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

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Volume 2012, Article ID 758694,11pages doi:10.1155/2012/758694

Research Article

A Note on Directional Wavelet Transform:

Distributional Boundary Values and Analytic

Wavefront Sets

Felipe A. Apolonio,

1

_{Daniel H. T. Franco,}

1

_{and F ´abio N. Fagundes}

2

1_{Departamento de F´}_ı_{sica, Universidade Federal de Vic¸osa, Avenida Peter Henry Rolfs s/n,}

Campus Universitário Viçosa, 36570-000 Vioçsa, MG, Brazil

2_{Universidade Federal de Mato Grosso, Campus Sinop, ICNHS, Avenida Alexandre Ferronato 1.200,}

Distrito Industrial, 78557-267 Sinop, MT, Brazil

Correspondence should be addressed to Daniel H. T. Franco,[email protected]

Received 9 February 2012; Revised 5 April 2012; Accepted 19 April 2012

Academic Editor: Brigitte Forster-Heinlein

Copyrightq2012 Felipe A. Apolonio et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 distributionf∈ 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 fyields a function on phase space whose high-frequency singularities are precisely the elements in the analytic wavefront set off.

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 purposessee, 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

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H örmander’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 in11is that these wavefront sets are equal to

the wavefront sets in the sense of H ¨ormander in the casesn1,2,4,8. Ifn∈N\ {1,2,4,8},

they combined results for dimensionsn 1,2,4,8 and characterized wavefront sets in

ξ-directions, whereξare presented as products of nonzero points ofRn1_{, . . . ,}_Rns_,_n1· · ·ns_n,

ni ∈ {1,2,4,8}, i 1, . . . , s. In the paper 7, Navarro introduced an analyzing wavelet

in SR2 _{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 SR2_{. It should be noted that the core of the construction of Navarro}

the irreducible group action onL2_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 ∈ SRn _{is obtained as}

a finite sum of boundary values of analytic functions arising from the complexification of

the variablexcorresponding to the location of the continuous wavelet transform off 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 inLemma 3.4andTheorem 3.5. InLemma 3.4, we

prove that the distributional wavelet transform with respect to the directional wavelet is

an analytic function of tempered growth. By usingLemma 3.4, we show that the tempered

distributions can be obtained as a finite sum of boundary values of analytic functions of

tempered growthTheorem 3.5. InSection 4, we applyTheorem 3.5in the study of analytic

wavefront set of tempered distributions. We show that, for a given distributionf ∈ SRn_,

the wavelet transform offwith respect to a conical wavelet is defined in such a way that the

directional wavelet transform off yields a function on phase space whose high-frequency

singularities are precisely the elements in the analytic wavefront set off. In21, H ¨ormander

introduced the notion of the analytic wavefront setWFafoffas a subset of the cotangent

spaceT∗X\0, whose projection toXcoincides with the analytic singular support off. His

definition relies on the use of the Fourier transform off. In this note, following Nishiwada

22,23, we present an alternative definition ofWFafin terms of generalized boundary

values of analytic functions arising from the complexification of the variablexcorresponding

to the location of the continuous wavelet transform offwith respect to directional waveletψ.

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 xy x1 y1, . . . , xnyn,λx λx1, . . . , λxn,x ≥ 0

meansx1 ≥ 0, . . . , xn ≥ 0,x, y x1y1· · ·xnyn and|x| |x1|· · ·|xn|. Moreover, we

defineα α1, . . . , αn∈Nn

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ofαis the corresponding1_-norm_|α|_α1_{· · ·}_αn,_α_β_denotes_α1_{β1, . . . , αn}_βn,_α_≥_β

meansα1≥β1, . . . , αn≥βn,α!α1!. . . αn!,xα_xα1

1 . . . x

αn

n , and

Dαϕx ∂

|α|_{ϕx1, . . . , xn}

∂xα1₁ ∂x₂α2, . . . , ∂xαn

n

. 2.1

We consider two n-dimensional spaces—x-space and k-space—with the Fourier

transform defined as follows

fk Ffxk

Rnd

n_{x fxe}−ik,x_, _2.2

while the Fourier inversion formula is

fx F−1_fk _x 1

2πn

Rnd

n_k _fke ik,x_. _2.3

The variablekwill always be taken real whilexwill also be complexified: when it is complex,

it will be notedzxiy.

Definition 2.1. A wavelet is a complex-valued function ψ in L2_Rn_{, d}n_{x, satisfying the} admissibility condition

Cψ 2πn

dn_k

|k|n ψk

2

<∞, 2.4

whereψis the Fourier transform ofψ.

Ifψ is suﬃciently regular—enough to takeψ ∈ L1_Rn_{, d}n_x_∩_L2_Rn_{, d}n_{x—then the}

admissibility condition above means that

ψ0 0⇐⇒

dnx ψx 0. 2.5

Definition 2.2. The continuous wavelet transform of a distributionfx∈ SRn_{with respect}

to some analyzing waveletψis defined as the following convolution:

Wψf x, fx, ψx,x

dnx ψx,xfx, 2.6

where

ψx,x 1

nψ

−1 x−x, 2.7

with ∈ R as the length scale at which we analyzefxand x ∈ Rn as the translation

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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

Wψf x, 1

2πn

dn_k_Wψ_{fk, e}ik,x_, _2.8

where

Wψfk, ψk fk, 2.9

is the Fourier transform ofWψf. With this we have the followingsee24, Lemma 8.2.6, page

441for the one-dimensional space.

Lemma 2.4. For anyf ∈L2_Rn_{, we have}

FWψf x, k,

Rn

dn_x_Wψ_f _x_,_e−ik,x

ψk fk.

2.10

Remark 2.5. The domain of a wavelet transform is usually theL2 space, but theLemma 2.4

can be extended toSRn_{, which is the dual space of}_SRn_{. In particular, see}_{Remark 3.2, the}

class of conical wavelets belongs to the spaceSRn_{. This allows us to define the directional}

wavelet transform of a distributionf ∈ SRn in such a way that the wavelet transform

off yields a function on phase space whose high-frequency singularities are precisely the

elements in the analytic wavefront set off.

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 inSRn_{. 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 zerois invariant under positive homotheties; that is, if for allλ > 0,

λC ⊂ C. A coneCis an open connected cone ifCis an open connected set. Moreover,Cis

called convex ifCC ⊂Cand proper if it contains no any straight lineobserve that ifCis

a cone, thenCis proper if and only ifx ∈ Cand_{x /}0 implie−x /∈ C. A coneCis called

compact inC—we writeCC—if the projection prCdef C∩Sn−1⊂prCdef C∩Sn−1, where

Sn−1is the unit sphere inRn_{. Being given a cone}_C_in_{x-space, we associate with}_C_{a closed}

convex coneC∗inkspace which is the setC∗ {k∈Rn | k, x ≥0,∀x∈C}. The coneC∗is

called the dual cone ofC.

Definition 3.1. A waveletψx is said to be directional if the eﬀective support of its Fourier

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vertex at the origin, or a finite union of disjoint such cones; in that case, one will usually call

ψmultidirectional.

Remark 3.2. According to19, for us to reach a genuinely directional waveletψx, it suﬃces

to consider a smooth functionψk with support in the strictly convex coneC∗in thekspace

of frequencies having an arbitrary large number of vanishing moments on the boundary of the

supporting cone and behaving insideC∗asPk1, . . . , kne−k,y, wherey∈Cand P·denotes

a polynomial innvariables. In this case, the resulting directional wavelet is also called conical.

We also observe that, for ally∈C, the exponentiale−k,y, considered as a function ofk∈C∗,

belongs to the space of rapidly decreasing functionsSRn_{. Then,}_P_{k1, . . . , kne}−k,y_{∈ SR}n

too. Since the inverse Fourier transform is a topological isomorphism ofSRn_onto_SRn_{, it}

follows that directional waveletψx∈ SRn.

In particular, from the above remark, it follows that for a directional wavelet2.8can

be rewritten as

Wψfz, 1

2πn

dnk Pkfke ik,z, 3.1

where we have introduced the complex variablezxiy∈Rn_{iC. Remember that here}

xcorresponds 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,Wψfx, , of a

distributionf ∈ SRn_{is naturally extensible as an analytic function in a domain in}_Cn_{, for}

each arbitrary but fixed >0. Still, sinceC∗is a regular set25, pages 98, 99, it follows that

suppWψf suppψ ⊆C∗.

LetΩbe an open subset inCn_{. Then we shall denote by}_OΩ_{the space of analytic}

functions inΩ. LetCbe a proper open convex cone, and letCbe an arbitrary compact cone

contained inC. Denote byTCthe subset ofCn_{consisting of all elements whose imaginary}

parts lie inC.TCis referred to as a tube domain. We will deal with tubes defined as the set

of all pointsz∈Cn_{such that}

T Czxiy∈Cn|x∈Rn, y∈C,y< δ, 3.2

whereδ >0 is an arbitrary number.

Definition 3.3. LetZbe a complex neighbourhood ofRn_{. An analytic function}_f_z_{∈ OZ}_∩

TCis said to be of tempered growth if there are an integerαand a constantM depending

onCsuch that

_f _x_iy_≤_M _C_y−α

, 3.3

for all pointzxiyinZ∩TC.

Lemma 3.4. For an arbitrary but fixed scale ∈ R, assume that the functionWψfz, , arising

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Proof. We start considering the formula

Wψf xiy, 1

2πn

C∗

dnk Pkfke ik,z. 3.4

By Theorem 7.13 of26, sincePkis a polynomial andfk a tempered distribution, then

Pkfk is also tempered. Note that, we can write

Pk Pmk Pm−1k · · ·P0, 3.5

where Pj is homogeneous of degree j and Pmk_/0 when _{k /}0. It follows that for

some constant M, we have that |Pk| ≤ Mm_|P_{k|. This implies that} _|P_k_{fk| ≤}

Mm_|Pk_{fk|. Still, the character tempered of} _Pk_fk _{implies that there exist an integer}

Nand a constantM1such thatPkfk satisfies the estimate

Pkfk ≤M11|k|N

. 3.6

Hence,

Pkfk ≤m_M₂₁_|k|N

. 3.7

Using the binomial theorem, the above estimate can be rewritten as

Pkfk ≤mM2

N

j0

cj|k|j_. _3.8

Now, letCbe a cone, such thatCC. Then there existsc >0 so thatk, y ≥c|ky|,

for allk∈C∗and for ally∈C. Hence forxiy∈Rn_iC_,

_Wψ_f _x_iy,_≤ m

2πn

C∗d

n_k _Pk_fk _e−k,y

≤ mM2

2πn

N

j0

cj

C∗d

n_k _|k|j

e−c|k||ly|.

3.9

Following Schwartz25, Proposition 32, page 39, we get the following:

_Wψf _x_iy,_≤ m_M

2

2πn

N

j0

cjσn−1

_∞

0

dt tnj−1e−c|y|t

≤M3 Cy−nN,

3.10

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Now, letCbe an open cone of the formC∪m_j₁Cj,m <∞, where eachCjis an proper

open convex cone. If we writeCC, we mean thatC∪m_j₁C_jwithC_jCj. Furthermore,

we define byC_j∗{k∈Rn_{| k, x ≥}_0,_∀x_∈_Cj}_{the dual cones of}_Cj_{, such that the dual cones}

C∗_j,j1, . . . , m, have the properties

Rn_\_∪m

j1Cj∗

,

C∗j∩C∗k , j /k, j, k1, . . . , m,

3.11

that are sets of Lebesgue measure zero. Moreover, assume thatPkfk can be written

asPkfk mj1λjkPkfk, where λjkdenotes the characteristic function ofC∗j,

j1, . . . , m. We shall consider the asymptotic property ofWψfz, as → 0 forzxiy∈

Z∩TC.

Theorem 3.5. Letf ∈ SRn. Thenfcan be expressed as a finite sum

f

m

j1

bC

j Wψfjz,

, 3.12

where each Wψfjz, , arising from the complexification of the variable x corresponding to the location of the continuous wavelet transform offj ∈ SRn_{with respect to directional wavelet}_ψ_,

is analytic inZ∩TC_jand of tempered growth, and wherebC

jWψfjz, denotes the boundary

value inSRn_.

Proof. The proof that eachWψfjz, is of tempered growth is obtained by the similar way as

inLemma 3.4. Letϕ∈ SRn_{. Choose now}_y0 _∈_C

jand writeyy0this defines a half-line

for 0_/y0∈C_j. Note that withyy0in3.1, then → 0 wheny → 0. Thus, we have

bC

j Wψfj

, ϕ lim

y→0

Rnd

n_xWψ_fj _x_iy,_ϕx

lim

y→0

Rn

dnx

1

2πn

C∗_j

dnkλjkPkfke ik,xiy

ϕx

lim

y→0

1

2πn

C∗_j

dnkλjkPkfk ϕ−ke −k,y

1

2πn

C∗_j

dnkλjkfkϕ−k

fj, ϕ.

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Hence, using the linearity off∈ SRn_{and the assumptions}_{3.11, we obtain that}

f, ϕ

m

j1

fj, ϕ

m

j1

bC

j Wψfj

, ϕ. 3.14

Thus, the limit of each Wψfjz, as C_j y → 0 exists in SRn; that is, f admits the

distributional boundary valuem_j₁bC

jWψfjin the sense of weak convergence. But from

27, Corollary 1, page 358, the latter implies strong convergence sinceSRn_{is Montel.}

4. Analytic Wavefront Set

From what we have seen in the previous section, a distributionf ∈ SRn_{is obtained as a}

finite sum of boundary values of analytic functionsWψfjz, , withj 1, . . . , m, arising

from the complexification of the variablexcorresponding to the location of the continuous

wavelet transform offjwith respect to directional waveletψ, 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 ∈ SRn_{, such that} _f _bCWψ_{fz, , where}_bCWψ_fz,_denotes

the strong boundary value in SRn _{of an analytic function} _Wψ_fz, _{arising from the}

complexification of the variable xcorresponding to the location of the continuous wavelet

transform offwith respect to directional waveletψ. Letq x0, k0. Then,q /∈WFafif and

only if there existMCandNfor which we have the estimate

_Wψ_fz,_≤_M _C_y−N

, zxiy∈Z∩T C. 4.1

WFafis called analytic wavefront set off.

Proposition 4.2. Letf ∈ SRn_{be the boundary value, in the distributional sense, of a function} Wψfz, analytic inZ∩TC, arising from the complexification of the variablexcorresponding to the location of the continuous wavelet transform offwith respect to directional waveletψand which satisfies the estimate3.3. Then nearx0the fibreWFaf|x0is contained inC∗.

Proof. LetWFaf|x0{k∈Rn_\₀_|_{x0, k}_∈_WFaf}_{be the fibre over}_x0_{. Let}_{C∗

j}j∈Lbe a

finite covering of closed properly convex cones ofC∗. DecomposePkfkas follows:

Pkfk

m

j1

λjkPkfk, 4.2

where λjk 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

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y → 0,y∈C_jCj. According toLemma 3.4, the family of functionsWψfjz, satisfies the following estimate:

_Wψ_fjz,_≤_M _C_y−N

, zxiy∈Z∩T C, 4.3

unlessk, Y <0 fork ∈C_j∗andY ∈ −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

WFa f⊂Rn×∪jC∗_j. 4.4

Then, by making a refinement of the covering and shrinking it toC∗, we obtain the desired

result.

Remark 4.3. It is remarked that in 21 the fiber over x0, WFaf|x0, is completely

characterized by sequences of typefN φNf, where{φN}is a bounded sequence inC∞₀ X

which is equal to 1 in a common neighborhood ofx0and satisfies the following estimate:

Dαβ_φN_≤_CαCN|β|_, _if_β_≤_N. _4.5

For the existence of such functions, we refer to Lemma 2.2 in21.

We can meetDefinition 4.1andProposition 4.2in the following proposition.

Proposition 4.4. Letf ∈ SRn_and_{x0, k0} _∈ _T∗_Rn_\_{0, where}_T∗_Rn_\_{0 :} _Rn_×_Rn_\₀_.

Thenx0, k0∈/WFafif 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

f

m

j1

bC

j Wψfjz,

, 4.6

with eachWψfjz, ∈ OZ∩TC_jbeing of tempered growth and analytic near tox0 for every

j satisfyingC_j ⊂ {y| k0, y ≥0}, and wherebC

jWψfjz, denotes the boundary value, in the

distributional sense, of analytic functionsWψfjz, arising from the complexification of the variable

xcorresponding to the location of the continuous wavelet transform off with respect to directional waveletψ.

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 off since the fibreWFaf|x0is contained in∪jC∗_j.

Moreover, the decomposition4.6is carried out in the space ofC∞functions, provided that

fisC∞.

Finally, we recall that in 28 H ¨ormander defined the wavefront set, WFf, for a

distribution as the set of points in the cotangent space which must be characteristic for

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Following Nishiwada22another characterization ofWFfcan be obtained based on the above results.

Proposition 4.5. LetUbe an open set inRn_,_{x0, k0}_∈_T∗_U_\_{0 and}_f _{∈ S}_U_{. Then}_{x0, k0}_∈_/

WFfif there exists a finite family{Cj}of proper open convex cones inRn_{, with}_j _{1, . . . , m}_{, a}

complex neighborhoodZofx0and a decomposition offnearx0

f

m

j1

bC

j Wψfjz,

inU, 4.7

withWψfjz, ∈ OZ∩TC_jbeing of tempered growth, such thatbC

jWψfjz, ∈C

∞_near

x0for everyjwithC_j⊂ {y| k, y ≥0}.

Proof. The proof is similar to the proof of the first part of Theorem 3.4 in23.

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 N´ıvel

SuperiorCAPESagency.

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

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