Int. J. Anal. Appl. (2022), 20:64

Inversion Formula for the Wavelet Transform on Abelian Group

C.P. Pandey1,∗, Khetjing Moungkang1, Sunil Kumar Singh2, M.M. Dixit1, Mopi Ado1

1Department of Mathematics, North Eastern Regional Institute of Science and Technology, Nirjuli,

Itanagar, Arunachal Pradesh, 791109, India
2Department of Mathematics, Babasaheb Bhimrao Ambedkar University, Lucknow, 226025, India

∗Corresponding author: drcppandey@gmail.com

Abstract. In this paper a reconstruction and inversion formula of the continuous wavelet transform on

abelian group for band-limited function is defined. This formula possesses a more explicit expression

than the well-known result. Also, Parseval and other interesting results on abelian group are obtained.

1. Introduction

A set S defines a group if an operator, +, holds the following properties:

• x + (y + z) = (x + y) + z ∀x,y,z ∈ S
• There exists an element 0 , such that x + 0 = 0 + x = x ∀x ∈ S
• For each∀x ∈ S there exists an inverse element x−1 = −x , such that x+(−x) = (−x)+x = 0.

S is a topological group if it has a group operation and a topology such that the maps α : G×G → G
and β : G ×G → G are continuous, where α(x,y) = x + y and β(x) = x−1.
If S is locally compact, that is every point in S is contained in a compact neighborhood, and its group

operation is commutative, then it is called locally compact abelian (LCA) group.

In order to define the Fourier transform on LCA groups, we should introduce the concept of integral

over these groups. Let M(X) be the space of all complex-valued regular measures on X where

||µ|| = |µ(S)| is finite. A Haar measure is a measure which is non negative, regular and invariant. The
corresponding integral is called the Haar integral, which is translation invariant.

Received: Oct. 2, 2022.

2010 Mathematics Subject Classification. 65T60, 43A25, 11F85.

Key words and phrases. wavelet transform; abelian group; Fourier transform.

https://doi.org/10.28924/2291-8639-20-2022-64
ISSN: 2291-8639

© 2022 the author(s).

https://doi.org/10.28924/2291-8639-20-2022-64


2 Int. J. Anal. Appl. (2022), 20:64

Let G be LCA group , we define an LP (G) space to be the space of all complex valued functions f

on G such that the integral
∫
G
|f |pdµ exists with respect to the Haar measure.

Definition 1.1. A complex function ω on a LCA group G [1] is called a character of G if |ω(x)| = 1
for all x ∈ G and if the functional equation ω(x + y) = ω(x)ω(y) for all (x,y) ∈ G is satisfied. The
set of all continuous characters of G form a group Ω , the dual group of G . Now it is customary to

write (x,ω) = ω(x)ω(x) satisfy the following properties [1,5]

• (0,ω) = (x, 0) = 1
• (−x,ω) = (x,−ω) = (x,ω)−1 = (x,ω)
• (x + y,ω) = (x,ω)(y,ω)
• (x,ω1 + ω2) = (x,ω1)(x,ω2)

Definition 1.2. The Fourier transform [2] of f ∈ L1(G) is denoted by f̂ (ω) defined by f̂ (ω) =∫
G
f (x)(−x,ω)dx , and its inverse Fourier transform is defined [1,5] by f (x) =

∫
G
f̂ (ω)(x,ω)dω,x ∈ G

The Fourier transform holds the following properties [4]:

• ‖f̂‖L∞(G) 6 ‖f‖L1(G)
• If f ∈ L1(G) ∩L2(G) , then ‖f̂‖L2(G) = ‖f‖L2(G)
• If the convolution of f and g is defined as

(f ∗g)(x) =
∫
G
f (x −y)g(y)dy then F ((f ∗g)) = F (f )F (g)

For f (x) ∈ L2(G), denote fb,a(x) = 1√|a|f
(
x−b
a

)
and suppf = clos{x ∈ G : f (x) 6= 0}. If suppf̂

is a bounded set, then we say f is band-limited.

The characteristic function on a set E is denoted by XE(x) .

In 1984, Morlet introduced first wavelet transform [7] that is defined as follows:

Let ψ ∈ L2(G) , the transform:

(Wψ)(b,a) =

∫
G

f (x)ψb,a(x)dx for any f ∈ L2(G) (1.1)

is said to be a wavelet transform.

When ψ ∈ L1 ∩ L2(G) and Cψ = 2π
∫
G
|ψ̂(ω)|2
|ω| < ∞ , the known inversion formula [8] is stated as

follows:

f (x) =
1

Cψ

∫
G

∫
G

(Wψf )(b,a)ψb,a(x)
dadb

|a|2
(1.2)

The above equality holds in L2(G) sense.

The aim of this paper is that for band-limited function we give another kind of inversion formula of

wavelet transform.

Theorem 1.1. Let ψ(x) ∈ L1 ∩L2(G). Take φ(x) ∈ L1 ∩L2(G) satisfying φ̂(ω) = O(|ω|−2). Then
for any f ∈ L1 ∩L2(G) and suppf̂ ⊆ [−Ω, Ω], the following inversion formula holds:

f (x) =
1

(2π)
3
2 (ϕ,ψ)

∫
H

∫
G

(Wψf )(b,a)(ϕb,a ∗h)(x)
dada

|a|
(1.3)



Int. J. Anal. Appl. (2022), 20:64 3

where h(x) satisfies ĥ(ω) = |ω|X[−Ω,Ω](ω) and the above equality holds in L2- sense.

2. Lemma

To prove theorem, we first give the following

Lemma: Let ψ(x),ϕ(x) and f (x) be stated in theorem. Then for any g ∈ L2(G) the following formula
is valid:

1

2π

∫
G

∫
G

(Wψf )(b,a)(Dϕg)(b,a)
dbda

|a|
= (ϕ,ψ)(f ,g),

where (Dϕg)(b,a) =
1
√

2π
(g,ϕ(b,a) ∗h)

(2.1)

Proof: By Parseval identity of Fourier transform, we have

(Wψf )(b,a) = |a|
1
2

∫
G

f̂ (ω)ψ̂(aω)(b,ω)dω (2.2)

Using the convolution formula [3] and Parseval identity, we also obtain from (2.1) that

(Dϕg)(b,a) = |a|
1
2

∫
G

ĝ(ω)ϕ̂(aω)ĥ(ω)(b,ω)dω (2.3)

Applying the inversion formula of Fourier transform, it follows from (2.2) and (2.3) that

1
√

2π|a|
1
2

(Wψf )(b,a) = (f̂ (ω)ψ̂(aω))
v (b) (2.4)

and
1

√
2π|a|

1
2

(Dϕg)(b,a) = (ĝ(ω)ϕ̂(aω)ĥ(ω))
v (b) (2.5)

Finally, again using Parseval identity, we get

1

2π|a|

∫
G

(Wψf )(b,a)(Dϕg)(b,a)db =

∫
G

f̂ (ω)ĝ(ω)ĥ(ω)ψ̂(aω)ϕ̂(aω)dω

Since suppf̂ ⊆ [−Ω, Ω] = supp ĥ(ω) and ĥ(ω) = |ω|X[−Ω,Ω](ω), we know that f̂ (ω)ĥ(ω) = |ω|f̂ (ω),
ω ∈ G.
Further,

1

2π|a|

∫
G

(Wψf )(b,a)(Dϕg)(b,a)db =

∫
G

f̂ (ω)ĝ(ω)|ω|ψ̂(aω)ϕ̂(aω)dω

In view of∫
G

∫
G

|f̂ (ω)ĝ(ω)ωψ̂(aω)ϕ̂(aω)|dωda =
∫
G

|f̂ (ω)ĝ(ω)|
(
|ω|
∫
G

|ψ̂(aω)ϕ̂(aω)|da
)
dω

=

(∫
G

|ψ̂(aω)ϕ̂(aω)|dω
)(∫

G

|f̂ (ω)ĝ(ω)|dω
)

6 ‖ϕ‖2‖ψ‖2‖f‖2‖g‖2



4 Int. J. Anal. Appl. (2022), 20:64

By Fubini theorem, we have

1

2π

∫
G

(∫
G

(Wψf )(b,a)(Dϕg)(b,a)

)
da

|a|
=

∫
G

(∫
G

f̂ (ω)ĝ(ω)|ω|ψ̂(aω)ϕ̂(aω)dω
)
da

=

∫
G

f̂ (ω)ĝ(ω

(
|ω|
∫
G

ψ̂(aω)ϕ̂(aω)da

)
dω

Again, noticing that

|ω|
∫
G

ψ̂(aω)ϕ̂(aω)da = (ϕ̂,ψ̂) = (ϕ,ψ),

For repeated integral, we get

1

2π

∫
G

(∫
G

(Wψf )(b,a)(Dϕg)(b,a)

)
da

|a|
= (ϕ,ψ)(f ,g).

In order to complete the proof of lemma, by Fubini theorem, we only need to prove that

K =

∫
G

∫
G

|(Wψf )(b,a)(Dϕg)(b,a)|
dadb

|a|
< ∞ (2.6)

we split the above integral into two parts, namely,

K =

(∫
H

+

∫
G−H

)(∫
|(Wψf )(b,a)(Dϕg)(b,a)|

db

|a|

)
da

= K1 + K2

(2.7)

Where H is a subgroup of G. First, we estimate K1.

Using Cauchy inequality, we get

K21 6
∫
H

(∫
G

|(Wψf )(b,a)|2
db

|a|

)
da ·

∫
H

(∫
G

|(Dϕg)(b,a)|2
db

|a|

)
da

= K11 ·K12

Applying (2.4) and Parseval identity, we have

K11 = 2π

∫
H

(∫
G

|f̂ (ω)|2|ψ̂(aω)|2dω
)
da

By ψ ∈ L1(G) we know that there is a M > 0 such that |ψ̂(ω)|6 M, so K11 6 4πM2‖f‖22.

On the other hand, applying (2.5) and Parseval identity, we also have

K12 = 2π

∫
H

(∫
G

|ĝ(ω)|2|ϕ̂(aω)|2|ĥ(ω)|2dω
)
da (2.8)

By ϕ ∈ L1(G) we know that there is an N > 0 such that |ϕ̂(ω)|6 N. Again noticing that |ĥ(ω)|6 Ω,
we have K12 6 4πN2Ω2‖g‖22. so K1 < ∞.
Next we estminate K2,

From (2.7), we know that for any given 0 < � < 1
2
,

K2 =

∫
G−H

(∫
G

|a|−1+
1
2 |(Wψf )(b,a)| · |a|−

1
2 |(Dϕg)(b,a)|db

)
da



Int. J. Anal. Appl. (2022), 20:64 5

Using Cauchy inequality, we get

K22 6

∫
G−H

(∫
G

|(Wψf )(b,a)|2
db

|a|2−�

)
da ·

∫
G−H

(∫
G

|(Dϕg)(b,a)|2
db

|a|�

)
da

= K21 ·K22

Since

|(Wψf )(b,a)| =

∣∣∣∣∣ 1√|a|
∫
G

f (x)ψ

(
x −b
a

)
dx

∣∣∣∣∣ 6 ‖f‖2‖ψ‖2
And

1
√
a

∫
G

|(Wψf )(b,a)|db 6
∫
G

∫
G

∣∣∣∣f (x)ψ
(
x −b
a

)∣∣∣∣ dxdba 6 ‖f‖1‖ψ‖1 (2.9)
We get

K21 6 ‖f‖2‖ψ‖2
∫
G−H

(∫
G

|(Wψf )(b,a)|2db
)

1

|a|2−�
da

6 ‖f‖2‖ψ‖2‖f‖1‖ψ‖1
∫
G−H

1

|a|
3
2
−�
da

=
4

1 − 2�
‖f‖2‖ψ‖2‖f‖1‖ψ‖1

Similar to the argument of (2.8), we have

K22 = 2π

∫
G−H

(∫
G

|ĝ(ω)|2|ϕ̂(aω)|2|ĥ(ω)|2|a|1−�
)
da

Further, by the definition of h(x),

K22 = 2π

∫
G−H
|a|−1−�

(∫
G

|aω|2|ĝ(ω)|2|ϕ̂(aω)|2dω
)
da

From ϕ̂(ω) = O(|ω|−2), we have |ϕ̂(aω)|2 6 M1(M1 is an absolute constant).
Further K22 6

4πM1
�
‖g‖22.

So K2 < ∞.
We finally obtain (2.6). The proof of lemma is completed.

3. Proof of theorem

From |(ϕb,a ∗h)(x)|6 ‖ϕb,a‖2‖h‖2 = ‖ϕ‖2‖h‖2 and (2.9), we have∫
H

∫
G

|(Wψf )(b,a)(ϕb,a ∗h)(x)|
dadb

|a|
6 ‖ϕ‖2‖h‖2

∫
H

1

|a|
1
2

(
1

|a|
1
2

∫
G

|(Wψf )(b,a)|db

)
da

6 4‖ϕ‖2‖h‖2‖ψ‖1‖f‖1

(3.1)

So, for all x ∈ G, we know that (Wψf )(b,a)(ϕb,a ∗h)(x) 1|a| ∈ L
1(H ×G).

Set

4H(x) =
1

(2π)
3
2 (ϕ,ψ)

∫
H

∫
G

(Wψf )(b,a)(ϕb,a ∗h)(x)
dadb

|a|
.



6 Int. J. Anal. Appl. (2022), 20:64

By the known result in theory of Hilbert space, we know that

‖f (x) −4H(x)‖2 = sup
‖g‖2=1

|(f ,g) − (4H,g)|. (3.2)

Again by (2.1), we get

(4H,g) =
1

(2π)
3
2 (ϕ,ψ)

∫
G

(∫
H

∫
G

(Wψf )(b,a)(ϕb,a ∗h)(x)g(x)
dadb

|a|

)
dx

=
1

(2π)(ϕ,ψ)

∫
H

∫
G

(Wψf )(b,a)(Dϕg)(b,a)
dbda

|a|
.

(3.3)

The reason for interchanging the order of the above integrals is stated as follows.

By (2.9) and ∫
G

|(ϕb,a ∗h)(x)g(x)|
1√
|a|
dx 6 ‖

1√
|a|

(ϕb,a ∗h)(x)‖2‖g‖2

6 ‖ϕ‖1‖h‖2‖g‖2

We get
∫
G

(∫
H

∫
G

(Wψf )(b,a)(ϕb,a ∗h)(x)g(x)
dadb

|a|

)
dx

=

∫
H

(∫
G

|(Wψf )(b,a)|

(∫
G

|(ϕb,a ∗h)(x)g(x)|
1√
|a|
dx

)
db√
|a|

)
da

6 2‖φ‖1‖h‖2‖g‖2‖f‖1‖ψ‖1

So, the order of integrals in (3.3) can be interchanged.

Using lemma and (3.3),

(f ,g) − (4H,g) =
1

(2π)(ϕ,ψ)

∫
G−H

∫
G

(Wψf )(b,a)(Dϕg)(b,a)
dbda

|a|

Further we get from (3.2)

‖(f ,g) − (4H,g)‖2 6 sup
‖g‖2=1

(
1

(2π)|(ϕ,ψ)|

∫
G−H

∫
G

(Wψf )(b,a)(Dϕg)(b,a)
dbda

|a|

)
= sup
‖g‖2=1

(
1

(2π)|(ϕ,ψ)|
I(h)

)
.

(3.4)

Where I(h) =
∫
G−H

∫
G
|(Wψf )(b,a)(Dϕg)(b,a)|dbda|a| ,∀h ∈ H

Using cauchy inequality, we can see that

I2(h) 6
∫
G−H

(∫
G

|(Wψf )(b,a)|2
db

|a|2−�

)
da ·

∫
G−H

(∫
G

|(Dϕg)(b,a)|2
db

|a|�

)
da

= I1(h)I2(h)

(3.5)

Imitating the estimates of K21 and K22 in lemma, we can get

I1(h) 6
4

1 − 2�
h
−1

2
+�‖f‖2‖ψ‖2‖f‖1‖ψ‖1



Int. J. Anal. Appl. (2022), 20:64 7

and

I2(h) 6
4πM1
�

h
−�‖g‖22.

From this and (3.4),(3.5), we know that

‖f (x) −4H(x)‖2 6 sup
‖g‖2=1

(
1

(2π)|(ϕ,ψ)|
I(h)

)

6
1

(2π)|(ϕ,ψ)|

(
16πM1

(1 − 2�)�
h
−1

2‖f‖2‖ψ‖2‖f‖1‖ψ‖1
)1

2

.

So,

lim
h→+∞

‖f (x) −4H(x)‖2 = 0.

This proof of Theorem is completed.

Acknowledgements: This work is supported by UGC grant No. 16-9(June 2018)/2019(NET/CSIR)

and CSIR grant No. 09/725(014)/2019-EMR-1.

Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi-

cation of this paper.

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https://doi.org/10.1006/acha.1995.1004
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https://doi.org/10.28919/aia/4067
https://doi.org/10.1007/978-1-4612-0097-0_6

	1. Introduction
	2. Lemma
	3. Proof of theorem
	References