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Curvelet transform on periodicdistributionsRajendran Subash Moorthya & Rajakumar Roopkumara

a Department of Mathematics, Alagappa University, Karaikudi 630004, IndiaPublished online: 15 Jul 2014.

To cite this article: Rajendran Subash Moorthy & Rajakumar Roopkumar (2014) Curvelet transformon periodic distributions, Integral Transforms and Special Functions, 25:11, 874-887, DOI:10.1080/10652469.2014.938237

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Integral Transforms and Special Functions, 2014Vol. 25, No. 11, 874–887, http://dx.doi.org/10.1080/10652469.2014.938237

Curvelet transform on periodic distributions

Rajendran Subash Moorthy and Rajakumar Roopkumar∗

Department of Mathematics, Alagappa University, Karaikudi 630 004, India

(Received 19 March 2014; accepted 19 June 2014)

The curvelet transform is defined on the spaces of infinitely differentiable periodic functions, periodicdistributions and square integrable periodic functions and its properties are studied.

Keywords: curvelet transform; periodic distributions

AMS Subject Classification: 44A15; 44A99

1. Introduction

When the French mathematician Joseph Fourier (1768–1830) was trying to solve a problem inheat conduction, he needed to express a function as an infinite series of sine and cosine func-tions. Previously, Daniel Bernoulli and Leonard Euler had used such series with the problemsinvolving vibrating strings and astronomy. It happens in the way that expressing a function as aFourier series is sometimes more advantageous than expanding it as a power series. In particular,astronomical phenomena are usually periodic, as are heartbeats, tides, and vibrating strings. So itmakes reasonable to express them in terms of periodic functions. In many applications, the nonhomogeneous term in linear differential equation is of periodic functions.

In abstract, many integral transforms are investigated on periodic functions and periodicdistributions. Holschneider [1] studied the wavelet transform on periodic distributions and Zayed[2] extended the wavelet transform to the space of periodic Beurling ultradistributions. Then thewavelet transform is extended to a larger class of generalized function space, called the periodicBoehmians in [3]. The Hilbert transform on periodic distributions is also discussed in [4]. In thisdirection, we have already studied the curvelet transform on Boehmians and on tempered distri-butions in [5,6]. Applying the properties of Fourier series and Fourier transform, we shall extendthe curvelet transform to periodic functions and periodic distributions, in this paper.

This paper is organized as follows. In Section 2, we recall some basic notations and resultsfrom [7,8]. We also present some essential results of Fourier transform and Fourier coefficients.In Section 3, we analyse the curvelet transform [9,10] over the suitable space of infinitely differ-entiable periodic functions on R2. In Section 4, we discuss the curvelet transform on a suitablespace of periodic distributions and its properties.

∗Corresponding author. Email: roopkumarr@rediffmail.com

c© 2014 Taylor & Francis

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Integral Transforms and Special Functions 875

2. Preliminaries

We use the notations N, N0, Z, R, C, respectively, to denote the set of all positive integers, non-negative integers, integers, real numbers and complex numbers. If x = (x1, x2), y = (y1, y2) ∈ R2

and m = (m1, m2) ∈ N20, then we use following standard notations: x · y = x1y1 + x2y2; |x| =√

x · x; |m|1 = m1 + m2; xm = xm11 xm2

2 ; and Dmx = (∂/∂x1)

m1(∂/∂x2)m2 .

Definition 2.1 [8, p.301] A complex-valued function f on R2 is said to be periodic with period2π if f (x1 + 2π , x2) = f (x1, x2) = f (x1, x2 + 2π), ∀x1, x2 ∈ R2.

Definition 2.2 The space P∞(R2) consists of all infinitely differentiable 2π -periodic func-tions on R2, together with the Fréchet space topology induced by the following family of seminorms:

‖f ‖P∞(R2);N = sup|m|1≤N

supx∈R2

|Dmx f (x)|, ∀ N ∈ N0.

Definition 2.3 With the reference to [11, p.13], for f ∈ P∞(R2), we have

f (x) =∑n∈Z2

f̂ (n) ein·x (in L 2 sense), (2.1)

where the Fourier coefficient of f denoted by f̂ (n), is defined by

f̂ (n) = 1

(2π)2

∫T2

f (x) e−in·x dx, ∀ n ∈ Z2.

Definition 2.4 Let S(Z2) be the collection of all double sequences r = (r(k)) where k ∈ Z2,satisfying

sup|m|1≤N

supk∈Z2

|kmr(k)| < ∞, ∀ N ∈ N0.

If ‖r‖S(Z2);N = sup|m|1≤N supk∈Z2 |kmr(k)|, ∀r ∈ S(Z2) then {‖ · ‖S(Z2);N : N ∈ N0} is a sequenceof semi norms which makes S(Z2) a Fréchet space.

For f ∈ P∞(R2), we associate a double sequence (f̂ (n)), n ∈ Z2 called the sequence of Fouriercoefficients defined by f̂ (n) = F[f (x)](n) = (1/2π)2

∫T2 f (x) e−in·x dx, ∀n ∈ Z2.

Theorem 2.5 The map F : P∞(R2) → S(Z2) is a homeomorphism.

Proof Let f ∈ P∞(R2) and m ∈ N20 with |m|1 ≤ N ∈ N0. With the help of integration by parts

and using the periodicity of f in each variable, we have

∣∣∣∣∫

T2kmf (x) e−ik·x dx

∣∣∣∣ =∣∣∣∣∫

T2Dm

x f (x) e−ik·x dx

∣∣∣∣ ≤ (2π)2‖f ‖P∞(R2);N .

Thus ‖Ff ‖S(Z2);N ≤ ‖f ‖P∞(R2);N , which shows that Ff ∈ S(Z2), and that F is continuous.For any sequence (r(n)) in S(Z2), we define f (x) = ∑

n∈Z2 r(n) ein·x. It is easy to see that f isperiodic. The inequality |r(n)| ≤ |(nm1+2

1 nm2+22 /n2

1n22)r(n)| ≤ ‖r‖S(Z2);4/n2

1n22, (n1, n2 �= 0) shows

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876 R. Subash Moorthy and R. Roopkumar

the uniform convergence of above sum. Therefore,∣∣∣∣∣Dmx

∑n∈Z2

r(n) ein·x∣∣∣∣∣ =

∣∣∣∣∣∑n∈Z2

nmr(n) ein·x∣∣∣∣∣ ≤

∑n∈Z

2,n1,n2 �=0

∣∣∣∣∣nm1+21 nm2+2

2

n21n2

2

r(n) ein·x∣∣∣∣∣ .

Hence, it follows that ∣∣∣∣∣Dmx

∑n∈Z2

r(n) ein·x∣∣∣∣∣ ≤ ‖r‖S(Z2),N+4

∑n∈Z

2,n1,n2 �=0

1

n21n2

2

, (2.2)

and so f is infinitely differentiable. Thus f ∈ P∞(R2). Furthermore, using

1

(2π)2

∫T2

eik·x e−in·x dx ={

1 if n = k,

0 if n �= k.

we get

F[f (x)](n) = r(n), (2.3)

Thus F is onto. From (2.2) we get that the inverse of F is continuous. �

For a given double sequence (r(n)) ∈ S(Z2), the inverse of F−1 of F is obtained from (2.3) asfollows:

F−1[r(n)](x) =∑n∈Z2

r(n) ein·x. (2.4)

The following lemma is an immediate from the definition of F.

Lemma 2.6 Let f , g ∈ P∞(R2). If (f ∗ g)(x) = ∫T2 f (x − y)g(y) dy and f̃ (x) = f (−x),

∀x ∈ R2, then F[(f ∗ g)(x)](n) = (2π)2[Ff (x)](n)[Fg(x)](n) and [Ff̃ (x)](n) = [Ff (x)](−n),∀n ∈ Z2.

Now from [7], we recall the testing functions space S(R2) of all infinitely differentiable func-tions on R2, together with the Fréchet space topology generated by the following family of seminorms:

‖f ‖S(R2);k,m = supx∈R2

∣∣xkDmx f (x)

∣∣ for all k, m ∈ N20. (2.5)

Furthermore, Fourier transform F on S(R2) is defined by F[f (x)](t) = ∫R2 f (x) e−ix·t dx,

t ∈ R2.

Theorem 2.7 [7] The Fourier transform F : S(R2) → S(R2) is continuous and hence it is ahomeomorphism. Furthermore, inverse of F is given by F−1(g)(x) = (1/(2π)2)

∫R2 g(t) eix·t dt,

x ∈ R2.

Definition 2.8 The periodization operator � from S(R2) to P∞(R2) can be defined by(�f )(x) = ∑

n∈Z2 f (x + 2πn), x ∈ R2.

Lemma 2.9 If f ∈ S(R2), then the series∑

n∈Z2 f (x + 2πn) converges uniformly on everycompact subset of R2.

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Integral Transforms and Special Functions 877

Proof Let f ∈ S(R2). Let K be any compact subset of R2 and M be such that |x| ≤ M , ∀x ∈ K.Using the well-known inequality,

|x + y|N ≤ 2N−1(|x|N + |y|N ), ∀ x, y ∈ R2, ∀ N ∈ N0,

we get

|f (x + 2πn)| ≤ 23(‖f ‖S(R2);4,0 + 22M 4‖f ‖S(R2);0,0)

n21n2

2

,

which concludes the proof of the lemma. �

Lemma 2.10 For any f ∈ S(R2), we have the Poisson summation formula,

∑n∈Z2

f (x + 2πn) = 1

(2π)2

∑n∈Z2

(Ff )(n) ein·x.

Proof For any function f ∈ S(R2), we associate a double sequence in S(Z2) in such a waythat (�f )(n) = f (n), n = (n1, n2) ∈ Z2. As � is a restriction operator, it is a continuous lin-ear function from S(R2) into S(Z2). By a direct computation, we see that F� = (1/(2π)2)�F,and hence � = (1/(2π)2)F−1�F. In other words, using (2.4), we have

∑n∈Z2 f (x + 2πn) =

(1/(2π)2)∑

n∈Z2(Ff )(n) ein·x. �

Lemma 2.11 If f ∈ S(R2), then the series∑

n∈Z2(Ff )(n) ein·x converges uniformly on everycompact subset of R2.

Proof Using Lemmas 2.9 and 2.10, proof of this lemma follows. �

Theorem 2.12 The periodization operator � : S(R2) → P∞(R2) is continuous.

Proof From Lemma 2.10, we have � = (1/(2π)2)F−1�F. Since F−1, �, F are continuous, weget that � continuous. �

3. The curvelet transform of P∞a0,b0

(R2)

First we recall the theory of curvelet transform from the literature [9,10] introduced by E. J.Candès and D. L., Donoho. If W , V are positive real-valued functions on R satisfying

supp W ⊂[

1

2, 2

],

∫ 2

1/2(W(ar))2 da

a= 1, ∀r > 0 (3.1)

supp V ⊂ [−1, 1],∫ 1

−1(V(u))2 du = 1, (3.2)

and 0 < a < b0 < π2, then a curvelet is defined by

γa,0,0(x) = F−1

[W(ar)V

(ω√

a

)a3/4

](x), x, r eiω ∈ R2 \ {0}, with ω ∈ (−π , π ].

In other words,

F[γa,0,0(x)](t) = W(a|t|2)V(

arg(t)√a

)a3/4, where arg(t) ∈ (−π , π ].

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For the scaling parameter a ∈ (0, b0) ⊂ (0, π2), position parameter b ∈ R2 and rotationparameter θ ∈ [0, 2π), a curvelet in [9,10] is defined by

γa,b,θ (x) = γa,0,θ (x − b) = γa,0,0(e−iθ (x − b)), ∀ x ∈ R2.

However, for the purpose of discussing the curvelet transform on P∞a0,b0

(R2) and on its dual space,throughout the paper, we assume that V and W are infinitely differentiable functions satisfyingthe admissibility conditions (3.1) and (3.2). Already, we pointed out a small mistake in the inver-sion theorem for curvelet transform, presented in [10], and corrected the condition ‘f̂ (ξ) = 0, if|ξ | < 2/b0’ as ‘f̂ (reiω) = 0, if 0 < r < 2/b0 or −π < ω <

√b0’. For more details we refer the

reader to [5].Next, we recall the theory of curvelet transform on a subspace of tempered distributions from

[6]. When we discuss the continuity of curvelet transform on a subspace of rapidly decreasingfunctions on R2, we observe that it is better to modify the S by U = (a0, b0) × R2 × (−π , π), forsome 0 < a0 < b0. The explanation for the fact that this modification does not affect the theoryor applications of curvelet transform is presented in [6].

Next, we introduce some other notations, which are required in the following sequel. Fora0, b0 ∈ R with 0 < 4a0 < b0 < π2, we define

ρa,0,0(x) =∑n∈Z2

γa,0,0(x + 2πn), x ∈ R2, ∀ a ∈ [a0, b0]

and

ρa,b,θ (x) = ρa,0,0(e−iθ (x − b)), ∀ x,b ∈ R2, ∀ a ∈ [a0, b0], ∀ θ ∈ [0, 2π ].

Clearly, we have

ρa,b,θ (x) = ρa,0,0(e−iθ (x − b)) =

∑n∈Z2

γa,b,θ (x + 2πn), ∀ x ∈ R2.

Definition 3.1 Let Y = [a0, b0] × R2 × [0, 2π). The curvelet transform � of any functionf ∈ P∞(R2) is defined by

(�f )(a, b, θ) = 〈ρa,b,θ (x), f (x)〉L 2(T2), ∀ (a, b, θ) ∈ Y,

where 〈ρa,b,θ (x), f (x)〉L 2(T2) = ∫T2 ρa,b,θ (x)f (x) dx.

First we note that for a given f ∈ P∞(R2), for each fixed a ∈ [a0, b0], θ ∈ [0, 2π ], (�f )(a, b, θ)

is a periodic function on R2. Indeed, we know that

ρa,b,θ (x1 + 2π , x2) = ρa,b,θ (x1, x2 + 2π), ∀ (a, b, θ) ∈ Y and ∀ (x1, x2) ∈ R2,

and hence we get

(�f )(a, (b1 + 2π , b2), θ) =∫ 2π

0

∫ 2π

0ρa,(b1+2π ,b2),θ (x1, x2)f ((x1, x2)) dx1 dx2

=∫ 2π

0

∫ 2π

0ρa,(b1,b2),θ (x1 − 2π , x2)f ((x1, x2)) dx1 dx2

=∫ 2π

0

∫ 2π

0ρa,(b1,b2),θ (x1, x2)f ((x1, x2)) dx1 dx2

= (�f )(a, (b1, b2), θ),

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Integral Transforms and Special Functions 879

and we also get

(�f )(a, (b1, b2), θ) = (�f )(a, (b1, b2 + 2π), θ), ∀ (b1, b2) ∈ R2,

in a similar way.

Definition 3.2 Let P(Y) denote the set of all infinitely differentiable functions G : Y → C

satisfying

G(a, (b1 + 2π , b2), θ) = G(a, (b1, b2), θ) = G(a, (b1, b2 + 2π), θ), ∀ (a, b1, b2, θ) ∈ Y.

It is a Fréchet space with the topology generated by the following family of semi norms:

‖G‖S(Y);α,k,β = sup(a,b,θ)∈Y

|∂αa ∂k

b ∂β

θ G(a, b, θ)|, ∀ α, β ∈ N0, ∀ k ∈ N20.

We now recall the multivariate Faa di Bruno’s formula from [12, p.503], which will be appliedin the proof of following lemma.

Let n ∈ N. Let g(x) be defined on a neighbourhood of x0 and have derivatives up to order n atx0, let f (y) be defined on a neighbourhood of y0 = g(x0) and have derivatives up to order n at y0.Then the nth derivative of the composition h(x) = f [g(x)] at x0 is given by the formula

Dnxh(x0) =

n∑k=1

Dkyf (y0)

∑I(n,k)

(n!)n∏

j=1

[Djxg(x0)]�j

(�j!)(j!)�j, (3.3)

where

I(n, k) =⎧⎨⎩(�1, �2, . . . �n) : �j ∈ N0,

n∑j=1

�j = k,n∑

j=1

j�j = n

⎫⎬⎭ .

Lemma 3.3 For all n ∈ Z2, α, β ∈ N0 and k ∈ N20, we have |nk∂α

a ∂β

θ (Fγa,b,θ )(n)| ≤ �k,α,β ,where �k,α,β is a constant depending on k, α, β.

Proof Let n = r eiω ∈ Z2 \ {0}, α, β ∈ N0 and k ∈ N20. For,

|∂αa ∂

β

θ (Fγa,b,θ )(n)| =∣∣∣∣∂α

a ∂β

θ W(ar)V

((ω − θ)√

a

)a3/4

∣∣∣∣=

∣∣∣∣∣∂αa W(ar)V (β)

((ω − θ)√

a

) (−1√a

)β

a3/4

∣∣∣∣∣=

∣∣∣∣∂αa

[W(ar)V (β)

((ω − θ)√

a

)a(3−2β)/4

]∣∣∣∣ . (3.4)

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880 R. Subash Moorthy and R. Roopkumar

Using the Leibnitz Formula and Faa di Bruno’s formula,[12] we obtain

∂αa

[W(ar)V (β)

((ω − θ)√

a

)a(3−2β)/4

]

=∑p≤α

cα,p∂α−pa W(ar)

∑q≤p

cp,q∂p−qa V (β)

((ω − θ)√

a

)∂q

a a(3−2β)/4

=∑p≤α

∑q≤p

cα,pcp,q∂α−pa W(ar)∂p−q

a V (β)

((ω − θ)√

a

)∂q

a a(3−2β)/4

=∑p≤α

∑q≤p

cα,pcp,qrα−pW (α−p)(ar)|p−q|∑λ=1

V (β+λ)

((ω − θ)√

a

) ∑I(p−q,λ)

(p − q)!

×|p−q|∏j=1

[∂ ja((ω − θ)/

√a)]λj

(λj!)(j!)λjQβ,qa(3−2β)/4−q, where Qβ,q =

q−1∏μ=1

(3 − 2β

4− μ

)

=∑p≤α

∑q≤p

cα,pcp,qrα−pW (α−p)(ar)|p−q|∑λ=1

V (β+λ)

((ω − θ)√

a

) ∑I(p−q,λ)

(p − q)!

×|p−q|∏j=1

[(ω − θ)Q1/2,ja−1/2−j]λj

(λj!)(j!)λjQβ,qa(3−2β)/4−q, here Q1/2,j =

j−1∏ν=1

(−1

2− ν

)

=∑p≤α

∑q≤p

cα,pcp,qrα−pW (α−p)(ar)|p−q|∑λ=1

V (β+λ)

((ω − θ)√

a

) ∑I(p−q,λ)

(p − q)!

×|p−q|∏j=1

Qλj

1/2,j(ω − θ)λa−λ/2−(p−q)

(λj!)(j!)λjQβ,qa(3−2β)/4−q

= a3/4∑p≤α

∑q≤p

cα,pcp,qrα−pW (α−p)(ar)|p−q|∑λ=1

V (β+λ)

((ω − θ)√

a

)

×∑

I(p−q,λ)

(p − q)!|p−q|∏j=1

Qλj

1/2,j(ω − θ)λa−λ/2−p−β/2

(λj!)(j!)λjQβ,q

= a3/4∑p≤α

∑q≤p

cα,pcp,q((ar)α−pW (α−p)(ar))|p−q|∑λ=1

V (β+λ)

((ω − θ)√

a

)

×∑

I(p−q,λ)

(p − q)!|p−q|∏j=1

Qλj

1/2,j((ω − θ)/√

a)λa−α−β/2

(λj!)(j!)λjQβ,q.

Since V and W are compactly supported smooth functions, we have |xαW (α)(x)| ≤ Aα ,|yβV (β)(y)| ≤ Bβ , for some constants Aα , Bβ depending on α, β respectively. From (3.4) we get

|nk∂αa ∂

β

θ (Fγa,b,θ )(n)| ≤ a3/4∑p≤α

∑q≤p

cα,pcp,q|ar||k|1+α−p|W (α−p)(ar)||p−q|∑λ=1

∑I(p−q,λ)

(p − q)!

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Integral Transforms and Special Functions 881

×∣∣∣∣∣(

ω − θ√a

)λ

V (β+λ)

((ω − θ)√

a

)∣∣∣∣∣|p−q|∏j=1

|Qλj

1/2,ja−α−β/2|

(λj!)(j!)λj|Qβ,q|

≤ b3/40

∑p≤α

∑q≤p

cα,pcp,qAk,α−p

|p−q|∑λ=1

∑I(p−q,λ)

(p − q)!Bβ,λ

×|p−q|∏j=1

|Qλj

1/2,ja−α−β/20 |

(λj!)(j!)λj|Qβ,q|

= �k,α,β (say). (3.5)

Hence the lemma. �

Lemma 3.4 For all x ∈ R2, we have |∂αa ∂

β

θ ρa,b,θ (x)| ≤ Cα,β , where Cα,β is a constant dependingon α, β.

Proof Let x ∈ R2. Using Lemmas 2.10 and 3.3,

|∂αa ∂

β

θ ρa,b,θ (x)| =∣∣∣∣∣∑n∈Z2

∂αa ∂

β

θ (Fγa,b,θ )(n) ein·x∣∣∣∣∣

≤∑n∈Z2

|n|4|∂αa ∂

β

θ (Fγa,b,θ )(n)|n2

1n22

≤ �4,α,β

∑n∈Z2

1

n21n2

2

= Cα,β (say).

where �4,α,β is defined as in (3.5). Hence the lemma. �

Lemma 3.5 For all k ∈ N20, ∂k

b ρa,b,θ = (−1)|k|1∂kx ρa,b,θ , where |k|1 = k1 + k2.

Proof By a direct computation, we get this lemma. �

Lemma 3.6 For all m ∈ Z2, ρ̂a,b,θ (m) = 1/(2π)2(Fγa,b,θ )(m).

Proof Using Lemmas 2.10 and 2.11, we have

ρ̂a,b,θ (m) = 1

(2π)2

∫T2

∑n∈Z2

γa,b,θ (x + 2πn) e−im·x dx

= 1

(2π)4

∫T2

∑n∈Z2

(Fγa,b,θ )(n) ein·x e−im·x dx

= 1

(2π)4

∑n∈Z2

(Fγa,b,θ )(n)

∫ 2π

0

∫ 2π

0ei(n−m)·x dx

= 1

(2π)2(Fγa,b,θ )(m).

�

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Definition 3.7 Let a0, b0 ∈ (0, π2) such that 4a0 < b0. We define

P∞a0,b0

(R2) ={

f ∈ P∞(R2) : f̂ (m) = 0 for all m satisfying |m|

<2

b0or

1

2a0< |m| or − π < arg(m) <

√b0

}.

One can observe that P∞a0,b0

(R2) consists of trigonometric polynomials, since the Fouriercoefficients of every element of P∞

a0,b0(R2) are zero for all but finitely m.

Theorem 3.8 The mapping � : P∞a0,b0

(R2) → P(Y) is continuous.

Proof Let α, β ∈ N0 and k ∈ N20. Then, using integration by parts, we have

|∂αa ∂k

b ∂β

θ �f (a, b, θ)| =∣∣∣∣∫

T2∂α

a ∂kb ∂

β

θ ρ̄a,b,θ (x)f (x) dx

∣∣∣∣=

∣∣∣∣∫

T2∂k

x ∂αa ∂

β

θ ρ̄a,b,θ (x)f (x) dx

∣∣∣∣ , (using Lemma 3.5)

=∣∣∣∣∫ 2π

0

∫ 2π

0∂k1

x1∂k2

x2∂α

a ∂β

θ ρ̄a,b,θ (x1, x2)f (x1, x2) dx1 dx2

∣∣∣∣=

∣∣∣∣∫ 2π

0

([f (x1, x2)∂

k1−1x1

∂k2x2

∂αa ∂

β

θ ρ̄a,b,θ (x1, x2)]2π0

− [∂x1 f (x1, x2)∂k1−2x1

∂k2x2

∂αa ∂

β

θ ρ̄a,b,θ (x1, x2)]2π0

...

− [∂k1−1x1

f (x1, x2)∂k2x2

∂αa ∂

β

θ ρ̄a,b,θ (x1, x2)]2π0

−∫ 2π

0∂k2

x2∂α

a ∂β

θ ρ̄a,b,θ (x1, x2)∂k1x1

f (x1, x2) dx1

)dx2

∣∣∣∣=

∣∣∣∣∫ 2π

0

∫ 2π

0∂k2

x2∂α

a ∂β

θ ρ̄a,b,θ (x1, x2)∂k1x1

f (x1, x2) dx1 dx2

∣∣∣∣ ,

(using the periodicity of f and ρa,b,θ )

=∣∣∣∣∫ 2π

0

∫ 2π

0∂α

a ∂β

θ ρ̄a,b,θ (x1, x2)∂k2x2

∂k1x1

f (x1, x2) dx1 dx2

∣∣∣∣=

∣∣∣∣∫

T2∂α

a ∂β

θ ρ̄a,b,θ (x)∂kx f (x) dx

∣∣∣∣ ≤ (2π)2Cα,β‖f ‖P∞(R2);N ,

by using Lemma 3.4. Hence the theorem follows. �

Definition 3.9 For any G ∈ P(Y) the adjoint curvelet transform �∗ is defined by

(�∗G)(x) =∫ b0

a0

∫T2

∫ 2π

0ρa,b,θ (x)G(a, b, θ)

da

a3db dθ , ∀ x ∈ R2.

Theorem 3.10 The adjoint curvelet transform �∗ : P(Y) → P∞a0,b0

(T2) is continuous.

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Integral Transforms and Special Functions 883

Proof Let k ∈ N20, x ∈ R2. Then using Lemma 3.5,

|∂kx (�∗G)(x)| =

∣∣∣∣∫ b0

a0

∫T2

∫ 2π

0∂k

x ρa,b,θ (x)G(a, b, θ)da

a3db dθ

∣∣∣∣=

∣∣∣∣∫ b0

a0

∫T2

∫ 2π

0∂k

b ρa,b,θ (x)G(a, b, θ)da

a3db dθ

∣∣∣∣=

∣∣∣∣∫ b0

a0

∫T2

∫ 2π

0ρa,b,θ (x)∂k

b G(a, b, θ)da

a3db dθ

∣∣∣∣≤ ‖G‖P(Y);0,k,0

∫ b0

a0

∫T2

∫ 2π

0|ρa,b,θ (x)|da

a3db dθ

≤ C0,0‖G‖P(Y);0,k,0

∫ b0

a0

∫T2

∫ 2π

0

da

a3db dθ

≤ C0,0C‖G‖P(Y);0,k,0, where C =∫ b0

a0

∫T2

∫ 2π

0

da

a3db dθ .

Hence �∗ is continuous. �

Definition 3.11 If f , g ∈ P∞a0,b0

(R2) then the convolution f ∗ g is defined by (f ∗ g)(x) =∫T2 f (x − y)g(y) dy, x ∈ R2.

Theorem 3.12 For all f ∈ P∞a0,b0

(R2), we have �∗�f = f .

Proof Let f ∈ P∞a0,b0

(R2). By Definition 3.9 of the adjoint curvelet transform,

(�∗�f )(x) =∫ 2π

0

∫T2

∫ b0

a0

�f (a, b, θ)ρa,b,θ (x)da

a3db dθ

=∫ 2π

0

∫ b0

a0

∫T2

ρa,b,θ (x)

∫T2

ρa,b,θ (y)f (y) dy dbda

a3dθ

=∫ 2π

0

∫ b0

a0

∫T2

ρa,b,θ (x)

∫T2

ρ̃a,0,θ (b − y)f (y) dy dbda

a3dθ

=∫ 2π

0

∫ b0

a0

∫T2

ρa,0,θ (x − b)( ¯̃ρa,0,θ ∗ f )(b) dbda

a3dθ

=∫ b0

a0

∫ 2π

0(ρa,0,θ ∗ ( ¯̃ρa,0,θ ∗ f ))(x)

da

a3dθ

=∫ b0

a0

∫ 2π

0((ρa,0,θ ∗ ¯̃ρa,0,θ ) ∗ f )(x)

da

a3dθ , ∀ x ∈ R2. (3.6)

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Using Fubini’s theorem and Lemma 2.6, we have,

�̂∗�f (m) =∫ 2π

0

∫ b0

a0

((ρa,0,θ ∗ ¯̃ρa,0,θ ) ∗ f )̂(m)da

a3dθ

= (2π)4∫ 2π

0

∫ b0

a0

ρ̂a,0,θ (m)ˆ̃̄ρa,0,θ (m)f̂ (m)

da

a3dθ

= (2π)4∫ 2π

0

∫ b0

a0

ρ̂a,0,θ (m) ¯̂ρa,0,θ (m)f̂ (m)da

a3dθ

= (2π)4∫ 2π

0

∫ b0

a0

|ρ̂a,0,θ (m)|2 f̂ (m)da

a3dθ

= (2π)4 f̂ (m)

∫ 2π

0

∫ b0

a0

|ρ̂a,0,θ (m)|2 da

a3dθ , ∀m ∈ N2

0. (3.7)

Let m ∈ N20 with |m| ∈ [2/b0, 1/2a0] and

√b0 < arg (m) < π . Then, we have supp V ⊆

[−1, 1] ⊆ [(arg(m) − 2π)/√

b0, arg (m)/√

b0] ⊆ [(arg (m) − 2π)/√

a, arg (m)/√

a], ∀a ∈[a0, b0] and supp W ⊆ [ 1

2 , 2] ⊆ [a0|m|, b0|m|]. Therefore, using Lemma 3.6, we obtain

(2π)4∫ 2π

0

∫ b0

a0

|ρ̂a,0,θ (m)|2 da

a3dθ =

∫ 2π

0

∫ b0

a0

|(Fγa,0,θ )(m)|2 da

a3dθ

=∫ 2π

0

∫ b0

a0

[W(a|m|)V

((arg(m) − θ)√

a

)a3/4

]2 da

a3dθ

=∫ b0

a0

[W(a|m|)a3/4]2∫ 2π

0

[V

(arg(m) − θ√

a

)]2

dθda

a3

=∫ b0

a0

(W(a|m|))2a3/2√a∫ arg(m)/

√a

[arg(m)−2π]/√

a(V(u))2 du

da

a3

=∫ b0|m|

a0|m|(W(v))2 dv

v= 1.

Using this observation in (3.7), we get �̂∗�f (m) = f̂ (m) for all m ∈ N20 with |m| ∈

[2/b0, 1/2a0] and√

b0 < arg m < π . For the remaining m, we get �̂∗�f (m) = f̂ (m) = 0. �

4. The curvelet transform of P′a0,b0

(R2)

We recall some well-known facts. The elements of P′a0,b0

(R2) are defined as set of all continuouslinear functionals on P∞

a0,b0(R2). That is, for any � ∈ P′

a0,b0(R2), there is a non-negative integer

n such that |�(f )| ≤ C‖f ‖P∞a0,b0

(R2);n, ∀f ∈ P∞a0,b0

(R2).

A sequence (�n) → 0 in P′a0,b0

(R2) as n → ∞ if and only if �n(f ) → 0, ∀f ∈ P∞a0,b0

(R2).Also we can identify any function f of P∞

a0,b0(R2) with the distribution defined by 〈f , ϕ〉L 2(T2) =∫

T2 f (x)ϕ(x) dx, ∀ϕ ∈ P∞a0,b0

(R2).

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Integral Transforms and Special Functions 885

By a routine procedure, one can verify that P∞a0,b0

(R2) is dense in P′a0,b0

(R2), and therefore, fora given � ∈ P′

a0,b0(R2), we can choose fn ∈ P∞

a0,b0(R2) (treating as distributions) such that

|〈fn, f 〉L 2(T2)| ≤ Cf , ∀ f ∈ P∞a0,b0

(R2), ∀ n ∈ N, (4.1)

where Cf is a constant depending on f .We consider P′(Y) as the space of continuous linear functionals on P(Y). If G, H are the

functions over Y, we define

〈G(a, b, θ), H(a, b, θ)〉L 2(Y) =∫ 2π

0

∫T2

∫ b0

a0

G(a, b, θ)H(a, b, θ)da

a3db dθ , (4.2)

provided the integral exists.

Definition 4.1 The curvelet transform � : P′a0,b0

(R2) → P′(Y) and the adjoint curvelet trans-form �∗ : P′(Y) → P′

a0,b0(R2) are, respectively, defined by

(��)(G) = �(�∗G), ∀G ∈ P(Y), ∀ � ∈ P′a0,b0

(R2).

(�∗ϒ)(f ) = ϒ(�f ), ∀ f ∈ P∞a0,b0

(R2), ∀ ϒ ∈ P′(Y).

Theorem 4.2 � and �∗ are the only possible continuous extensions of � restricted to P∞a0,b0

(R2)

and �∗ restricted to P(Y).

Proof Let �n ∈ P′a0,b0

(R2) be such that �n → 0 as n → ∞. For all G ∈ P(Y), (��n)(G) =�n(�

∗G) → 0 as n → ∞ which implies that � is a continuous map from P′a0,b0

(R2) to P′(Y).For f ∈ P∞

a0,b0(R2), and for a G ∈ P(Y),

(�f )(G) =∫

T2f (x)

∫Y

ρa,b,θ (x)G(a, b, θ)da

a3db dθ dx

=∫

Y

G(a, b, θ)

∫T2

f (x)ρa,b,θ (x) dxda

a3db dθ (by Fubini’s theorem)

=∫

Y

G(a, b, θ)�f (a, b, θ)da

a3db dθ

= 〈�f (a, b, θ), G(a, b, θ)〉L 2(Y).

Since G is arbitrary, we have proved that the Definition 4.1 is consistent with Definition 3.1. Bythe density of P∞

a0,b0(R2) in P′

a0,b0(R2), we have that the extension is unique. �

Theorem 4.3 Let � ∈ P′a0,b0

(R2), For all G ∈ P(Y) we have (��)(G) = 〈Y, G〉, where thefunction Y ∈ C∞(Y) such that Y(a, b, θ) = �(ρa,b,θ ).

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886 R. Subash Moorthy and R. Roopkumar

Proof Let � ∈ P′a0,b0

(R2). Using (4.1) we can find a sequence (fn) in P∞a0,b0

(R2) such that(fn) → � as n → ∞ in P′

a0,b0(R2), and

|〈fn, ϕ〉L (T2)| ≤ Cϕ , ∀ ϕ ∈ P∞a0,b0

(R2), n ∈ N, for some Cϕ > 0.

Then, using Fubini’s theorem, we obtain

(��)(G) = �(�∗G)

= limn→∞

∫T2

fn(x)(�∗G)(x) dx

= limn→∞

∫T2

∫Y

fn(x)ρa,b,θ (x)G(a, b, θ)da

a3db dθ dx

= limn→∞

∫Y

〈fn, ρa,b,θ 〉L (T2)G(a, b, θ)da

a3db dθ .

Therefore, |〈fn, ρa,b,θ 〉L (T2)G(a, b, θ)| ≤ Cρa,b,θ |G(a, b, θ)| ∈ L 1(Y). Hence, by applying thedominated convergence theorem, we have

(��)(G) =∫

Y

limn→∞〈fn, ρa,b,θ 〉L (T2)G(a, b, θ)

da

a3db dθ

=∫

Y

�(ρa,b,θ )G(a, b, θ)da

a3db dθ

= 〈Y(a, b, θ), G(a, b, θ)〉L 2(Y) where Y(a, b, θ) = �(ρa,b,θ ).

Hence the theorem. �

Theorem 4.4 Let f ∈ P∞a0,b0

(R2) and � be a distribution in P′a0,b0

(R2). Then �(f ) =〈Y, �f 〉L 2(Y), where Y = �� as given by Theorem 4.3.

Proof From Theorem 3.12, we have (��)(�f ) = �(�∗�f ) = �(f ). Since �f ∈ P(Y), usingthe previous theorem, we get that (��)(�f ) can be written as 〈Y(a, b, θ), �f (a, b, θ)〉L 2(Y),which concludes the theorem. �

Acknowledgements

The authors are indebted to the anonymous referee whose valuable comments and suggestions helped to improve thequality of this paper.

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