown that a tempered distribution is obtained as a finite sum of boundary values of analytic functions arising from the complexification of the translational parameter of the wavelet transform. Moreover, we show that for a given distribution , the continuous wavelet transform of with respect to a conical wavelet is defined in such a way that the directional wavelet transform of yields a function on phase space whose high-frequency singularities are precisely the elements in the analytic wavefront set of .
Wavelets, as well as the theory of distributions, have found applications in various fields of pure and applied mathematics, physics, and engineering. The requirements of modern mathematics, mathematical physics and engineering, have brought the necessity to incorporate ideas from wavelet analysis to the distribution theory, and reciprocally. Over the last decades, a number of authors have studied the relationship between wavelets and the theory of distributions for different purposes (see, e.g., [1–17] and references therein). Particularly related to this note, in [5, 7, 11], the authors investigate the singularities of tempered distributions via the wavelet transform. In , Moritoh introduced a class of wavelet transform as a continuous and microlocal version of the Littlewood-Paley decomposition and compared the wavefront sets defined by his wavelet transform and Hörmander's wavefront sets. More recently, Pilipović and Vuletić  gave some more precise information concerning the work of Moritoh. They consider a special wavelet transform of Moritoh and gave new definitions of wavefront sets of tempered distributions via that wavelet transform. The major result in  is that these wavefront sets are equal to the wavefront sets in the sense of Hörmander in the cases . If , they combined results for dimensions and characterized wavefront sets in -directions, where are presented as products of nonzero points of , , , . In the paper , Navarro introduced an analyzing wavelet in and an irreducible group action with the property that the associated wavelet transform of a tempered distribution is singular along the wavefront set of the distribution. The main result relates the notion of the wavefront set and the wavelet transform of distributions in . It should be noted that the core of the construction of Navarro (the irreducible group action on ) parallels that of Kutyniok and Labate , where they consider a different notion of wavefront set based on the concept of continuous shearlet transform.
In this short note, by using a particular class of directional wavelets (namely, the conical wavelets, which are wavelets strictly supported in a proper convex cone in the -space of frequencies [19, 20]), it is shown that a distribution is obtained as a finite sum of boundary values of analytic functions arising from the complexification of the variable corresponding to the location of the continuous wavelet transform of with respect to directional wavelet . Furthermore, we characterize the analytic wavefront set of a tempered distribution in terms of the behavior of its directional wavelet transform. The main results of this work are given in Lemma 3.4 and Theorem 3.5. In Lemma 3.4, we prove that the distributional wavelet transform with respect to the directional wavelet is an analytic function of tempered growth. By using Lemma 3.4, we show that the tempered distributions can be obtained as a finite sum of boundary values of analytic functions of tempered growth (Theorem 3.5). In Section 4, we apply Theorem 3.5 in the study of analytic wavefront set of tempered distributions. We show that, for a given distribution , the wavelet transform of with respect to a conical wavelet is defined in such a way that the directional wavelet transform of yields a function on phase space whose high-frequency singularities are precisely the elements in the analytic wavefront set of . In , Hörmander introduced the notion of the analytic wavefront set of as a subset of the cotangent space , whose projection to coincides with the analytic singular support of . His definition relies on the use of the Fourier transform of . In this note, following Nishiwada [22, 23], we present an alternative definition of in terms of generalized boundary values of analytic functions arising from the complexification of the variable corresponding to the location of the continuous wavelet transform of with respect to directional wavelet .
2. A Glance at the Wavelets: Definitions and Basic Properties
We shall recall in this section some definitions and basic properties of the wavelets. But before we establish some notation. We will use the standard multi-index notation. Let (resp. ) be the real (resp. complex) -space whose generic points are denoted by (resp. ), such that , , means , and . Moreover, we define , where is the set of nonnegative integers, such that the length of is the corresponding -norm , denotes , means , , , and
We consider two -dimensional spaces—-space and -space—with the Fourier transform defined as follows while the Fourier inversion formula is The variable will always be taken real while will also be complexified: when it is complex, it will be noted .
Definition 2.1. A wavelet is a complex-valued function in , satisfying the admissibility condition where is the Fourier transform of .
If is sufficiently regular—enough to take —then the admissibility condition above means that
Definition 2.2. The continuous wavelet transform of a distribution with respect to some analyzing wavelet is defined as the following convolution: where with as the length scale at which we analyze and as the translation parameter corresponding to the position of the analyzing wavelet .
Remark 2.3. Throughout this article the Fourier transform, the wavelet transform, and boundary values of analytic functions are always interpreted in a distributional sense.
Note that the wavelet transform may also be written as where is the Fourier transform of . With this we have the following (see [24, Lemma , page 441] for the one-dimensional space).
Lemma 2.4. For any , we have
Remark 2.5. The domain of a wavelet transform is usually the space, but the Lemma 2.4 can be extended to , which is the dual space of . In particular, see Remark 3.2, the class of conical wavelets belongs to the space . This allows us to define the directional wavelet transform of a distribution in such a way that the wavelet transform of yields a function on phase space whose high-frequency singularities are precisely the elements in the analytic wavefront set of .
3. Distributional Boundary Values
In this section, the directional wavelet transform is used to show that analytic functions which satisfy a tempered growth condition obtain distributional boundary values in . Before that, in order to define a directional wavelet, we need some terminology and simple facts concerning cones.
An open set is called a cone if (unless specified otherwise, all cones will have their vertices at zero) is invariant under positive homotheties; that is, if for all , . A cone is an open connected cone if is an open connected set. Moreover, is called convex if and proper if it contains no any straight line (observe that if is a cone, then is proper if and only if and implie ). A cone is called compact in —we write —if the projection , where is the unit sphere in . Being given a cone in -space, we associate with a closed convex cone in space which is the set . The cone is called the dual cone of .
Definition 3.1. A wavelet is said to be directional if the effective support of its Fourier transform is contained in a convex and proper cone in the -space of frequencies, with vertex at the origin, or a finite union of disjoint such cones; in that case, one will usually call multidirectional.
Remark 3.2. According to , for us to reach a genuinely directional wavelet , it suffices to consider a smooth function with support in the strictly convex cone in the space of frequencies having an arbitrary large number of vanishing moments on the boundary of the supporting cone and behaving inside as , where and denotes a polynomial in variables. In this case, the resulting directional wavelet is also called conical. We also observe that, for all , the exponential , considered as a function of , belongs to the space of rapidly decreasing functions . Then, too. Since the inverse Fourier transform is a topological isomorphism of onto , it follows that directional wavelet .
In particular, from the above remark, it follows that for a directional wavelet (2.8) can be rewritten as where we have introduced the complex variable . Remember that here corresponds to the location of the continuous wavelet transform of with respect to . Consequently, it is clear from (3.1) that the directional wavelet transform, , of a distribution is naturally extensible as an analytic function in a domain in , for each arbitrary but fixed . Still, since is a regular set [25, pages 98, 99], it follows that .
Let be an open subset in . Then we shall denote by the space of analytic functions in . Let be a proper open convex cone, and let be an arbitrary compact cone contained in . Denote by the subset of consisting of all elements whose imaginary parts lie in . is referred to as a tube domain. We will deal with tubes defined as the set of all points such that where is an arbitrary number.
Definition 3.3. Let be a complex neighbourhood of . An analytic function is said to be of tempered growth if there are an integer and a constant depending on such that for all point in .
Lemma 3.4. For an arbitrary but fixed scale , assume that the function , arising from the complexification of the variable corresponding to the location of the continuous wavelet transform of with respect to directional wavelet , is analytic in . Then is of tempered growth as a function of .
Proof. We start considering the formula
By Theorem 7.13 of , since is a polynomial and a tempered distribution, then is also tempered. Note that, we can write
where is homogeneous of degree and when . It follows that for some constant , we have that . This implies that . Still, the character tempered of implies that there exist an integer and a constant such that satisfies the estimate
Using the binomial theorem, the above estimate can be rewritten as
Now, let be a cone, such that . Then there exists so that , for all and for all . Hence for ,
Following Schwartz [25, Proposition 32, page 39], we get the following: where is the area of the unit sphere in .
Now, let be an open cone of the form , , where each is an proper open convex cone. If we write , we mean that with . Furthermore, we define by the dual cones of , such that the dual cones , , have the properties that are sets of Lebesgue measure zero. Moreover, assume that can be written as , where denotes the characteristic function of , . We shall consider the asymptotic property of as for .
Theorem 3.5. Let . Then can be expressed as a finite sum where each , arising from the complexification of the variable corresponding to the location of the continuous wavelet transform of with respect to directional wavelet , is analytic in and of tempered growth, and where denotes the boundary value in .
Proof. The proof that each is of tempered growth is obtained by the similar way as in Lemma 3.4. Let . Choose now and write (this defines a half-line for ). Note that with in (3.1), then when . Thus, we have Hence, using the linearity of and the assumptions (3.11), we obtain that Thus, the limit of each as exists in ; that is, admits the distributional boundary value in the sense of weak convergence. But from [27, Corollary 1, page 358], the latter implies strong convergence since is Montel.
4. Analytic Wavefront Set
From what we have seen in the previous section, a distribution is obtained as a finite sum of boundary values of analytic functions , with , arising from the complexification of the variable corresponding to the location of the continuous wavelet transform of with respect to directional wavelet , and where a tempered growth condition was described to characterize such boundary values. We now translate growth condition in terms of the analytic wavefront set.
Definition 4.1. Let , such that , where denotes the strong boundary value in of an analytic function arising from the complexification of the variable corresponding to the location of the continuous wavelet transform of with respect to directional wavelet . Let . Then, if and only if there exist and for which we have the estimate is called analytic wavefront set of .
Proposition 4.2. Let be the boundary value, in the distributional sense, of a function analytic in , arising from the complexification of the variable corresponding to the location of the continuous wavelet transform of with respect to directional wavelet and which satisfies the estimate (3.3). Then near the fibre is contained in .
Proof. Let be the fibre over . Let be a finite covering of closed properly convex cones of . Decompose as follows: where denotes the characteristic function of , . Then, by Theorem 3.5, the decomposition (4.2) will induce a representation of in the form of a sum of boundary values of functions , such that in the strong topology of as , . According to Lemma 3.4, the family of functions satisfies the following estimate: unless for and , with . Then, the cones of “bad” directions responsible for the singularities of these boundary values are contained in the dual cones of the base cones. So we have the inclusion Then, by making a refinement of the covering and shrinking it to , we obtain the desired result.
Remark 4.3. It is remarked that in  the fiber over , , is completely characterized by sequences of type , where is a bounded sequence in which is equal to 1 in a common neighborhood of and satisfies the following estimate: For the existence of such functions, we refer to Lemma 2.2 in .
Proposition 4.4. Let and , where . Then if and only if there exists a finite family of proper open convex cones in , a complex neighborhood of in and a decomposition of with each being of tempered growth and analytic near to for every satisfying , and where denotes the boundary value, in the distributional sense, of analytic functions arising from the complexification of the variable corresponding to the location of the continuous wavelet transform of with respect to directional wavelet .
Note that the above proposition shows that a decomposition of a tempered distribution into a sum of boundary values of analytic functions is equivalent to a decomposition of analytic wavefront set of since the fibre is contained in . Moreover, the decomposition (4.6) is carried out in the space of functions, provided that is .
Finally, we recall that in  Hörmander defined the wavefront set, , for a distribution as the set of points in the cotangent space which must be characteristic for every pseudodifferential operator such that . It is clear that . Following Nishiwada  another characterization of can be obtained based on the above results.
Proposition 4.5. Let be an open set in , and . Then if there exists a finite family of proper open convex cones in , with , a complex neighborhood of and a decomposition of near with being of tempered growth, such that near for every with .
Proof. The proof is similar to the proof of the first part of Theorem 3.4 in .
The authors are specially indebted to the referees for valuable comments, suggestions, and constructive criticisms, which improved the presentation of the results. F. A. Apolonio and F. N. Fagundes are supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) agency.
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