Wavelet transform of Beurling-Bjorck type...

Rend. Sem. Mat. Univ. Padova 1xx (201x) Rendiconti del Seminario Matematico

della Universita di Padova

c© European Mathematical Society

Wavelet transform of Beurling-Bjorck type

ultradistributions

R. S. Pathak ∗ – Abhishek Singh∗∗

Abstract – Wavelet transform of a distribution in M′

ωinvolving wavelet of infraex-

ponential decay (subexponential decay) is studied. An inversion formula is obtained

which is valid in the weak topology of D′

. A discussion on extension of the results toultradistribution space of compact support is also given.

Mathematics Subject Classification (2010). 44A15, 42C40, 46F12.

Keywords. Beurling-Bjorck ultradistributions, infraexponential decaying wavelets, Wavelettransform.

1. Introduction

Wavelet analysis has been used for intrinsic characterizations of importantfunction and distribution spaces ([10], [11]). Recently, the wavelet transform hasbeen extended to distributions, and inversion formulae have been established indistribution setting by Pathak [13, 14], Pathak et al [16, 17, 18] and Pandey [12]using duality arguments.

Wavelets of subexponential decay whose Fourier transform have compact sup-port i.e. band limited wavelets, were investigated by Dziubanski and Hernandez[7]. Pathak and Singh [17] extended the work of Dziubanski and Hernandez andstudied wavelets with more general decay (infraexponential decay) whose Fourier

∗Research supported by grant (No. 2084) from Department of Science and Technology, India.∗∗Research supported by grant (No. F. 4-2/2006(BSR)/13-663/2012) from University Grants

Commission, India, under the Dr. D.S. Kothari Post Doctoral Fellowship.

R. S. Pathak, DST Centre for Interdisciplinary Mathematical Sciences, Faculty of Science,Banaras Hindu University, Varanasi- 221 005, India

E-mail: [email protected]

Abhishek Singh, DST Centre for Interdisciplinary Mathematical Sciences, Faculty of Science,Banaras Hindu University, Varanasi- 221 005, India

E-mail: [email protected] (∗∗Corresponding author)

2 R. S. Pathak – Abhishek Singh

transforms have compact support. The aim of the present paper is to develop thetheory of wavelet transform involving these wavelets using ultradistribution theoryof Beurling [3] and Bjorck [4].

Now, we recall definitions and properties of the desired test function and ul-tradistribution spaces from [4], [6] and [9]. Let M be the set of all real-valuedfunctions ω on R which can be represented as ω(x) = σ(|x|), where σ(t) is an in-creasing continuous concave function on [0,∞) satisfying the following conditions[9, p. 14]:

(1) (α) 0 = ω(0) ≤ ω(ξ + η) ≤ ω(ξ) + ω(η), ∀ξ, η ∈ R

(2) (β)

∫

R

σ(ξ)

(1 + |ξ|)2dξ <∞,

(γ) there exists real number p and positive real number q such that

(3) σ(ξ) ≥ p+ q log(1 + t), t ≥ 0.

Let ω ∈ M. We denote by Mω the set of all functions ψ(t) ∈ C∞(R) whichsatisfy

(4) Pk,λ(ψ) = supt∈R

{

eλω(t)|Dkψ(t)|}

<∞

for all non-negative integers k and all non-negative real λ. The topology on Mω

is defined by the semi-norms {Pk,λ}. It can be readily seen that Mω is a vectorspace. A sequence {ψν}

∞ν=1 is a Cauchy sequence in Mω if for each non-negative

integerm and k, Pk,λ(ψµ−ψν) → 0 as µ, ν → ∞ independently of each other. Thespace Mω is a sequentially complete space and therefore it is a complete countablymultinormed space and so a Frechet space. The dual of Mω is denoted by Mω

′

; it is a distribution space [6, p. 170]. The Schwartz space D(R) consisting ofC∞−functions of compact support is a subspace of Mω(R) and M

′

ω ⊂ D′

.Suppose that the Fourier transform of ψ ∈ Mω, defined by

ψ(ξ) =

∫ ∞

−∞

e−ixξψ(x)dx,

satisfies

(5) πk,λ(ψ) = supξ∈R

{

eλω(ξ)∣

∣

∣Dkψ(ξ)

∣

∣

∣

}

<∞, λ ≥ 0, k ∈ N0.

Then the space of all functions ψ ∈ L1(R) such that ψ, ψ ∈ C∞(R) and (4), (5)hold, is denoted by Sω the topology of Sω is defined by the seminorms Pk,λ andπk,λ [4, p. 377].

Let K be a compact subset of R. The space Dω (K) is the set of all ψ in L1(R)such that ψ has support in K and

(6) ‖ψ‖λ :=

∫

R

|ψ(ξ)|eλω(ξ)dξ <∞, ∀λ > 0.

Wavelet transform of Beurling-Bjorck type ultradistributions 3

Let {Kn} be a sequence of compact set in R such that⋃∞

n=1Kn = R and Kn iscontained in the interval of Kn+1 for all n. Then Dω(R) = lim ind Dω(Kn). SinceDω ⊂ Sω and the topology of Dω is stronger than that induced on Dω by Sω, itfollows that the restriction of any f ∈ S

′

ω to Dω is in D′

ω. The elements of D′

ω arecalled ultradistributions [6].

Now we recall from [5] some definitions and results related to wavelet transformneeded in the present investigation.

Let ψ ∈ L2(R). Define

(7) ψb,a(t) :=1

√

|a|ψ

(

t− b

a

)

, t ∈ R, b ∈ R, a ∈ R0 = R\{0}.

The wavelet transform W (b, a) of f ∈ L2(R) with respect to the wavelet ψb,a(t) ∈L2(R) is defined by

(8) W (b, a) :=

∫

R

f(t)ψb,a(t)dt

and the corresponding wavelet inversion formula is given by

(9)1

Cψ

∫

R

∫

R0

1√

|a|W (b, a)ψ

(

x− b

a

)

dbda

a2= f(x),

where

Cψ =

∫

R

|ψ(w)|2

|w|dw <∞ [5, p. 9].

In the present work we shall investigate properties of the wavelet ψb,a(t) ∈

Mω(R). Wavelet transform of f ∈ M′

ω will be studied and the inversion formula(9) will be extended to distribution space M

′

ω. It has been noted by Constantinescuet al [6, p. 169] that the Schwartz space S (R) and Gelfand-Shilov space Sα,A arespecial cases of the space Mω. Therefore, some of the results obtained in thispaper are more general than those derived in [13] and [16].

2. Wavelet Transform on M′

ω

In this section, certain basic properties of the wavelets in Mω and wavelettransform of f ∈ M

′

ω are obtained.

Lemma 2.1. If ψ ∈ Mω, then ψ(t−ba ) ∈ Mω for arbitrary but fixed b, a ∈ R, a 6=

0.

Proof. Let a and b be fixed real numbers. Then for k = 0, 1, 2, ...,

sup−∞<t<∞

∣

∣

∣

∣

eλω(t)Dkψ

(

t− b

a

)∣

∣

∣

∣

= sup−∞<t<∞

∣

∣

∣

∣

eλω(t−b

a )ψ(k)

(

t− b

a

)(

1

ak

)∣

∣

∣

∣

∣

∣

∣

∣

eλω(t)

eλω(t−b

a )

∣

∣

∣

∣

.


Then by Property (α) we get

sup−∞<t<∞

∣

∣

∣

∣

eλω(t)Dkψ

(

t− b

a

)∣

∣

∣

∣

≤1

|a|kPk,λ(ψ) sup

−∞<t<∞

∣

∣

∣

∣

∣

eλω(t−b

a+ b

a )

eλω(t−b

a )

∣

∣

∣

∣

∣

, by (4)

≤ Pk,λ(ψ)

(

1

|a|k

)

eλω(b

a ) <∞, by (1)

for all fixed real numbers b and a 6= 0.

In what follows we shall assume that ψ ∈ Mω(R) is the basic function gener-ating the wavelet ψb,a(t) given by (7). Since function ψ

(

t−ba

)

belongs to Mω for

fixed b and a 6= 0 as a function of t under conditions of Lemma 2.1, for f ∈ M′

ω

the wavelet transform W (b, a) of f is defined by

W (b, a) =1

√

|a|

⟨

f(t), ψ

(

t− b

a

)

⟩

, a 6= 0, a, b ∈ R

=⟨

f(t), ψb,a(t)⟩

.

(10)

Theorem 2.2. For real b and a 6= 0 let W (b, a) be defined by ( 10), then underconditions of Lemma 2.1, there exists m ∈ N0 such that

|W (b, a)| ≤ C(m,ψ)(

|a|−m−1/2 exp(mω(b/a))

, for some m ∈ N0.

Proof. To very f ∈ M′

ω there exists a non-negative integer m and a constantC > 0 such that, for all ψ ∈ Mω,

|W (b, a)| =

∣

∣

∣

∣

∣

1√

|a|

⟨

f(t), ψ

(

t− b

a

)

⟩∣

∣

∣

∣

∣

≤C√

|a|max

0≤k≤msupb,t∈R

∣

∣

∣

∣

∣

emω(t

a )Dkt ψ

(

t− b

a

)

∣

∣

∣

∣

∣

=C√

|a|max

0≤k≤msupb,t∈R

∣

∣

∣

∣

∣

emω(t−b

a )ψ(k)

(

t− b

a

)(

1

ak

)

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

emω(t

a )

emω(t−b

a )

∣

∣

∣

∣

∣

=C√

|a|

[

supb,t∈R

∣

∣

∣

∣

∣

emω(t

a )

emω(t−b

a )

∣

∣

∣

∣

∣

]

(

1

|a|m

)

max0≤k≤m

Pk,m(ψ)

≤ C max0≤k≤m

Pk,m(ψ)emω(b/a)

|a|m+1/2

by using property (α), as in the Lemma 2.1. This gives the required result.


3. Inversion of the wavelet transform on M′

ω

In order to study properties of the wavelet transform of f ∈ M′

ω we obtain anappropriate structure formula for f ∈ M

′

ω.

Assume that f ∈ M′

ω then as in the proof of the above theorem there exists anon-negative integer m and constant C > 0 such that for all φ ∈ Mω,

(11) | 〈f, φ〉 | ≤ C max0≤k≤m

supt∈R

∣

∣

∣emω(t)Dk

t φ(t)∣

∣

∣.

Then following [8, p. 112] we have following two cases:Case I. For t > 0,

emω(t)∣

∣Dkt φ(t)

∣

∣ = emω(t)∣

∣

∣

∣

∫ ∞

t

d

dz

(

Dkzφ(z)

)

dz

∣

∣

∣

∣

≤ max0≤k≤m

∫ ∞

t

emω(z)∣

∣

∣

∣

d

dz

(

Dkzφ(z)

)

dz

∣

∣

∣

∣

≤ max0≤k≤m+1

∫ ∞

−∞

∣

∣

∣emω(z)Dk

zφ(z)∣

∣

∣dz

≤ max0≤k≤m+1

∫ ∞

−∞

∣

∣

∣e−ω(z)e(m+1)ω(z)Dk

zφ(z)∣

∣

∣dz

≤ max0≤k≤m+1

∥

∥

∥e−ω(z)

∥

∥

∥

2

∥

∥

∥e(m+1)ω(z)Dk

zφ(z)∥

∥

∥

2.

Case II. For t < 0,

emω(t)∣

∣Dkt φ(t)

∣

∣ = emω(t)∣

∣

∣

∣

∫ t

−∞

d

dz

(

Dkzφ(z)

)

dz

∣

∣

∣

∣

≤ max0≤k≤m

∫ t

−∞

emω(z)∣

∣

∣

∣

d

dz

(

Dkzφ(z)

)

dz

∣

∣

∣

∣

≤ max0≤k≤m+1

∫ ∞

−∞

∣

∣

∣emω(z)Dk

zφ(z)∣

∣

∣dz

≤ max0≤k≤m+1

∫ ∞

−∞

∣

∣

∣e−ω(z)e(m+1)ω(z)Dk

zφ(z)∣

∣

∣dz

≤ max0≤k≤m+1

∥

∥

∥e−ω(z)

∥

∥

∥

2

∥

∥

∥e(m+1)ω(z)Dk

zφ(z)∥

∥

∥

2.

Therefore, ∃ C > 0, such that

(12) |〈f, φ〉| ≤ C max0≤k≤m+1

∥

∥

∥e(m+1)ω(t)Dk

t φ(t)∥

∥

∥

2, φ ∈ Mω.


Now, we show that the above L2−norm exists finitely. Indeed, for some µ > 0, wehave

max0≤k≤m+1

(∫ ∞

−∞

|e(m+1)ω(t)Dkt φ(t)|

2dt

)1

2

= max0≤k≤m+1

(∫ ∞

−∞

|e−µω(t)e(m+1+µ)ω(t)Dkt φ(t)|

2dt

)1

2

≤ max0≤k≤m+1

Pk,m+1+µ(φ)

(∫ ∞

−∞

e−2µ[p+q log(1+t)]dt

)1

2

, by (3)

we can choose µ large so that the last integral is finite. Thus

max0≤k≤m+1

∥

∥

∥e(m+1)ω(t)Dk

t φ(t)∥

∥

∥

2<∞.

Now, applying Hahn-Banach theorem and Riesz representation theorem to (12)we get gk belonging to the space L2(R) such that

| 〈f, φ〉 | =m+1∑

k=0

⟨

gk(t), e(m+1)ω(t)Dk

t φ(t)⟩

=

m+1∑

k=0

⟨

Dkt

(

e(m+1)ω(t)gk(t))

, φ(t)⟩

.

Therefore desired structure formula is

(13) f =

m+1∑

k=0

Dkt

(

e(m+1)ω(t)gk(t))

,

where gk ∈ L2(R) and k = 0, 1, 2, 3, ....

Theorem 3.1. Let f ∈ M′

ω, ψ ∈ Mω and W (b, a) be defined by ( 10). Then

DrbW (b, a) =

∂rW

∂br=

⟨

f(t),∂r

∂br1√

|a|ψ

(

t− b

a

)

⟩

,

and

DraW (b, a) =

∂rW

∂ar=

⟨

f(t),∂r

∂ar1

√

|a|ψ

(

t− b

a

)

⟩

.

Proof. Using the structure formula for f as given in (13) and following [12] we


have

W (b, a) = 〈f(t), ψb,a(t)〉

=

⟨

m+1∑

k=0

Dkt

(

e(m+1)ω(t)gk(t))

, ψb,a(t)

⟩

=

⟨

gk(t),m+1∑

k=0

(

e(m+1)ω(t))

(−1)kDkt

(

ψ(

t−ba

)

√

|a|

)⟩

.

Therefore, we have

∂rW

∂br(b, a) =

m+1∑

k=0

∫ ∞

−∞

gk(t)e(m+1)ω(t) ∂

r

∂br(−1)k

[

Dkt ψb,a(t)

]

dt

=

m+1∑

k=0

∫ ∞

−∞

gk(t)e(m+1)ω(t)(−1)k+rDk+r

t ψb,a(t)dt

=m+1∑

k=0

∫ ∞

−∞

gk(t)e(m+1)ω(t)(−1)kDk

t

∂r

∂brψb,a(t)dt

=m+1∑

k=0

⟨

Dkt

(

e(m+1)ω(t)gk(t))

,∂r

∂brψb,a(t)

⟩

=

⟨

f(t),∂r

∂brψb,a(t)

⟩

[by structure formula (13)].

Similarly result for differentiation with respect to a can be proved.

In order to derive inversion formula for the wavelet transform of f ∈ M′

ω wedefine function gν(t) as follows [1]:

gν(t) =

g(t), −ν ≤ t ≤ ν

0, elsewhere.

Also define fν ∈ M′

ω by

(14) 〈fν , φ〉 =

m+1∑

k=0

⟨

gν(t),(

e(m+1)ω(t)Dkt φ(t)

)⟩

, φ ∈ Dω,

then gν → g in L2(R) as ν → ∞ therefore, < fν , φ >−→< f, φ > as ν → ∞.

Theorem 3.2. Assume that the wavelet transform W (b, a) of f ∈ M′

ω withrespect to ψ ∈ Mω is defined by ( 10). Then

(15) limN→∞R→∞

⟨

1

Cψ

∫ R

−R

∫ N

−N

W (b, a)ψb,a(x)dbda

a2, φ(x)

⟩

= 〈f, φ〉 ,

for each φ ∈ Dω and b, a ∈ R, a 6= 0 where ψb,a(x) is given by ( 7).


Proof. Using the structure formula for fν as given in (14) we have

(Wfν)(b, a) =⟨

fν(t), ψb,a(t)⟩

=

∫ ∞

−∞

gν(t)

m+1∑

k=0

e(m+1)ω(t)Dkt ψb,a(t)dt.(16)

We wish to derive the inversion formula

1

Cψ

∫ ∞

−∞

∫ ∞

−∞

(Wfν)(b, a)ψb,a(x)dbda

a2= fν ,

interpreting convergence in the weak topology of D′

ω, i.e.

J ≡

⟨

1

Cψ

∫ ∞

−∞

∫ ∞

−∞

(Wfν)(b, a)ψb,a(x)dbda

a2, φ(x)

⟩

= 〈fν , φ〉 , ∀φ ∈ Dω.

From (16) we have

J =

⟨

1

Cψ

∫ ∞

−∞

∫ ∞

−∞

∫ ∞

−∞

gν(t)

m+1∑

k=0

e(m+1)ω(t)Dkt ψb,a(t)ψb,a(x)

dtdbda

a2, φ(x)

⟩

=

⟨

1

Cψ

∫ ∞

−∞

[

∫ ∞

−∞

{

∫ ∞

−∞

gν(t)

m+1∑

k=0

e(m+1)ω(t)(−1)kDkbψb,a(t)

}

ψb,a(x)dt

]

dbda

a2, φ(x)

⟩

as Dtψb,a(t) = −Dbψb,a(t). Therefore, by integration by parts with respect to bwe have

J =

⟨

1

Cψ

∫ ∞

−∞

∫ ∞

−∞

∫ ∞

−∞

gν(t)

m+1∑

k=0

e(m+1)ω(t)ψb,a(t)Dkbψb,a(x)

dtdbda

a2, φ(x)

⟩

=

⟨

1

Cψ

∫ ∞

−∞

∫ ∞

−∞

∫ ∞

−∞

gν(t)

m+1∑

k=0

e(m+1)ω(t)ψb,a(t)(−1)kDkxψb,a(x)

dtdbda

a2, φ(x)

⟩

.

Hence, by distributional differentiation,

J =

⟨

1

Cψ

∫ ∞

−∞

∫ ∞

−∞

∫ ∞

−∞

gν(t)

m+1∑

k=0

e(m+1)ω(t)ψb,a(t)ψb,a(x)dtdbda

a2, Dk

xφ(x)

⟩

(17)

The integrand

Dkxφ(x)ψb,a(x)ψb,a(t)gν(t)

e(m+1)ω(t)

a2


is absolutely integrable with respect to x and t in the x, t-plane and so Fubini’stheorem is applicable with respect to integration by x and t. Therefore (17) yields

J =1

Cψ

m+1∑

k=0

∫ ∞

−∞

∫ ∞

−∞

∫ ∞

−∞

∫ ∞

−∞

Dkxφ(x)ψb,a(x)ψb,a(t)gν(t)e

(m+1)ω(t) dxdtdbda

a2

=1

Cψ

m+1∑

k=0

∫ ∞

−∞

∫ ∞

−∞

∫ ∞

−∞

[

Wψ {Dkxφ(x)}(b, a)ψb,a(t)

dbda

a2

]

gν(t)e(m+1)ω(t)dt

=

m+1∑

k=0

∫ ∞

−∞

Dkt φ(t)gν(t)e

(m+1)ω(t)dt [by inversion formula (9)]

=m+1∑

k=0

⟨

gν(t), (−1)ke(m+1)ω(t)Dkt φ(t)

⟩

=

⟨

m+1∑

k=0

Dkt

(

e(m+1)ω(t)gν(t))

, φ(t)

⟩

= 〈fν , φ〉 [by structure formula (14)]

−→ 〈f, φ〉 as ν → ∞.

This completes the proof of the theorem.

4. Wavelet transform on S′

ω

In this section we assume that wavelets are of infraexponential decay so thattheir Fourier transforms are of compact support. To deal with such wavelets wesuppose that ψ ∈ Sω(R), then ψb,a ∈ Sω for fixed a, b ∈ R, a 6= 0. Now, weextend the wavelet transform in Fourier space defined by

(18) W (b, a) =1

2π

∫

R

eibω f(ω)ψ(aω)dω.

Assume that f(ω) ∈ D′

ω(R) and ψ(ω) ∈ Sω(R) is of compact support, then

ψ(aω)f(ω) ∈ E′

ω(R) [2, pp. 121-127]. Now, we define generalized wavelet transform

of f ∈ Z′

ω(R) [2, pp. 127] as generalized Fourier transform of f(·)ψ(a·):

W (b, a) =1

2π

⟨

f(ω), ψ(aω)eibω⟩

=1

2π

⟨

f(ω)ψ(aω), eibω⟩

=1

2π

⟨

1

af(u/a)ψ(u), eiζu/a

⟩

=1

2πa

⟨

ga(u), eiζu/a

⟩

.(19)

where ga(u) = f(u/a)ψ(u), a 6= 0.

Assume that suppψ(u) = [−β, β], α > 0. Then suppga(u) = [−β, β], β > 0.


Theorem 4.1. If ψ(u)f(u/a) ∈ E′

ω(R), its wavelet transform is a C∞ functionin R given by

(20) W (b, a) =1

2πa

⟨

gα(u), eibu/a

⟩

.

Moreover, W (b, a) can be extended to the complex space C as an entire analyticfunction given by

(21) W (ζ, a) =1

2πa

⟨

gα(u), eiζu/a

⟩

.

Proof. In (20), ga is a distribution with compact support while eibu/a is a C∞-function of u. Thus, the right-hand side of (20) is well defined. Also, by [2,Theorem 4.6, p. 124] it follows that the right-hand side of (20) is a C∞-functionof b ∈ R and W (b, a) can be extended to the complex plane as an entire analyticfunction given by (21).

Further proof is very similar to that given in [2, p. 124] in the case of Fouriertransform of distributions.

Remark 4.2. Relation (21) can be used to study Paley-Wiener-Schwartz the-orem for wavelet transform of ultradistribution of compact support.

References

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[11] Y. Meyer, Wavelets and Operators, Cambridge Univ. Press, Cambridge, 1992.

[12] J. N. Pandey, Wavelet transforms of Schwartz distributions, J. Comput. Anal. Appl.13 (1) (2011), pp. 47-83.

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[15] R. S. Pathak, Integral Transforms of Generalized Functions and Their Applications,Gordon and Breach Science Publishers, Amsterdam, 1997.

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[17] R. S. Pathak – S. K. Singh, Infraexponential decay of wavelets, Proc. Nat. Acad. Sci.India Sect. A 78 (2008), pp. 155-162.

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[19] A. H. Zemanian, Generalized Integral Transformations, Interscience Publishers, NewYork, 1996.

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