# A Note on Directional Wavelet Transform: Distributional Boundary Values and Analytic Wavefront Sets

Authors: Felipe A. Apolonio; Daniel H. T. Franco; Fábio N. Fagundes
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

By using a particular class of directional wavelets (namely, the conical wavelets, which are wavelets strictly supported in a proper convex cone in thek-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 f∈𝒮′(ℝn), the continuous wavelet transform of f with respect to a conical wavelet is defined in such a way that the directional wavelet transform of f yields a function on phase space whose high-frequency singularities are precisely the elements in the analytic wavefront set of f.


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## Body

## 1. Introduction

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 [5], 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 [11] 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 [11] is that these wavefront sets are equal to the wavefront sets in the sense of Hörmander in the cases n=1,2,4,8. If n∈ℕ∖{1,2,4,8}, they combined results for dimensions n=1,2,4,8 and characterized wavefront sets in ξ-directions, where ξ are presented as products of nonzero points of ℝn1,…,ℝns, n1+⋯+ns=n, ni∈{1,2,4,8}, i=1,…,s. In the paper [7], Navarro introduced an analyzing wavelet in 𝒮(ℝ2) 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 𝒮′(ℝ2). It should be noted that the core of the construction of Navarro (the irreducible group action on L2(ℝ2)) parallels that of Kutyniok and Labate [18], 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 thek-space of frequencies [19, 20]), it is shown that a distribution f∈𝒮′(ℝn) is obtained as a finite sum of boundary values of analytic functions arising from the complexification of the variable x corresponding to the location of the continuous wavelet transform of f with respect to directional wavelet ψ. 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 f∈𝒮′(ℝn), the wavelet transform of f with respect to a conical wavelet is defined in such a way that the directional wavelet transform of f yields a function on phase space whose high-frequency singularities are precisely the elements in the analytic wavefront set of f. In [21], Hörmander introduced the notion of the analytic wavefront set WFa(f) of f as a subset of the cotangent space T*(X)∖0, whose projection to X coincides with the analytic singular support of f. His definition relies on the use of the Fourier transform of f. In this note, following Nishiwada [22, 23], we present an alternative definition of WFa(f) in terms of generalized boundary values of analytic functions arising from the complexification of the variable x corresponding to the location of the continuous wavelet transform of f with respect to directional wavelet ψ.


## 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. Letℝn (resp. ℂn) be the real (resp. complex) n-space whose generic points are denoted by x=(x1,…,xn) (resp. z=(z1,…,zn)), such that x+y=(x1+y1,…,xn+yn), λx=(λx1,…,λxn), x≥0 means x1≥0,…,xn≥0, 〈x,y〉=x1y1+⋯+xnyn and  |x|=|x1|+⋯+|xn|. Moreover, we define α=(α1,…,αn)∈ℕon, where ℕo is the set of nonnegative integers, such that the length of α is the corresponding ℓ1-norm |α|=α1+⋯+αn, α+β denotes (α1+β1,…,αn+βn), α≥β means (α1≥β1,…,αn≥βn), α!=α1!…αn!, xα=x1α1…xnαn, and
(2.1)Dαφ(x)=∂|α|φ(x1,…,xn)∂x1α1∂x2α2,…,∂xnαn.We consider twon-dimensional spaces—x-space and k-space—with the Fourier transform defined as follows
(2.2)f^(k)=F[f(x)](k)=∫Rndnx‍f(x)e-i〈k,x〉,
while the Fourier inversion formula is
(2.3)f(x)=F-1[f^(k)](x)=1(2π)n∫Rn‍dnkf^(k)ei〈k,x〉.
The variable k will always be taken real while x will also be complexified: when it is complex, it will be noted z=x+iy.Definition 2.1.
A wavelet is a complex-valued functionψ in L2(ℝn,dnx), satisfying the admissibility condition
(2.4)Cψ=(2π)n∫‍dnk|k|n|ψ^(k)|2<∞,
where ψ^ is the Fourier transform of ψ.Ifψ is sufficiently regular—enough to take ψ∈L1(ℝn,dnx)∩L2(ℝn,dnx)—then the admissibility condition above means that
(2.5)ψ^(0)=0⟺∫dnxψ(x)=0.Definition 2.2.
The continuous wavelet transform of a distributionf(x)∈𝒮′(ℝn) with respect to some analyzing wavelet ψ is defined as the following convolution:
(2.6)Wψf(x′,l)=〈f(x),ψx′,l(x)¯〉=∫dnxψx′,l(x)¯f(x),
where
(2.7)ψx′,l(x)=1lnψ(l-1(x-x′)),
with ℓ∈ℝ+ as the length scale at which we analyze f(x) and x′∈ℝn as the translation parameter corresponding to the position of the analyzing wavelet ψ.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(2.8)Wψf(x′,l)=1(2π)n∫dnkWψf^(k,l)ei〈k,x′〉,
where
(2.9)Wψf^(k,l)=ψ^(lk)f^(k),
is the Fourier transform of 𝕎ψf. With this we have the following (see [24, Lemma 8.2.6, page 441] for the one-dimensional space).Lemma 2.4.
For anyf∈L2(ℝn), we have
(2.10)F[Wψf(x′,l)](k,l)=∫RndnxWψf(x′,l)e-i〈k,x′〉=ψ^(lk)f^(k).Remark 2.5.
The domain of a wavelet transform is usually theL2 space, but the Lemma 2.4 can be extended to 𝒮′(ℝn), which is the dual space of 𝒮(ℝn). In particular, see Remark 3.2, the class of conical wavelets belongs to the space 𝒮(ℝn). This allows us to define the directional wavelet transform of a distribution f∈𝒮′(ℝn) in such a way that the wavelet transform of f yields a function on phase space whose high-frequency singularities are precisely the elements in the analytic wavefront set of f.


## 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 in𝒮′(ℝn). Before that, in order to define a directional wavelet, we need some terminology and simple facts concerning cones.An open setC⊂ℝn is called a cone if C (unless specified otherwise, all cones will have their vertices at zero) is invariant under positive homotheties; that is, if for all λ>0, λC⊂C. A cone C is an open connected cone if C is an open connected set. Moreover, C is called convex if C+C⊂C and proper if it contains no any straight line (observe that if C is a cone, then C is proper if and only if x∈C and x≠0 implie -x∉C). A cone C′ is called compact in C—we write C′⋐C—if the projection prC¯′≝C¯′∩Sn-1⊂prC≝C∩Sn-1, where Sn-1 is the unit sphere in ℝn. Being given a cone C in x-space, we associate with C a closed convex cone C* in k space which is the set C*={k∈ℝn∣〈k,x〉≥0,∀x∈C}. The cone C* is called the dual cone of C.Definition 3.1.
A waveletψ(x) is said to be directional if the effective support of its Fourier transform ψ^(k) is contained in a convex and proper cone in the k-space of frequencies, with vertex at the origin, or a finite union of disjoint such cones; in that case, one will usually call ψ multidirectional.Remark 3.2.
According to [19], for us to reach a genuinely directional wavelet ψ(x), it suffices to consider a smooth function ψ^(k) with support in the strictly convex cone C* in the k space of frequencies having an arbitrary large number of vanishing moments on the boundaryof the supporting cone and behaving inside C* as P(k1,…,kn)e-〈k,y〉, where y∈Cand  P(·) denotes a polynomial in n variables. In this case, the resulting directional wavelet is also called conical. We also observe that, for all y∈C, the exponential e-〈k,y〉, considered as a function of k∈C*, belongs to the space of rapidly decreasing functions 𝒮(ℝn). Then, P(k1,…,kn)e-〈k,y〉∈𝒮(ℝn) too. Since the inverse Fourier transform is a topological isomorphism of 𝒮(ℝn) onto 𝒮(ℝn), it follows that directional wavelet ψ(x)∈𝒮(ℝn).In particular, from the above remark, it follows that for a directional wavelet (2.8) can be rewritten as
(3.1)Wψf(z,l)=1(2π)n∫dnkP(lk)f^(k)ei〈k,z〉,
where we have introduced the complex variable z=x+iℓy∈ℝn+iC. Remember that here x corresponds to the location of the continuous wavelet transform of f with respect to ψ. Consequently, it is clear from (3.1) that the directional wavelet transform, 𝕎ψf(x,ℓ), of a distribution f∈𝒮′(ℝn) is naturally extensible as an analytic function in a domain in ℂn, for each arbitrary but fixed ℓ>0. Still, since C* is a regular set [25, pages 98, 99], it follows that supp(𝕎ψf^)=supp(ψ^)⊆C*.LetΩ be an open subset in ℂn. Then we shall denote by 𝒪(Ω) the space of analytic functions in Ω. Let C be a proper open convex cone, and let C′ be an arbitrary compact cone contained in C. Denote by T(C′) the subset of ℂn consisting of all elements whose imaginary parts lie in C′. T(C′) is referred to as a tube domain. We will deal with tubes defined as the set of all points z∈ℂn such that
(3.2)T(C′)={z=x+iy∈Cn∣x∈Rn,y∈C′,|y|<δ},
where δ>0 is an arbitrary number.Definition 3.3.
LetZ be a complex neighbourhood of ℝn. An analytic function f(z)∈𝒪(Z∩T(C′)) is said to be of tempered growth if there are an integer α and a constant M depending on C′ such that
(3.3)|f(x+iy)|≤M(C′)|y|-α,
for all point z=x+iy in Z∩T(C′).Lemma 3.4.
For an arbitrary but fixed scaleℓ∈ℝ+, assume that the function 𝕎ψf(z,ℓ), arising from the complexification of the variable x corresponding to the location of the continuous wavelet transform of f∈𝒮′(ℝn) with respect to directional wavelet ψ, is analytic in Z∩T(C′). Then 𝕎ψf(z,ℓ) is of tempered growth as a function of z.Proof.
We start considering the formula(3.4)Wψf(x+ily,l)=1(2π)n∫C*dnkP(lk)f^(k)ei〈k,z〉.
By Theorem 7.13 of [26], since P(ℓk) is a polynomial and f^(k) a tempered distribution, then P(ℓk)f^(k) is also tempered. Note that, we can write
(3.5)P(lk)=Pm(lk)+Pm-1(lk)+⋯+P0,
where Pj is homogeneous of degree j and Pm(ℓk)≠0 when k≠0. It follows that for some constant M, we have that |P(ℓk)|≤Mℓm|P(k)|. This implies that |P(ℓk)f^(k)|≤Mℓm|P(k)f^(k)|. Still, the character tempered of P(k)f^(k) implies that there exist an integer N and a constant M1 such that P(k)f^(k) satisfies the estimate
(3.6)|P(k)f^(k)|≤M1(1+|k|)N.
Hence,
(3.7)|P(lk)f^(k)|≤lmM2(1+|k|)N.
Using the binomial theorem, the above estimate can be rewritten as(3.8)|P(lk)f^(k)|≤lmM2∑j=0N‍cj|k|j.
Now, letC′ be a cone, such that C′⋐C. Then there exists c>0 so that 〈k,ℓy〉≥c|k∥ℓy|, for all k∈C* and for all y∈C′. Hence for x+iℓy∈ℝn+iC′,
(3.9)|Wψf(x+ily,l)|≤lm(2π)n∫C*dnk|P(k)f^(k)|e-〈k,ly〉≤lmM2(2π)n∑j=0Ncj‍∫C*dnk|k|je-c|k||ly|.
Following Schwartz [25, Proposition 32, page 39], we get the following:
(3.10)|Wψf(x+ily,l)|≤lmM2(2π)n∑j=0Ncjσn-1‍∫0∞dttn+j-1e-c|ly|t≤M3(C′)|ly|-(n+N),
where σn-1 is the area of the unit sphere in ℝn.Now, letC be an open cone of the form C=∪j=1mCj, m<∞, where each Cj is an proper open convex cone. If we write C′⋐C, we mean that C′=∪j=1mCj′ with Cj′⋐Cj. Furthermore, we define by Cj*={k∈ℝn∣〈k,x〉≥0,∀x∈Cj} the dual cones of Cj, such that the dual cones Cj*, j=1,…,m, have the properties
(3.11)Rn∖(∪j=1mCj*),Cj*∩Ck*,j≠k,j,k=1,…,m,
that are sets of Lebesgue measure zero. Moreover, assume that P(ℓk)f^(k) can be written as P(ℓk)f^(k)=∑j=1m‍λj(k)P(ℓk)f^(k), where λj(k) denotes the characteristic function of Cj*, j=1,…,m. We shall consider the asymptotic property of 𝕎ψf(z,ℓ) as ℓ→0 for z=x+iℓy∈Z∩T(C′).Theorem 3.5.
Letf∈𝒮′(ℝn). Then f can be expressed as a finite sum
(3.12)f=∑j=1m‍bCj′(Wψfj(z,l)),
where each 𝕎ψfj(z,ℓ), arising from the complexification of the variable x corresponding to the location of the continuous wavelet transform of fj∈𝒮′(ℝn) with respect to directional wavelet ψ, is analytic in Z∩T(Cj′) and of tempered growth, and where bCj′(𝕎ψfj(z,ℓ)) denotes the boundary value in 𝒮′(ℝn).Proof.
The proof that each𝕎ψfj(z,ℓ) is of tempered growth is obtained by the similar way as in Lemma 3.4. Let φ∈𝒮(ℝn). Choose now y0∈Cj′ and write y=ℓy0 (this defines a half-line for 0≠y0∈Cj′). Note that with y=ℓy0 in (3.1), then ℓ→0 when y→0. Thus, we have
(3.13)〈bCj′(Wψfj),φ〉=lim⁡y→0∫RndnxWψfj(x+iy,l)φ(x)=lim⁡y→0∫Rndnx(1(2π)n∫Cj*dnkλj(k)P(lk)f^(k)ei〈k,x+iy〉)φ(x)=lim⁡y→01(2π)n∫Cj*dnkλj(k)P(lk)f^(k)φ^(-k)e-〈k,y〉=1(2π)n∫Cj*dnkλj(k)f^(k)φ^(-k)=〈fj,φ〉.
Hence, using the linearity of f∈𝒮′(ℝn) and the assumptions (3.11), we obtain that
(3.14)〈f,φ〉=∑j=1m‍〈fj,φ〉=∑j=1m‍〈bCj′(Wψfj),φ〉.
Thus, the limit of each 𝕎ψfj(z,ℓ) as Cj′∋y→0 exists in 𝒮′(ℝn); that is, f admits the distributional boundary value ∑j=1m‍bCj′(𝕎ψfj) in the sense of weak convergence. But from [27, Corollary 1, page 358], the latter implies strong convergence since 𝒮(ℝn) is Montel.


## 4. Analytic Wavefront Set

From what we have seen in the previous section, a distributionf∈𝒮′(ℝn) is obtained as a finite sum of boundary values of analytic functions 𝕎ψfj(z,ℓ), with j=(1,…,m), arising from the complexification of the variable x corresponding to the location of the continuous wavelet transform of fj with respect to directional wavelet ψ, 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.
Letf∈𝒮′(ℝn), such that f=bC(𝕎ψf(z,ℓ)), where bC(𝕎ψf(z,ℓ)) denotes the strong boundary value in 𝒮′(ℝn) of an analytic function 𝕎ψf(z,ℓ) arising from the complexification of the variable x corresponding to the location of the continuous wavelet transform of f with respect to directional wavelet ψ. Let q=(x0,k0). Then, q∉WFa(f) if and only if there exist M(C′) and N for which we have the estimate
(4.1)|Wψf(z,l)|≤M(C′)|ly|-N,z=x+ily∈Z∩T(C′).WFa(f) is called analytic wavefront set of f.Proposition 4.2.
Letf∈𝒮′(ℝn) be the boundary value, in the distributional sense, of a function 𝕎ψf(z,ℓ) analytic in Z∩T(C′), arising from the complexification of the variable x corresponding to the location of the continuous wavelet transform of f with respect to directional wavelet ψ and which satisfies the estimate (3.3). Then near x0 the fibre WFa(f)|x0 is contained in C*.Proof.
LetWFa(f)|x0={k∈(ℝn∖0)∣(x0,k)∈WFa(f)} be the fibre over x0. Let {Cj*}j∈L be a finite covering of closed properly convex cones of C*. Decompose P(ℓk)f^(k) as follows:
(4.2)P(lk)f^(k)=∑j=1m‍λj(k)P(lk)f^(k),
where λj(k) denotes the characteristic function of Cj*, j∈L. Then, by Theorem 3.5, the decomposition (4.2) will induce a representation of f in the form of a sum of boundary values of functions 𝕎ψfj(z,ℓ), such that 𝕎ψfj(z,ℓ)→fj in the strong topology of 𝒮′(ℝn) as y→0, y∈Cj′⋐Cj. According to Lemma 3.4, the family of functions 𝕎ψfj(z,ℓ) satisfies the following estimate:
(4.3)|Wψfj(z,l)|≤M(C′)|ly|-N,z=x+ily∈Z∩T(C′),
unless 〈k,Y〉<0 for k∈Cj* and Y∈-Cj′, with |Y|<δ. 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
(4.4)WFa(f)⊂Rn×(∪jCj*).
Then, by making a refinement of the covering and shrinking it to C*, we obtain the desired result.Remark 4.3.
It is remarked that in [21] the fiber over x0, WFa(f)|x0, is completely characterized by sequences of type fN=ϕNf, where {ϕN} is a bounded sequence in C0∞(X) which is equal to 1 in a common neighborhood of x0 and satisfies the following estimate:
(4.5)|Dα+βϕN|≤Cα(CN)|β|,if|β|≤N.
For the existence of such functions, we refer to Lemma 2.2 in [21].We can meet Definition4.1 and Proposition 4.2 in the following proposition.Proposition 4.4.
Letf∈𝒮′(ℝn) and (x0,k0)∈T*(ℝn)∖0, where T*(ℝn)∖0:=ℝn×(ℝn∖0). Then (x0,k0)∉WFa(f) if and only if there exists a finite family {Cj} of proper open convex cones in ℝn, a complex neighborhood Z of x0 in ℂn and a decomposition of f(4.6)f=∑j=1m‍bCj′(Wψfj(z,l)),
with each 𝕎ψfj(z,ℓ)∈𝒪(Z∩T(Cj′)) being of tempered growth and analytic near to x0 for every j satisfying Cj′⊂{y∣〈k0,y〉≥0}, and where bCj′(𝕎ψfj(z,ℓ)) denotes the boundary value, in the distributional sense, of analytic functions 𝕎ψfj(z,ℓ) arising from the complexification of the variable x corresponding to the location of the continuous wavelet transform of f with respect to directional wavelet ψ.Note that the above proposition shows that a decomposition of a tempered distributionf into a sum of boundary values of analytic functions is equivalent to a decomposition of analytic wavefront set of f since the fibre WFa(f)|x0 is contained in ∪jCj*. Moreover, the decomposition (4.6) is carried out in the space of C∞ functions, provided that f is C∞.Finally, we recall that in [28] Hörmander defined the wavefront set, WF(f), for a distribution as the set of points in the cotangent space which must be characteristic for every pseudodifferential operator P such that Pf∈C∞. It is clear that WF(f)⊂WFa(f). Following Nishiwada [22] another characterization of WF(f) can be obtained based on the above results.Proposition 4.5.
LetU be an open set in ℝn, (x0,k0)∈T*(U)∖0 and f∈𝒮′(U). Then (x0,k0)∉WF(f) if there exists a finite family {Cj} of proper open convex cones in ℝn, with j=1,…,m, a complex neighborhood Z of x0 and a decomposition of f near x0(4.7)f=∑j=1mbCj′‍(Wψfj(z,l))inU,
with 𝕎ψfj(z,ℓ)∈𝒪(Z∩T(Cj′)) being of tempered growth, such that bCj′(𝕎ψfj(z,ℓ))∈C∞ near x0 for every j with Cj′⊂{y∣〈k,ℓy〉≥0}.Proof.
The proof is similar to the proof of the first part of Theorem 3.4 in [23].



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