CONVOLUTION FOR THE DISCRETE WAVELET TRANSFORM

December 16, 2011 15:1 WSPC/S0219-6913 181-IJWMIP 00437

International Journal of Wavelets, Multiresolutionand Information ProcessingVol. 9, No. 6 (2011) 905–922c© World Scientific Publishing CompanyDOI: 10.1142/S0219691311004377

CONVOLUTION FOR THE DISCRETEWAVELET TRANSFORM

R. S. PATHAK

DST Centre for Interdisciplinary Mathematical Sciences

Banaras Hindu UniversityVaranasi — 221 005, India

[email protected]

Received 22 June 2010

Revised 2 February 2011

Translation and convolution associated with the discrete wavelet transform are investi-gated using properties of Calderon–Zygmund operator and Riesz fractional integral oper-ator. Dual convolution is also studied. The wavelet convolution is applied to approximatefunctions belonging to certain Lp-spaces.

Keywords: Wavelets; wavelet transform; translation; convolution.

AMS Subject Classification: 42C40, 42A38, 44A35

1. Introduction

A theory of convolution associated with the continuous wavelet transform has beendeveloped by Pathak and Pathak11 using the technique of Hirschman Jr.7 Convo-lution associated with the discrete wavelet transform has been defined and some ofits properties have been stated formally in Ref. 10. Conditions of validity of vari-ous results involving convolution for the discrete wavelet transform, called waveletconvolution henceforth, will be obtained using the theory of Calderon–Zygmundoperator5 and Riesz fractional integral operator.9 The discrete wavelet transformhas many scientific and engineering applications. It has recently been applied onpersonal identity verification with ECG signal.3

Let us recall the properties of Calderon–Zygmund operator and associated func-tion spaces,5 which will be used in the sequel.

Definition 1.1. Let the kernel K(x, y) satisfy

|K(x, y)| ≤ C

|x− y| (1.1)

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http://dx.doi.org/10.1142/S0219691311004377


906 R. S. Pathak

and ∣∣∣∣ ∂∂xK(x, y)∣∣∣∣ +

∣∣∣∣ ∂∂yK(x, y)∣∣∣∣ ≤ C

|x− y|2 (1.2)

for some constant C > 0. Then the operator T defined by

(Tf )(x) :=∫ ∞

−∞K(x, y)f(y)dy (1.3)

is called Calderon–Zygmund operator provided it is a bounded operator on L2(R).

Definition 1.2. Let f be a function defined on R such that for some C > 0 andall α > 0,

|{x : |f(x)| ≥ α}| ≤ C

α. (1.4)

The infimum of all C for which this holds, for all α > 0, is denoted by ‖f‖L1weak

.

Important properties of the Calderon–Zygmund operator T are contained in thefollowing theorem whose proof can be found in Ref. 5, pp. 291–296.

Theorem 1.1. If T is an integral operator with kernel K(x, y) satisfying (1.1) and(1.2), and if T is bounded from L2(R) to L2(R), then T is a bounded operator fromL1(R) to L1

weak(R), and T can be extended as a bounded operator from Lp(R) toLp(R) for all p with 1 < p <∞.

Another interesting result that we shall need in our investigation is on Rieszfractional integral due to Okikioulu.9

Theorem 1.2. Let f ∈ Lr(R), (r > 1), 0 < α < 1/r and let 1/s = 1/r − α. Let

Iαf(x) :=∫ ∞

−∞|t− x|α−1f(t)dt, (1.5)

where 0 < α < 1. Then Iα(f) ∈ Ls(R), and

‖Iαf‖s ≤ K‖f‖r (1.6)

for some K > 0.

Now, we recall the definition of the discrete wavelet transform given in Ref. 5;see Ref. 4 also.

Definition 1.3. For j, k ∈ Z, let

θj,k(x) = a− j

20 θ(a−j0 x− kb0), a0 > 0, b0 ∈ R. (1.7)

Then the discrete wavelet transform of f is defined by

aj,k = (Wθf)(j, k) := 〈f, θj,k〉 =∫ ∞

−∞f(x)θj,k(x)dx. (1.8)

The series∑∞

j,k=−∞ aj,kθj,k(x) is called the wavelet series of f .

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Convolution for the Discrete Wavelet Transform 907

It is well known that the function f ∈ L2(R) is completely determined by itswavelet series (i.e. wavelet coefficients) if the wavelets form an orthonormal basisin L2(R).5 Thus

f(x) = W−1θ [aj,k](x) :=

∑j,k

aj,kθj,k(x) in L2(R), (1.9)

aj,k = f(j, k) = 〈f, θj,k〉 (1.10)

and

‖f‖2 =∑j,k

|aj,k|2. (1.11)

The aim of the paper is to define translation and convolution associated withtransform (1.8) in Bochner’s form1 and to establish certain existence and approxi-mation theorems involving wavelet convolutions. A dual convolution involving dis-crete wavelet transforms is also studied.

2. Wavelet Convolution on L2(R)

In order to define convolution associated with the discrete wavelet transform weexploit the basic property of the convolution operation, viz., the transform of theconvolution of two functions is equal to the product of their transforms. Therefore,following Bochner,1 convolution of two functions f, g ∈ L2(R) is defined usingtheory of the discrete wavelet transform. A motivation for this definition can alsobe found in Ref. 10.

Assume that θ, φ ∈ L2(R) and f(x) is given by (1.9). Define

g(x) :=∑j,k

bj,kφj,k(x) in L2(R), (2.1)

where

bj,k = 〈g, φj,k〉. (2.2)

Now, define the associated function, called wavelet translation, by

(τxf)(y) := f(x; y) =∑j,k

aj,kψj,k(x)φj,k(y), (2.3)

provided the series converges. If we substitute the value of aj,k given by (1.10), weget

f(x; y) =∫ ∞

−∞D(x, y, z)f(z)dz, (2.4)

provided the integral exists, where

D(x, y, z) :=∑j,k

ψj,k(x)φj,k(y)θj,k(z). (2.5)

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908 R. S. Pathak

If θ(x) = φ(x) = ψ(x), and these are real-valued then D(x, y, z) is symmetric inx, y and z.

The wavelet convolution of f and g is defined formally by

(f#g)(x) : = W−1ψ [(Wθf)(j, k)(Wφg)(j, k)](x) (2.6)

=∑j,k

aj,kbj,kψj,k(x), (2.7)

where W−1ψ is defined by (1.9).

Orthonormality of {ψj,k} yields the fundamental result:

Wψ(f#g)(j, k) = (Wθf)(j, k)(Wφg)(j, k). (2.8)

If we substitute the value of bj,k from (2.2), then (2.7) gives

(f#g)(x) :=∫ ∞

−∞f(x; y)g(y)dy. (2.9)

In case {θj,k} form an orthonormal system in L2(R), then from (2.5) it followsthat ∫ ∞

−∞D(x, y, z)θj,k(z)dz = ψj,k(x)φj,k(y). (2.10)

If we assume that ψ = θ = φ ∈ L1(R) ∩ L2(R) is a scaling function of amultiresolution analysis, then using

∑k

φ(x+ k) = 1,∑k

φ(x − k)φ(y − k) = Φ(x, y),∫ ∞

−∞Φ(x, y)dy = 1

(2.11)

given in Ref. 12, pp. 32 and 186, and writing

D0(x, y, z) =∑k

ψ0,k(x)φ0,k(y)θ0,k(z), (2.12)

we get ∫ ∞

−∞D0(x, y, z)dz = Φ(x, y) (2.13)

and ∫ ∞

−∞

∫ ∞

−∞D0(x, y, z)dzdy = 1, (2.14)

∫ ∞

−∞f(x; y)dy =

∫ ∞

−∞f(z)Φ(x, z)dz, (2.15)

∫ ∞

−∞

∫ ∞

−∞f(x; y)dydx =

∫ ∞

−∞f(z)dz. (2.16)

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Also, by orthonormality of {φj,k} from (2.3) we have∫ ∞

−∞f(x; y)φj,k(y)dy = aj,kψj,k(x) (2.17)

and orthonormalities of {φj,k} and {ψj,k} yield:∫ ∞

−∞

∫ ∞

−∞|f(x; y)|2dxdy =

∑j,k

|aj,k|2 = ‖f‖22. (2.18)

Theorem 2.1. Let f, g, h ∈ L2(R) and {θj,k}, {φj,k}, {ψj,k} be orthonormalwavelets in L2(R). Then

(f#g)(x) =∑j,k

aj,kbj,kψj,k(x) ∈ L2(R), (2.19)

‖f#g‖2 ≤ ‖f‖2‖g‖2, (2.20)

where {aj,k}, {bj,k} are given by (1.10) and (2.2) respectively, and

Wψ(f#g)(m,n) = (Wθf)(m,n)(Wφg)(m,n), (2.21)

f#g = g#f, (2.22)

(f#g)#h = f#(g#h), (2.23)

when θ(x) = φ(x) = ψ(x).In case {ψj,k} constitute a frame and {ψj,k} is the dual frame, then

Wψ(f#g)(m,n) = (Wθf)(m,n)(Wφg)(m,n). (2.24)

Proof. By orthonormality of {ψj,k} from (2.7) we have∫ ∞

−∞|(f#g)(x)|2dx =

∑j,k

|aj,kbj,k|2

≤∑j,k

|aj,k|2∑m,n

|bm,n|2

≤ ‖f‖2‖g‖2.

Since f#g ∈ L2(R), using (2.19) we have

Wψ(f#g)(m,n) = 〈f#g, ψm,n〉

=∫ ∞

−∞

∑j,k

aj,kbj,kψj,k(x)ψm,n(x)dx

= am,nbm,n,

by orthonormality of {ψj,k}.

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910 R. S. Pathak

Clearly, (2.22) and (2.23) follow from (2.19) and (2.21) respectively, and (2.24)is a consequence of the Daubechies result given in Ref. 5, p. 54.

3. Boundedness of D(x, y, z)

Imposing conditions on discrete wavelets certain existence theorems for D(x, y, z)are established. In what follows, to simplify the analysis, we shall assume thata0 = 2 and b0 = 1; although results can be established for general a0, b0 withoutmuch difficulty.

Theorem 3.1. (i) Assume that there exists a constant C > 0 such that

|θ(x)|, |φ(x)| ≤ C(1 + |x|)−2−ε and |ψ(x)| ≤ C(1 + |x|)−1−ε, ε > 0.

Then

|D(x, y, z)| ≤ C|x− y|− 32 for some C > 0. (3.1)

(ii) If |φ(x)| ≤ C(1 + |x|)−2−ε and |θ(x)|, |ψ(x)| ≤ C(1 + |x|)−1−ε, then

|D(x, y, z)| ≤ C|x− y|− 34 |y − z|− 3

4 , for some C > 0. (3.2)

(iii) If |θ(x)| ≤ C(1 + |x|)−2−ε and |φ(x)|, |ψ(x)| ≤ C(1 + |x|)−1−ε, then

|D(x, y, z)| ≤ C|x− y|− 34 |x− z|− 3

4 , for some C > 0. (3.3)

(iv) If |ψ(x)| ≤ C(1 + |x|)−2−ε and |θ(x)|, |φ(x)| ≤ C(1 + |x|)−1−ε, then

|D(x, y, z)| ≤ C|x − z|− 34 |y − z|− 3

4 , for some C > 0. (3.4)

Proof. (i) Following the technique of proof of Daubechies given in Ref. 5, pp. 291–296, from (2.5) we have

|D(x, y, z)| ≤ C∑j,k

2−32 j(1 + |2j − k|)−2−ε

× (1 + |2−jy − k|)−2−ε(1 + |2−jz − k|)−1−ε. (3.5)

Assume that there exists j0 ∈ Z such that 2j0 ≤ |x− y| ≤ 2j0+1. Then

|D(x, y, z)| ≤ C

∞∑j=j0

+j0−1∑j=−∞

∑

k

2−32 j(1 + |2−jx− k|)−2−ε

× (1 + |2−jy − k|)−2−ε(1 + |2−jz − k|)−1−ε

= T1 + T2 (say).

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Now,

T1 ≤ C

∞∑j=j0

2−32 j

∑k

(1 + |2−jx− k|)−2−ε

≤ C′∞∑r=0

2−32 (r+j0)

≤ C′2−32 j0

∞∑r=0

2−( 32 )r

≤ D|x− y|− 32 , for some D > 0.

Next,

T2 ≤j0−1∑j=−∞

2−( 32 )j(1 + |2−jx− k|)−2−ε(1 + |2−jy − k|)−2−ε

=∞∑

j=−j0+1

232 j

∑k

(1 + |2jx− k|)−2−ε(1 + |2jy − k|)−2−ε

≤ 24+4ε∞∑

j=−j0+1

232 j

∑k

(2 + |2jx− k|)−2−ε(2 + |2jy − k|)−2−ε. (3.6)

Let us find k0 ∈ Z so that k0 ≤ 2j(x+ y)/2 ≤ k0 + 1, and set l = k− k0. Thenfollowing Daubechies in Ref. 5, pp. 297 and 310, and putting a = aj(x − y)/2,we get

∑k[(2+ |2jx−k|)(2+ |2jy−k|)]−2−ε ≤ ∑

l[(1+ |a−l|)(1+ |a+l|)]−2−ε ≤C(1 + |a|)−2−ε. Therefore,

T2 ≤ C

∞∑j=−j0+1

232 j(1 + |2j(x− y)/2|)−2−2ε

= C

∞∑j′=1

2( 32 )(j′−j0)(1 + |2j′−j0(x− y)/2|−2−2ε

≤ C∞∑j′=1

23(j′−j0)

2

(1 + 2j

′−j0(

12

)2j0

)−2−2ε

≤ C2−32 j0

∞∑j′=1

23j′2 (1 + 2j

′−1)−2−2ε

≤ C|x− y|− 32 .

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912 R. S. Pathak

(ii) Inequality (3.5) yields:

|D(x, y, z)| ≤ C∑j,k

2−34 j(1 + |2−jx− k|)−1−ε(1 + |2−jy − k|)−1− ε

2

× 2−34 j(1 + |2−jy − k|)−1− ε

2 (1 + |2−jz − k|)−1−ε

≤ C∑j,k

2−34 j(1 + |2−jx− k|)−1−ε(1 + |2−jy − k|)−1− ε

2

×∑m,n

2−34 j(1 + |2−my − n|)−1− ε

2 (1 + |2−mz − n|)−1−ε.

Now, following the above technique it can be shown that

|D(x, y, z)| ≤ C|x− y|− 34 |y − z|− 3

4 .

The proofs of (iii) and (iv) are similar.

Theorem 3.2. (i) Assume that |θ(x)|, |φ(x)|, |ψ(x)| ≤ C(1 + |x|)−1−ε. Then∫ ∞−∞ |D(x, y, z)|dx ≤ C|y − z|−1/2,

∫ ∞−∞ |D(x, y, z)|dy ≤ C|z − x|−1/2 and∫ ∞

−∞ |D(x, y, z)|dz ≤ C|x− y|−1/2.

(ii) If in addition {θj,k} (resp. {φj,k} or {ψj,k}) is orthonormal in L2(R), then∫ ∞−∞ |D(x, y, z)|2dx ≤ C|y − z|−2,

∫ ∞−∞ |D(x, y, z)|2dy ≤ C|z − x|−2 and∫ ∞

−∞ |D(x, y, z)|2dz ≤ C|x− y|−2.

Proof. (i) Using the technique of proof of Theorem 3.1(i) we have

∫ ∞

−∞|D(x, y, z)|dz ≤ C

∑j,k

2−32 j(1 + |2−jx− k|)−1−ε(1 + |2−jy − k|)−1−ε

×∫ ∞

−∞(1 + |2−jz − k|)−1−εdz

≤ C∑j,k

2−j2 (1 + |2−jx− k|)−1−ε(1 + |2−jy − k|)−1−ε

×∫ ∞

−∞(1 + |u|)−1−εdu

≤ C′ ∑j,k

2−j2 (1 + |2−jx− k|)−1−ε(1 + |2−jy − k|)−1−ε

≤ C′|x− y|− 12 for some C′ > 0.

Similarly, other inequalities can be proved.

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(ii) Applying orthonormality of {ψj,k} to (2.5) we get

∫ ∞

−∞|D(x, y, z)|2dx =

∑j,k

|φj,k(y)|2|θj,k(z)|2

≤ C∑j,k

[2−j(1 + |2−jy − k|)−1−ε(1 + |2−jz − k|)−1−ε]2

≤ C∑j,k

2−2j(1 + |2−jy − k|)−1−ε(1 + |2−jz − k|)−1−ε

≤ C|y − z|−2

for some C > 0; see Ref. 5, p. 298.

The proofs of other inequalities are similar.

4. Wavelet Convolution on Lp(R)

In this section we show that the convolution transform involving certain decayingwavelets is a Calderon–Zygmund operator. Then using Theorems 3.1 and 3.2 certainLp-boundedness results are obtained. Finally, we show that the product of twodiscrete wavelet transforms is a wavelet transform of the convolution.

First we obtain a boundedness result for the wavelet translation which will beused in the sequel. In this section also we assume that a0 = 2 and b0 = 1.

Lemma 4.1. Let f, θ ∈ L2(R). Assume that |ψ(x)|, |φ(x)| ≤ C(1 + |x|)−1−ε. Then

|(τxf)(y)| ≤ C|x− y|−1 for some C > 0. (4.1)

Proof. In view of the definition (2.4),

|(τxf)(y)| =∣∣∣∣∫ ∞

−∞D(x, y, z)f(z)dz

∣∣∣∣≤

∑j,k

|ψj,k(x)φj,k(y)|∫ ∞

−∞|θj,k(z)f(z)|dz

≤∑j,k

|ψj,k(x)φj,k(y)|‖θj,k‖2‖f‖2

≤ ‖θ‖2‖f‖2

∑j,k

(1 + |2−jx− k|)−1−ε(1 + |2−jy − k|)−1−ε

≤ C|x− y|−1;

see Ref. 5, p. 296.

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914 R. S. Pathak

Theorem 4.1. Let f, g ∈ L2(R) and let |ψ(x)|, |ψ′(x)|, |φ(x)|, |φ′(x)| ≤ C(1 +|x|)−2−ε, C > 0; then

(Tg)(x) := (f#g)(x) =∫ ∞

−∞f(x; y)g(y)dy (4.2)

is a Calderon–Zygmund operator. Hence T : L1(R) → L1weak(R), and T can be

extended to be a bounded operator from Lp(R) to Lp(R)∀ p, 1 < p <∞.

Proof. Since |aj,k| = |〈f, θj,k〉| ≤ ‖f‖2‖θ‖2, we have

|f(x; y)| ≤∑j,k

|aj,k| |ψj,k(x)| |φj,k(y)|

≤ ‖f‖2‖θ‖2C∑j,k

2−j(1 + |2−jx− k|)−1−ε(1 + |2−jy − k|)−1−ε

≤ C′/|x− y|.Also,∣∣∣∣ ∂∂xf(x; y)

∣∣∣∣ =∑j,k

|aj,k2−jψ′j,k(x)φj,k(y)|

≤ ‖f‖ ‖θ‖C∑j,k

2−2j(1 + |z−j − k|)−1−ε(1 + |2−jy − k|)−1−ε

≤ C′/|x− y|2;see Ref. 5, p. 298.

Similarly, | ∂∂y f(x; y)| ≤ C′/|x− y|2.Moreover, by Theorem 2.1, T maps g ∈ L2(R) to f#g ∈ L2(R). Hence

f#g defined by (4.2) is a Calderon–Zygmund operator; and conclusion is true byTheorem 1.1.

Theorem 4.2. Let f ∈ Lr(R), g ∈ Lr′(R), 1/r + 1/r′ = 3/2. Assume that |θ(x)|,

|φ(x)|, |ψ(x)| ≤ C(1 + |x|)−1−ε. Then

‖f#g‖1 ≤ C‖f‖r ‖g‖r′, for some C > 0. (4.3)

Proof. Using Theorem 3.2(i) we have∫ ∞

−∞|(f#g)(x)| dx ≤ C

∫ ∞

−∞

∫ ∞

−∞

|f(z)g(y)||y − z|1/2 dzdy.

Now, applying Hardy–Littlewood–Sobolev inequality8 we arrive at the desiredresult.

Theorem 4.3. Let f ∈ L∞(R) and g ∈ Lr(R), 1 < r < 2. Assume that θ, φ and ψsatisfy conditions of Theorem 3.1(iii). Then

‖f#g‖s ≤ C‖f‖∞‖g‖r, 1s

=1r− 1

2> 0. (4.4)

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Proof. Using inequality (3.3) we get

|(f#g)(x)| ≤∫ ∞

−∞

∫ ∞

−∞|D(x, y, z)f(z)g(y)|dzdy (4.5)

≤ C

∫ ∞

−∞

∫ ∞

−∞

|f(z)g(y)||x− y| 34 |x− z| 34 dzdy. (4.6)

We know that ∫ ∞

−∞|x− y|− 3

4 |x− z|− 34 dx = M |y − z|− 1

2 ,

where M = Γ(1/4)[

Γ(1/4)√π

+√π

Γ(3/4)

]; see Ref. 2, pp. 397–398. Therefore

|(f#g)(x)| ≤ CM ‖f‖∞∫ ∞

−∞

|g(y)||y − z| 12 dy

≤ CM ‖f‖∞I 12(|g|)(x).

Now, applying Theorem 1.2, we get

‖f#g‖s ≤ CM ‖f‖∞K‖g‖r, 1s

=1r− 1

2.

5. Continuity and Other Properties of the Wavelet Convolution

Theorem 5.1. Let f, g ∈ L1(R) and |θ(x)|, |φ(x)|, |ψ(x)| ≤ C(1+ |x|)−1−ε, C > 0.Assume that φj,k(x) = 2−

j2φ(2−jx − k), ψj,k(x) = 2−

j2ψ(2−jx − k) but θj,k(x) =

2−ρ|j|θ(2−jx− k) with ρ > 1/2. Then

‖(f#g)(x)‖∞ ≤ C‖f‖1‖g‖1 for some C > 0 (5.1)

and (f#g)(x) is continuous on R provided ψ(x) is continuous.

Proof. For f ∈ L1(R), from (1.10) we have

|aj,k| ≤∣∣∣∣∫ ∞

−∞f(z)C

2−ρ|j|

(1 + |2−jz − k|)1+ε dz∣∣∣∣

≤ C2−ρ|j|‖f‖1. (5.2)

Similarly, for g ∈ L1(R), (2.2) gives

|bj,k| ≤∫ ∞

−∞|g(y)|C2−

j2 (1 + |2−jy − k|)−1−εdy

≤ C2|j|2 ‖g‖1. (5.3)

Therefore, for f, g ∈ L1(R) we can define

(f#g)(x) :=∑j,k

aj,kbj,kψj,k(x). (5.4)

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916 R. S. Pathak

Indeed using (5.2) and (5.3) we have

|(f#g)(x)| ≤∑j,k

|aj,k| |bj,k| |ψj,k(x)|

≤ C∑j

2−ρ|j|+|j|2 ‖f‖1‖g‖1

∑k

1(1 + |2−jx− k|)1+ε

≤ C‖f‖1‖g‖1 for some C > 0 and ρ >12;

so that

|(f#g)(x) − (f#g)(x0)| ≤ 2C‖f‖1‖g‖1. (5.5)

Hence, if ψj,k(x) is continuous at x = x0, then

limx→x0

[(f#g)(x) − (f#g)(x0)] =∑j,k

aj,kbj,k limx→x0

[ψj,k(x) − ψj,k(x0)]

= 0.

Theorem 5.2. Assume that θ, φ, ψ, θj,k, φj,k and ψj,k are the same as in Theorem5.1. If f ∈ L1(R) and g ∈ Lp(R), p ≥ 1, then f#g ∈ Lp(R) and

‖f#g‖p ≤ C‖f‖1‖g‖p, for some C > 0 and ρ > 1. (5.6)

Proof. Let us choose at first p = 1. Then

|f(x)| =

∣∣∣∣∣∣∑j,k

〈f(z), θj,k(z)〉θj,k(x)∣∣∣∣∣∣

≤∑j,k

2−2ρ|j|C∫ ∞

−∞|f(z)|dz

∑k

1(1 + |2−jz − k|)1+ε

1(1 + |2−jx− k|)1+ε

≤ C‖f‖1

∑j

2−2ρ|j|

<∞, for ρ > 0; (5.7)

see Ref. 5, p. 310.Also, from (2.3) it follows that for |φ(x)| ≤ (1 + |x|)−1−ε and φj,k(x) =

2−j2φ(2−jx− k),

|f(x; y)| ≤∑j

2−ρ|j|−jC∫ ∞

−∞|f(z)|dz

∑k

(1 + |2−jz − k|)−1−ε

× (1 + |2−jx− k|)−1−ε(1 + |2−jy − k|)−1−ε (5.8)

≤ C‖f‖1

∑j

2−ρ|j|+|j| <∞ for ρ > 1. (5.9)

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Moreover, integrating (5.8) with respect to y, we get∫ ∞

−∞|f(x; y)|dy ≤

∑j

2−ρ|j|C‖f‖1

∫ ∞

−∞(1 + |v|)−1−εdv

≤ C‖f‖1 for some C > 0 and ρ > 0. (5.10)

Similarly, ∫ ∞

−∞|f(x; y)|dx ≤ C‖f‖1 for some C > 0 and ρ > 0. (5.11)

Therefore, using Fubini’s theorem and changing order of integration we get∫ ∞

−∞|(f#g)(x)|dx ≤

∫ ∞

−∞dx

∫ ∞

−∞|g(y)f(x; y)|dy

≤(∫ ∞

−∞|g(y)|dy

) (∫ ∞

−∞|f(x; y)|dx

)

≤ C‖g‖1‖f‖1.

Next, consider the case p > 1. Then for 1/p+ 1/p′ = 1, from (2.9) we have

|(f#g)(x)| ≤∫ ∞

−∞|g(y)||f(x; y)| 1p |f(x; y)| 1

p′dy

≤(∫ ∞

−∞|g(y)|p|f(x; y)|dy

) 1p

(∫ ∞

−∞|f(x; y)|dy

) 1p′

;

so that using (5.10),

|(f#g)(x)|p ≤ (C‖f‖1)p

p′∫ ∞

−∞|g(y)|p|f(x; y)|dy.

Hence by (5.11),∫ ∞

−∞|(f#g)(x)|pdx = (C‖f‖1)

p

p′∫ ∞

−∞|g(y)|pdy

∫ ∞

−∞|f(x; y)|dx

= (C‖f‖1)1+(p/p′)‖g‖pp.

Corollary 5.1. Assume that θ, φ, ψ, θj,k, φj,k and ψj,k are as in Theorem 5.1. Iff, g, h ∈ L1(R), then for ρ > 1,

(i) (f#g)(x) = (g#f)(x); (5.12)

(ii) ((f#g)#h)(x) = (f#(g#h))(x), (5.13)

when θ(x) = φ(x) = ψ(x).

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918 R. S. Pathak

Proof. (i) By Theorem 5.1, f#g and g#f exist and are equal in view ofDefinition (5.4).

(ii) Again, using (5.4), (1.8) and (2.8), for θ(x) = φ(x) = ψ(x), we have

((f#g)#h)(x) =∑j,k

〈f#g, θj,k〉〈h, φj,k〉ψj,k(x)

=∑j,k

〈f, θj,k〉〈g, θj,k〉〈h, φj,k〉ψj,k(x)

=∑j,k

〈f, θj,k〉〈g#h, φj,k〉ψj,k(x)

= (f#(g#h))(x).

By Theorem 5.2, f#g and g#h ∈ L1(R) for ρ > 1. Hence

|((f#g)#h)(x)| ≤ C‖f#g‖1‖h‖1 ≤ C‖f‖1‖g‖1‖h‖1.

Similarly,

|(f#(g#h))(x)| ≤ C‖f‖1‖g‖1‖h‖1 for ρ > 1.

Thus both sides of (5.13) exist.

6. Dual Wavelet Convolution

In this section following Gasper6 we define convolution of two discrete waveletconvolution transformations, called dual wavelet convolution, and show that theinverse wavelet transform of the convolution is equal to the product of the functions.In this section we shall use the definition of θj,k etc. as given in (1.7).

Let {ψm,n}, {θj1,k1} and {φj2,k2} be orthonormal systems in L2(R), wherem,n; j1, k1; j2, k2 ∈ Z. Define the basic function

D(j1, k1; j2, k2;m,n) :=∫ ∞

−∞θj1,k1(x)φj2,k2(x)ψm,n(x)dx. (6.1)

The integral exists, if θ(x) ∈ L∞(R) ∩ L2(R) and φ, ψ ∈ L2(R).Indeed by Schwartz’s inequality, under the above assumptions, we have

|D(j1, k1; j2, k2;m,n)| ≤(∫ ∞

−∞|θj1,k1(x)φj2 ,k2(x)|2dx

) 12

(∫ ∞

−∞|ψm,n(x)|2dx

) 12

≤ a− j1

20 ‖θ‖∞‖φ‖2‖ψ‖2. (6.2)

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From (6.1) it follows that D(j1, k1; j2, k2;m,n) is the wavelet transform ofθj1,k1(x)φj2 ,k2(x); hence by (1.9) and (1.11), we have∑

m,n

D(j1, k1; j2, k2;m,n)ψm,n(x) = θj1,k1(x)φj2 ,k2(x), (6.3)

∑m,n

|D(j1, k1; j2, k2;m,n)|2 = ‖θj1,k1(·)φj2,k2(·)‖22. (6.4)

In view of (1.10) we write

f(j1, k1) = 〈f, θj1,k1〉 (6.5)

and

g(j2, k2) = 〈g, φj2,k2〉. (6.6)

Now, we define the convolution of f and g by

(f#g)(m,n) :=∑j1,k1

∑j2,k2

D(j1, k1; j2, k2;m,n)f(j1, k1)g(j2, k2). (6.7)

Taking inverse discrete wavelet transform of (6.7) we have

W−1ψ [(f#g)(m,n)](x)

=∑m,n

(f#g)(m,n)ψm,n(x)

=∑j1,k1

∑j2,k2

[∑m,n

D(j1, k1; j2, k2;m,n)ψm,n(x)

]f(j1, k1)g(j2, k2)

=∑j1,k1

∑j2,k2

θj1,k1(x)θj2,k2(x)f (j1, k1)g(j2, k2) on using (6.3),

=∑j1,k1

f(j1, k1)θj1,k1(x)∑j2,k2

g(j2, k2)θj2,k2(x)

= f(x)g(x), by (1.9).

From the above we conclude the following:

Theorem 6.1. Let f ∈ L∞(R) ∩ �L2(R), g ∈ L2(R), θ ∈ L∞(R) ∩ L2(R), andφ, ψ ∈ L2(R). Assume that D(·; ·; ·), f and g are defined by (6.1), (6.5) and (6.6)respectively. Then the dual convolution (f#g)(m,n) defined by (6.7) satisfies

(f#g)(m,n) = (fg) (m,n) (6.8)

and

f#g = g#f . (6.9)

It would be interesting to investigate other properties and applications of thedual convolution.

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920 R. S. Pathak

7. Applications

In this section certain L1-approximation result using the aforesaid theory ofwavelet convolution is obtained. Theory of projection operator is applied to obtainLp-approximation of the wavelet transformation. In the first case we choose j = 0,in the wavelet ψj,k(x) = 2−

j2ψ(2−jx− k) and write ψ0,k(x) = ψk(x) = ψ(x− k).

Theorem 7.1. Let gα ∈ L1(R) ∩ L2(R), α > 0, and∫ ∞−∞ gα(x)dx = 1. Let f ∈

L2(R) and θ, φ be orthonormal wavelets in L2(R), and in addition let φ ∈ L1(R).Set hα(x) = (φ(0))−1gα(x), where φ(0) �= 0 and φ(x) = φ(−x). Then

limα→0+

∫ ∞

−∞[(f#hα)(x) − f(x)]θ(x)dx = 0. (7.1)

Proof. From (2.7) with g(y) = (φ(0))−1gα(y), we have

(f#hα)(x) =∑j,k

aj,k(φ(0))−1〈gα(y), φj,k(y)〉θj,k(x).

Then by orthonormality of {θj,k},∫ ∞

−∞(f#hα)(x)θ0,0(x)dx = a0,0(φ(0))−1〈gα, φ0,0〉.

Therefore,

limα→0+

∫ ∞

−∞(f#hα)(x)θ(x)dx

=∫ ∞

−∞f(x)θ(x)dx(φ(0))−1 lim

α→0+

∫ ∞

−∞gα(y)φ(y)dy

=∫ ∞

−∞f(x)θ(x)dx(φ(0))−1 lim

α→0+(gα ∗ (φ)(0))

=∫ ∞

−∞f(x)θ(x)dx,

by approximate identity.4

An example of gα is given by

gα(x) = (4πα)−12 e−

x24α , α > 0, x ∈ R.

Theorem 7.2. Let θ(x) = ψ(x) = φ(x) and |φ(x)| ≤ C(1 + |x|)−2. Assume thatf ∈ Lp(R) if 1 ≤ p <∞, or f ∈ C0(R) if p = ∞. Then

limj→−∞

∥∥∥∥2j2

∫ ∞

−∞

∫ ∞

−∞f(y)D(x, y, 2jz)dzdy − f

∥∥∥∥p

= 0. (7.2)

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Also, for 1 < p <∞ and for any f ∈ Lp(R),∥∥∥∥2j2

∫ ∞

−∞

∫ ∞

−∞f(y)D(x, y, 2jz)dzdy

∥∥∥∥p

→ 0 as j → ∞. (7.3)

Proof. From (2.10) we have∫ ∞

−∞D(x, y, z)

∑k

2−j2φ(2−jz − k)dz =

∑k

φj,k(x)φj,k(y);

so that∫ ∞

−∞f(y)2

j2

∫ ∞

−∞D(x, y, 2jz)

∑k

φ(z − k)dzdy =∫ ∞

−∞

∑k

φj,k(x)φj,k(y)f(y)dy.

Since∑k φ(z − k) = 1, in view of the result (8.5) of Wojtaszczyk,12 we have

2j2

∫ ∞

−∞

∫ ∞

−∞f(y)D(x, y, 2jz)dzdy = (Pjf)(x).

Now, invoking Theorem 8.4 and Proposition 8.5 of Wojtaszczyk12 we arrive at(7.2) and (7.3).

Acknowledgment

The author is thankful to the referee for his valuable comments and suggestions.The work is supported by Department of Science and Technology, Government ofIndia under Grant No. 2084.

References

1. S. Bochner, Sturm–Liouville and heat equations whose eigenfunctions are ultraspher-ical polynomials or associated Bessel functions, in Proc. of the Conference on Differ-ential Equations, University of Maryland (1955) 23–48.

2. P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation (Academic Press,New York, 1971).

3. C.-C. Chiu, C.-M. Chuang and C.-Y. Hsu, Discrete wavelet transform applied onpersonal identity verification with ECG signal, Int. J. Wavelets Multiresolut. Anal.Inf. Process. 7(3) (2009) 341–355.

4. C. K. Chui, An Introduction to Wavelets (Academic Press, New York, 1992).5. I. Daubechies, Ten Lectures on Wavelets (SIAM, Philadelphia, Pennsylvania, USA,

1992).6. G. Gasper, Banach algebras for Jacobi series and positivity of a kernel, Ann. Math.

95(2) (1972) 261–280.7. I. I. Hirschman Jr., Variation diminishing Hankel transforms, J. Anal. Math.

8 (1960/61) 307–336.8. E. H. Lieb and M. Loss, Analysis (Narosa Publishing House, New Delhi, 1998).

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922 R. S. Pathak

9. G. O. Okikiolu, Fourier transform and the operator Hα, Proc. Camb. Philos. Soc.62 (1966) 73–78.

10. R. S. Pathak, The Wavelet Transform (Atlantis Press/World Scientific, 2009).11. R. S. Pathak and A. Pathak, On convolution for wavelet transform, Int. J. Wavelets,

Multiresolut. Anal. and Inf. Process. 6(5) (2008) 739–747.12. P. Wojtaszczyk, A Mathematical Introduction to Wavelets (Cambridge University

Press, 1997).

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CONVOLUTION FOR THE DISCRETE WAVELET TRANSFORM