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

Abstract

By using a particular class of directional wavelets (namely, the conical wavelets, which are wavelets strictly supported in a proper convex cone in the 𝑘-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^′(ℝ^𝑛), 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.

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 ℝ^𝑛1,…,ℝ^𝑛𝑠, 𝑛₁+⋯+𝑛_𝑠=𝑛, 𝑛_𝑖∈{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 the 𝑘-space of frequencies [19, 20]), it is shown that a distribution 𝑓∈𝒮^′(ℝ^𝑛) 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^′(ℝ^𝑛), 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 𝑊𝐹_𝑎(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 𝑊𝐹_𝑎(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. Let ℝ^𝑛 (resp. ℂ^𝑛) be the real (resp. complex) 𝑛-space whose generic points are denoted by 𝑥=(𝑥₁,…,𝑥_𝑛) (resp. 𝑧=(𝑧₁,…,𝑧_𝑛)), such that 𝑥+𝑦=(𝑥₁+𝑦₁,…,𝑥_𝑛+𝑦_𝑛), 𝜆𝑥=(𝜆𝑥₁,…,𝜆𝑥_𝑛), 𝑥≥0 means 𝑥₁≥0,…,𝑥_𝑛≥0, ⟨𝑥,𝑦⟩=𝑥₁𝑦₁+⋯+𝑥_𝑛𝑦_𝑛 and |𝑥|=|𝑥₁|+⋯+|𝑥_𝑛|. Moreover, we define 𝛼=(𝛼₁,…,𝛼_𝑛)∈ℕ^𝑛_𝑜, where ℕ_𝑜 is the set of nonnegative integers, such that the length of 𝛼 is the corresponding ℓ¹-norm |𝛼|=𝛼₁+⋯+𝛼_𝑛, 𝛼+𝛽 denotes (𝛼₁+𝛽₁,…,𝛼_𝑛+𝛽_𝑛), 𝛼≥𝛽 means (𝛼₁≥𝛽₁,…,𝛼_𝑛≥𝛽_𝑛), 𝛼!=𝛼₁!…𝛼_𝑛!, 𝑥^𝛼=𝑥^𝛼1₁…𝑥^𝛼𝑛_𝑛, and 𝐷^𝛼𝜑(𝑥)=𝜕^|𝛼|𝜑𝑥₁,…,𝑥_𝑛𝜕𝑥^𝛼1₁𝜕𝑥^𝛼2₂,…,𝜕𝑥^𝛼𝑛_𝑛. (2.1)

We consider two 𝑛-dimensional spaces—𝑥-space and 𝑘-space—with the Fourier transform defined as follows 𝑓(𝑘)=ℱ𝑓(𝑥)(𝑘)=_ℝ𝑛𝑑^𝑛𝑥𝑓(𝑥)𝑒^{−𝑖⟨𝑘,𝑥⟩}, (2.2) while the Fourier inversion formula is 𝑓(𝑥)=ℱ⁻¹𝑓(𝑘𝑥)=1)((2𝜋)^𝑛_ℝ^𝑛𝑑^𝑛𝑘𝑓(𝑘)𝑒^{𝑖⟨𝑘,𝑥⟩}. (2.3) 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 function 𝜓 in 𝐿²(ℝ^𝑛,𝑑^𝑛𝑥), satisfying the admissibility condition 𝐶_𝜓=(2𝜋)^𝑛𝑑^𝑛𝑘|𝑘|^𝑛𝜓(𝑘)²<∞, (2.4) where 𝜓 is the Fourier transform of 𝜓.

If 𝜓 is sufficiently regular—enough to take 𝜓∈𝐿¹(ℝ^𝑛,𝑑^𝑛𝑥)∩𝐿²(ℝ^𝑛,𝑑^𝑛𝑥)—then the admissibility condition above means that 𝜓(0)=0⟺𝑑^𝑛𝑥𝜓(𝑥)=0. (2.5)

Definition 2.2. The continuous wavelet transform of a distribution 𝑓(𝑥)∈𝒮^′(ℝ^𝑛) with respect to some analyzing wavelet 𝜓 is defined as the following convolution: 𝕎_𝜓𝑓𝑥^′,ℓ=𝑓(𝑥),𝜓_𝑥′,ℓ(𝑥)=𝑑^𝑛𝑥𝜓_𝑥′,ℓ(𝑥)𝑓(𝑥), (2.6) where 𝜓_𝑥′,ℓ(𝑥)=1ℓ^𝑛𝜓ℓ⁻¹𝑥−𝑥^′, (2.7) with ℓ∈ℝ₊ as the length scale at which we analyze 𝑓(𝑥) and 𝑥^′∈ℝ^𝑛 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 𝕎_𝜓𝑓𝑥^′=1,ℓ(2𝜋)^𝑛𝑑^𝑛𝑘𝕎_𝜓𝑓(𝑘,ℓ)𝑒^{𝑖⟨𝑘,𝑥′⟩}, (2.8) where 𝕎_𝜓𝑓(𝑘,ℓ)=𝜓(ℓ𝑘)𝑓(𝑘), (2.9) is the Fourier transform of 𝕎_𝜓𝑓. With this we have the following (see [24, Lemma 8.2.6, page 441] for the one-dimensional space).

Lemma 2.4. For any 𝑓∈𝐿²(ℝ^𝑛), we have ℱ𝕎_𝜓𝑓𝑥^′,ℓ(𝑘,ℓ)=_ℝ𝑛𝑑^𝑛𝑥𝕎_𝜓𝑓𝑥^′,ℓ𝑒^{−𝑖⟨𝑘,𝑥′⟩}=𝜓(ℓ𝑘)𝑓(𝑘). (2.10)

Remark 2.5. The domain of a wavelet transform is usually the 𝐿² space, but the Lemma 2.4 can be extended to 𝒮^′(ℝ^𝑛), which is the dual space of 𝒮(ℝ^𝑛). In particular, see Remark 3.2, the class of conical wavelets belongs to the space 𝒮(ℝ^𝑛). This allows us to define the directional wavelet transform of a distribution 𝑓∈𝒮^′(ℝ^𝑛) 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 in 𝒮^′(ℝ^𝑛). Before that, in order to define a directional wavelet, we need some terminology and simple facts concerning cones.

An open set 𝐶⊂ℝ^𝑛 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=^′def𝐶^′∩𝑆^𝑛−1⊂pr𝐶=^def𝐶∩𝑆^𝑛−1, where 𝑆^𝑛−1 is the unit sphere in ℝ^𝑛. 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∈ℝ^𝑛∣⟨𝑘,𝑥⟩≥0,∀𝑥∈𝐶}. The cone 𝐶^∗ is called the dual cone of 𝐶.

Definition 3.1. A wavelet 𝜓(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 boundary of the supporting cone and behaving inside 𝐶^∗ as 𝑃(𝑘₁,…,𝑘_𝑛)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(ℝ^𝑛). Then, 𝑃(𝑘₁,…,𝑘_𝑛)𝑒^{−⟨𝑘,𝑦⟩}∈𝒮(ℝ^𝑛) too. Since the inverse Fourier transform is a topological isomorphism of 𝒮(ℝ^𝑛) onto 𝒮(ℝ^𝑛), it follows that directional wavelet 𝜓(𝑥)∈𝒮(ℝ^𝑛).

In particular, from the above remark, it follows that for a directional wavelet (2.8) can be rewritten as 𝕎_𝜓𝑓()=1𝑧,ℓ(2𝜋)^𝑛𝑑^𝑛𝑘𝑃(ℓ𝑘)𝑓(𝑘)𝑒^{𝑖⟨𝑘,𝑧⟩}, (3.1) where we have introduced the complex variable 𝑧=𝑥+𝑖ℓ𝑦∈ℝ^𝑛+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, 𝕎_𝜓𝑓(𝑥,ℓ), of a distribution 𝑓∈𝒮^′(ℝ^𝑛) is naturally extensible as an analytic function in a domain in ℂ^𝑛, for each arbitrary but fixed ℓ>0. Still, since 𝐶^∗ is a regular set [25, pages 98, 99], it follows that supp(𝕎_𝜓𝑓)=supp(𝜓)⊆𝐶^∗.

Let Ω be an open subset in ℂ^𝑛. 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 ℂ^𝑛 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∈ℂ^𝑛 such that 𝑇𝐶^′=𝑧=𝑥+𝑖𝑦∈ℂ^𝑛∣𝑥∈ℝ^𝑛,𝑦∈𝐶^′,𝑦<𝛿, (3.2) where 𝛿>0 is an arbitrary number.

Definition 3.3. Let 𝑍 be a complex neighbourhood of ℝ^𝑛. 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 𝑓𝑥+𝑖𝑦≤𝐌𝐶^′𝑦^−𝛼, (3.3) for all point 𝑧=𝑥+𝑖𝑦 in 𝑍∩𝑇(𝐶^′).

Lemma 3.4. For an arbitrary but fixed scale ℓ∈ℝ₊, assume that the function 𝕎_𝜓𝑓(𝑧,ℓ), arising from the complexification of the variable 𝑥 corresponding to the location of the continuous wavelet transform of 𝑓∈𝒮^′(ℝ^𝑛) with respect to directional wavelet 𝜓, is analytic in 𝑍∩𝑇(𝐶^′). Then 𝕎_𝜓𝑓(𝑧,ℓ) is of tempered growth as a function of 𝑧.

Proof. We start considering the formula 𝕎_𝜓𝑓=1𝑥+𝑖ℓ𝑦,ℓ(2𝜋)^𝑛_𝐶^∗𝑑^𝑛𝑘𝑃(ℓ𝑘)𝑓(𝑘)𝑒^{𝑖⟨𝑘,𝑧⟩}. (3.4) 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 𝑃(ℓ𝑘)=𝑃_𝑚(ℓ𝑘)+𝑃_𝑚−1(ℓ𝑘)+⋯+𝑃₀, (3.5) where 𝑃_𝑗 is homogeneous of degree 𝑗 and 𝑃_𝑚(ℓ𝑘)≠0 when 𝑘≠0. It follows that for some constant 𝐌, we have that |𝑃(ℓ𝑘)|≤𝐌ℓ^𝑚|𝑃(𝑘)|. This implies that |𝑃(ℓ𝑘)𝑓(𝑘)|≤𝐌ℓ^𝑚|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 𝑃(𝑘)𝑓(𝑘)≤𝐌₁(1+|𝑘|)^𝑁. (3.6) Hence, 𝑃(ℓ𝑘)𝑓(𝑘)≤ℓ^𝑚𝐌₂(1+|𝑘|)^𝑁. (3.7)
Using the binomial theorem, the above estimate can be rewritten as 𝑃(ℓ𝑘)𝑓(𝑘)≤ℓ^𝑚𝐌₂^𝑁_𝑗=0𝑐_𝑗|𝑘|^𝑗. (3.8)
Now, let 𝐶^′ 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∈ℝ^𝑛+𝑖𝐶^′, 𝕎_𝜓𝑓≤ℓ𝑥+𝑖ℓ𝑦,ℓ^𝑚(2𝜋)^𝑛_𝐶^∗𝑑^𝑛𝑘𝑃(𝑘)𝑓(𝑘)𝑒^{−⟨𝑘,ℓ𝑦⟩}≤ℓ^𝑚𝐌₂(2𝜋)^𝑛𝑁_𝑗=0𝑐_𝑗𝐶∗𝑑^𝑛𝑘|𝑘|^𝑗𝑒^{−𝑐|𝑘||𝑙𝑦|}. (3.9)
Following Schwartz [25, Proposition 32, page 39], we get the following: 𝕎_𝜓𝑓≤ℓ𝑥+𝑖ℓ𝑦,ℓ^𝑚𝐌₂(2𝜋)^𝑛𝑁_𝑗=0𝑐_𝑗𝜎^𝑛−1∞₀𝑑𝑡𝑡^{𝑛+𝑗−1}𝑒^{−𝑐|ℓ𝑦|𝑡}≤𝐌₃𝐶^′ℓ𝑦^{−(𝑛+𝑁)}, (3.10) where 𝜎^𝑛−1 is the area of the unit sphere in ℝ^𝑛.

Now, let 𝐶 be an open cone of the form 𝐶=∪^𝑚_𝑗=1𝐶_𝑗, 𝑚<∞, where each 𝐶_𝑗 is an proper open convex cone. If we write 𝐶^′⋐𝐶, we mean that 𝐶^′=∪^𝑚_𝑗=1𝐶^′_𝑗 with 𝐶^′_𝑗⋐𝐶_𝑗. Furthermore, we define by 𝐶^∗_𝑗={𝑘∈ℝ^𝑛∣⟨𝑘,𝑥⟩≥0,∀𝑥∈𝐶_𝑗} the dual cones of 𝐶_𝑗, such that the dual cones 𝐶^∗_𝑗, 𝑗=1,…,𝑚, have the properties ℝ^𝑛⧵∪^𝑚_𝑗=1𝐶^∗_𝑗,𝐶^∗_𝑗∩𝐶^∗_𝑘,𝑗≠𝑘,𝑗,𝑘=1,…,𝑚, (3.11) that are sets of Lebesgue measure zero. Moreover, assume that 𝑃(ℓ𝑘)𝑓(𝑘) can be written as 𝑃(ℓ𝑘)𝑓(𝑘)=∑^𝑚_𝑗=1𝜆_𝑗(𝑘)𝑃(ℓ𝑘)𝑓(𝑘), where 𝜆_𝑗(𝑘) denotes the characteristic function of 𝐶^∗_𝑗, 𝑗=1,…,𝑚. We shall consider the asymptotic property of 𝕎_𝜓𝑓(𝑧,ℓ) as ℓ→0 for 𝑧=𝑥+𝑖ℓ𝑦∈𝑍∩𝑇(𝐶^′).

Theorem 3.5. Let 𝑓∈𝒮^′(ℝ^𝑛). Then 𝑓 can be expressed as a finite sum 𝑓=^𝑚_𝑗=1𝑏_𝐶′𝑗𝕎_𝜓𝑓_𝑗(𝑧,ℓ), (3.12) where each 𝕎_𝜓𝑓_𝑗(𝑧,ℓ), arising from the complexification of the variable 𝑥 corresponding to the location of the continuous wavelet transform of 𝑓_𝑗∈𝒮^′(ℝ^𝑛) with respect to directional wavelet 𝜓, is analytic in 𝑍∩𝑇(𝐶^′_𝑗) and of tempered growth, and where 𝑏_𝐶′𝑗(𝕎_𝜓𝑓_𝑗(𝑧,ℓ)) denotes the boundary value in 𝒮^′(ℝ^𝑛).

Proof. The proof that each 𝕎_𝜓𝑓_𝑗(𝑧,ℓ) is of tempered growth is obtained by the similar way as in Lemma 3.4. Let 𝜑∈𝒮(ℝ^𝑛). Choose now 𝑦₀∈𝐶^′_𝑗 and write 𝑦=ℓ𝑦₀ (this defines a half-line for 0≠𝑦₀∈𝐶^′_𝑗). Note that with 𝑦=ℓ𝑦₀ in (3.1), then ℓ→0 when 𝑦→0. Thus, we have 𝑏_𝐶′𝑗𝕎_𝜓𝑓_𝑗,𝜑=lim_{𝑦→0ℝ𝑛}𝑑^𝑛𝑥𝕎_𝜓𝑓_𝑗𝑥+𝑖𝑦,ℓ𝜑(𝑥)=lim_{𝑦→0ℝ𝑛}𝑑^𝑛𝑥1(2𝜋)^𝑛_𝐶^∗_𝑗𝑑^𝑛𝑘𝜆_𝑗(𝑘)𝑃(ℓ𝑘)𝑓(𝑘)𝑒^{𝑖⟨𝑘,𝑥+𝑖𝑦⟩}𝜑(𝑥)=lim_𝑦→01(2𝜋)^𝑛_𝐶^∗_𝑗𝑑^𝑛𝑘𝜆_𝑗(𝑘)𝑃(ℓ𝑘)𝑓(𝑘)𝜑(−𝑘)𝑒^{−⟨𝑘,𝑦⟩}=1(2𝜋)^𝑛_𝐶^∗_𝑗𝑑^𝑛𝑘𝜆_𝑗(𝑘)𝑓(𝑘)𝜑()=𝑓−𝑘_𝑗,𝜑. (3.13) Hence, using the linearity of 𝑓∈𝒮^′(ℝ^𝑛) and the assumptions (3.11), we obtain that 𝑓,𝜑=^𝑚_𝑗=1𝑓_𝑗,𝜑=^𝑚_𝑗=1𝑏_𝐶′𝑗𝕎_𝜓𝑓_𝑗,𝜑. (3.14) Thus, the limit of each 𝕎_𝜓𝑓_𝑗(𝑧,ℓ) as 𝐶^′_𝑗∋𝑦→0 exists in 𝒮^′(ℝ^𝑛); that is, 𝑓 admits the distributional boundary value ∑^𝑚_𝑗=1𝑏_𝐶′𝑗(𝕎_𝜓𝑓_𝑗) in the sense of weak convergence. But from [27, Corollary 1, page 358], the latter implies strong convergence since 𝒮(ℝ^𝑛) is Montel.

4. Analytic Wavefront Set

From what we have seen in the previous section, a distribution 𝑓∈𝒮^′(ℝ^𝑛) is obtained as a finite sum of boundary values of analytic functions 𝕎_𝜓𝑓_𝑗(𝑧,ℓ), with 𝑗=(1,…,𝑚), arising from the complexification of the variable 𝑥 corresponding to the location of the continuous wavelet transform of 𝑓_𝑗 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. Let 𝑓∈𝒮^′(ℝ^𝑛), such that 𝑓=𝑏_𝐶(𝕎_𝜓𝑓(𝑧,ℓ)), where 𝑏_𝐶(𝕎_𝜓𝑓(𝑧,ℓ)) denotes the strong boundary value in 𝒮^′(ℝ^𝑛) of an analytic function 𝕎_𝜓𝑓(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∉𝑊𝐹_𝑎(𝑓) if and only if there exist 𝐌(𝐶^′) and 𝑁 for which we have the estimate 𝕎_𝜓𝑓(𝑧,ℓ)≤𝐌𝐶^′ℓ𝑦^−𝑁,𝑧=𝑥+𝑖ℓ𝑦∈𝑍∩𝑇𝐶^′. (4.1)𝑊𝐹_𝑎(𝑓) is called analytic wavefront set of 𝑓.

Proposition 4.2. Let 𝑓∈𝒮^′(ℝ^𝑛) be the boundary value, in the distributional sense, of a function 𝕎_𝜓𝑓(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 𝑊𝐹_𝑎(𝑓)|_𝑥0 is contained in 𝐶^∗.

Proof. Let 𝑊𝐹_𝑎(𝑓)|_𝑥0={𝑘∈(ℝ^𝑛⧵0)∣(𝑥₀,𝑘)∈𝑊𝐹_𝑎(𝑓)} be the fibre over 𝑥₀. Let {𝐶^∗_𝑗}_𝑗∈𝐿 be a finite covering of closed properly convex cones of 𝐶^∗. Decompose 𝑃(ℓ𝑘)𝑓(𝑘) as follows: 𝑃(ℓ𝑘)𝑓(𝑘)=^𝑚_𝑗=1𝜆_𝑗(𝑘)𝑃(ℓ𝑘)𝑓(𝑘), (4.2) where 𝜆_𝑗(𝑘) denotes the characteristic function of 𝐶^∗_𝑗, 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 𝕎_𝜓𝑓_𝑗(𝑧,ℓ), such that 𝕎_𝜓𝑓_𝑗(𝑧,ℓ)→𝑓_𝑗 in the strong topology of 𝒮^′(ℝ^𝑛) as 𝑦→0, 𝑦∈𝐶^′_𝑗⋐𝐶_𝑗. According to Lemma 3.4, the family of functions 𝕎_𝜓𝑓_𝑗(𝑧,ℓ) satisfies the following estimate: 𝕎_𝜓𝑓_𝑗(𝑧,ℓ)≤𝐌𝐶^′ℓ𝑦^−𝑁,𝑧=𝑥+𝑖ℓ𝑦∈𝑍∩𝑇𝐶^′, (4.3) unless ⟨𝑘,𝑌⟩<0 for 𝑘∈𝐶^∗_𝑗 and 𝑌∈−𝐶^′_𝑗, 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 𝑊𝐹_𝑎𝑓⊂ℝ^𝑛×∪_𝑗𝐶^∗_𝑗. (4.4) 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, 𝑊𝐹_𝑎(𝑓)|_𝑥0, is completely characterized by sequences of type 𝑓_𝑁=𝜙_𝑁𝑓, where {𝜙_𝑁} is a bounded sequence in 𝐶^∞₀(𝑋) which is equal to 1 in a common neighborhood of 𝑥₀ and satisfies the following estimate: 𝐷^𝛼+𝛽𝜙_𝑁≤𝐶_𝛼(𝐶𝑁)^|𝛽|,if𝛽≤𝑁. (4.5) For the existence of such functions, we refer to Lemma 2.2 in [21].

We can meet Definition 4.1 and Proposition 4.2 in the following proposition.

Proposition 4.4. Let 𝑓∈𝒮^′(ℝ^𝑛) and (𝑥₀,𝑘₀)∈𝑇^∗(ℝ^𝑛)⧵0, where 𝑇^∗(ℝ^𝑛)⧵0∶=ℝ^𝑛×(ℝ^𝑛⧵0). Then (𝑥₀,𝑘₀)∉𝑊𝐹_𝑎(𝑓) if and only if there exists a finite family {𝐶_𝑗} of proper open convex cones in ℝ^𝑛, a complex neighborhood 𝑍 of 𝑥₀ in ℂ^𝑛 and a decomposition of 𝑓𝑓=^𝑚_𝑗=1𝑏_𝐶′𝑗𝕎_𝜓𝑓_𝑗(𝑧,ℓ), (4.6) with each 𝕎_𝜓𝑓_𝑗(𝑧,ℓ)∈𝒪(𝑍∩𝑇(𝐶^′_𝑗)) being of tempered growth and analytic near to 𝑥₀ for every 𝑗 satisfying 𝐶^′_𝑗⊂{𝑦∣⟨𝑘₀,𝑦⟩≥0}, and where 𝑏_𝐶′𝑗(𝕎_𝜓𝑓_𝑗(𝑧,ℓ)) denotes the boundary value, in the distributional sense, of analytic functions 𝕎_𝜓𝑓_𝑗(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 distribution 𝑓 into a sum of boundary values of analytic functions is equivalent to a decomposition of analytic wavefront set of 𝑓 since the fibre 𝑊𝐹_𝑎(𝑓)|_𝑥0 is contained in ∪_𝑗𝐶^∗_𝑗. 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)⊂𝑊𝐹_𝑎(𝑓). Following Nishiwada [22] another characterization of 𝑊𝐹(𝑓) can be obtained based on the above results.

Proposition 4.5. Let 𝑈 be an open set in ℝ^𝑛, (𝑥₀,𝑘₀)∈𝑇^∗(𝑈)⧵0 and 𝑓∈𝒮^′(𝑈). Then (𝑥₀,𝑘₀)∉WF(𝑓) if there exists a finite family {𝐶_𝑗} of proper open convex cones in ℝ^𝑛, with 𝑗=1,…,𝑚, a complex neighborhood 𝑍 of 𝑥₀ and a decomposition of 𝑓 near 𝑥₀𝑓=^𝑚_𝑗=1𝑏_𝐶′𝑗𝕎_𝜓𝑓_𝑗(𝑧,ℓ)in𝑈, (4.7) with 𝕎_𝜓𝑓_𝑗(𝑧,ℓ)∈𝒪(𝑍∩𝑇(𝐶^′_𝑗)) being of tempered growth, such that 𝑏_𝐶′𝑗(𝕎_𝜓𝑓_𝑗(𝑧,ℓ))∈𝐶^∞ near 𝑥₀ for every 𝑗 with 𝐶^′_𝑗⊂{𝑦∣⟨𝑘,ℓ𝑦⟩≥0}.

Proof. The proof is similar to the proof of the first part of Theorem 3.4 in [23].

Acknowledgment

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.