DSpace at VNU: Conlutions for the fourier with geometric variable and applications

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Math. Nachr. 283, No. 12, 1758 – 1770 (2010) / DOI 10.1002/mana.200710192

Convolutions for the Fourier transforms with geometric variables and
applications
Bui Thi Giang1 , Nguyen Van Mau2 , and Nguyen Minh Tuan∗3
1
2
3

Dept. of Basic Science, Institute of Cryptography Science, 141 Chien Thang str., Thanh Tri dist., Hanoi, Vietnam
Dept. of Mathematical Analysis, University of Hanoi, 334 Nguyen Trai str., Thanh Xuan dist., Hanoi, Vietnam
Dept. of Math., Univ. of Edu, Ha Noi National Univ., G7 Build., 144 Xuan Thuy Rd., Cau Giay dist., Ha Noi Vietnam
Received 6 November 2007, revised 11 November 2009, accepted 11 November 2009
Published online 29 October 2010
Key words Convolution, generalized convolution, factorization identity, Gaussian function, Hermite function
MSC (2000) Primary: 43A32, 44A35; Secondary: 44-99, 44A15
This paper gives a general formulation of convolutions for arbitrary linear operators from a linear space to a
commutative algebra, constructs three convolutions for the Fourier transforms with geometric variables and
four generalized convolutions for the Fourier-cosine, Fourier-sine transforms. With respect to applications, by
using the constructed convolutions normed rings on L1 (Rn ) are constructed, and explicit solutions of integral
equations of convolution type are obtained.
c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1

Introduction and summary of results

The theory of convolutions of integral transforms has been studied for a long time, and is applied to many fields
of mathematics (see [4,15–18]). The generalized convolutions for integral transforms and their applications were
first studied by Churchill in 1941, then the idea of construction of convolutions was formulated by Vilenkin
in 1958 (see [7, 9, 33]). In 1967, the construction methods for generalized convolutions of arbitrary integral
transforms were proposed by Kakichev, and in 1990 the concept of generalized convolutions for linear operators
was introduced by the same author (see [19, 20]). In 1997, some convolutions for integral transforms were
obtained, and in 1998 the generalized convolutions for the Fourier-cosine and Fourier-sine integral transforms
were presented (see [21, 22]).
In recent years, many papers devoted to those transforms have been published containing convolutions, generalized convolutions, polyconvolutions and their applications (see [5, 6, 10–12, 25–27, 31, 32]). Generally
speaking, each of the convolutions is a new transform which has become an object of study (see [1, 19]). In our
view, the integral transforms of Fourier type deserve special interest.
The main purpose of this paper is to construct generalized convolutions for the Fourier transforms with geometric variables: shift, similarity and inversion, and consider some applications.

The paper is divided into three sections and organized as follows.
Section 2 consists of three subsections. The general formulation of convolutions is stated in Subsection 2.1. In
Subsection 2.2, there are six convolutions for the transforms of Fourier type sorted according to those transforms
with the geometric variables: shift, similarity, inversion. In Subsection 2.3, there are four generalized convolutions of the Fourier-cosine and Fourier-sine transforms. As usual, there exist different convolutions for the same
transform, and a transform may be a convolution for different transforms.
Section 3 deals with applications of the constructed convolutions. In Subsection 3.1, the linear space L1 (Rn ),
equipped with each of the convolutions, becomes a normed ring. All normed rings in this subsection have
no unit, most of them are commutative and could be used in the theories of Banach algebra. Subsection 3.2
contains the most important results of this section where Fredholm integral equations of the first and second
∗

Corresponding author: e-mail: , , Phone: +84 4 37548092, Fax: +84 4 37548092

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Math. Nachr. 283, No. 12 (2010) / www.mn-journal.com

1759

kind are considered simultaneously. By using the constructed convolutions, we provide necessary and sufficient
conditions of the solvability of the integral equations of convolution type and obtain explicit solutions. Observing
the procedures for obtaining the solutions of the first equation in Subsection 3.2, there is the perhaps surprising
fact: although the non-injective transforms Tc , Ts are applied to Equation (3.2), the obtained function is still
the solution of this equation. The comparison with previously published works concerning integral equations of
convolution type is provided at the end of this paper.

2

Convolutions

2.1 General definition of convolutions
The concept of generalized convolutions with weight is a nice idea based on the so-called factorization identity.
Let U1 , U2 , U3 be linear spaces on the field of scalars K, and let V be a commutative algebra on K. Suppose
that K1 ∈ L(U1 , V ), K2 ∈ L(U2 , V ), K3 ∈ L(U3 , V ) are linear operators from U1 , U2 , U3 to V, respectively.
Let δ denote an element in the algebra V.
Definition 2.1 (See also [5, 19, 20]) A bilinear map ∗ : U1 × U2 :−→ U3 is called a convolution with
weight-element δ for K3 , K1 , K2 (in that order) if K3 (∗(f, g)) = δK1 (f )K2 (g) for any f ∈ U1 , g ∈ U2 .
We call K3 (∗(f, g)) = δK1 (f )K2 (g) the factorization identity of the convolution. In the sequel, we write
∗(f, g) := f

δ

∗

K3 ,K1 ,K2

g. If δ is the unit of V, we say briefly the convolution for K3 , K1 , K2 . If U1 = U2 = U3
δ

and K1 = K2 = K3 , the convolution is denoted simply f ∗ g, and f ∗ g if δ is the unit of V . As the notation
K1

f

δ

∗

K1

δ

K3 ,K1 ,K2

g already defines the factorization identity K3 f ∗ g
K1

= δK1 (f )K2 (g), it is sufficient to formulate

the convolution expressions in the theorems. In the next sections, we consider U = U1 = U2 = U3 = L1 (Rn )
with the Lebesgue integral, V is the algebra of all measurable functions (real or complex) defined on Rn . For
x, y ∈ Rn , let x, y denote the scalar product, and |x|2 = x, x .
2.2
2.2.1

Convolutions for the Fourier transforms with geometric variables
Convolutions for the Fourier transform with shift

Let h ∈ Rn be fixed. Let F denote the Fourier and inverse transforms as
(F f )(x) :=

1
n
(2π) 2

e−i

x,y

Rn

f (y) dy,

(F −1 f )(x) :=

1
n
(2π) 2

ei

x,y

Rn

f (y) dy.

Consider the following integral transforms
1
n
(2π) 2
1
(Fh−1 f )(x) :=
n
(2π) 2

(Fh f )(x) :=

e−i

x+h,y

f (y) dy,

ei

x,y+h

f (y)dy.

Rn

Rn

We call Fh the Fourier transform with shift, and Fh−1 its inverse transform. The inversion formula of Fh is proved
2
1
by Lemma 3.7. Put γ1 (x) = e− 2 |x| .
Theorem 2.2 If f, g ∈ L1 (Rn ), then each of the integral expressions (2.1), (2.2) below defines a convolution:
1
n
(2π) 2
γ1
1
f ∗ g (x) :=
Fh
(2π)n

f ∗ g (x) :=

Fh

www.mn-journal.com

Rn

f (x − y)g(y) dy,
1

Rn

Rn

f (u)g(v)e− 2 |x−u−v|

(2.1)
2

+i h,x−u−v

du dv.

(2.2)

c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


1760

Bui Thi Giang, Nguyen Van Mau, and Nguyen Minh Tuan: Convolutions for Fourier transforms

P r o o f. The proof of the convolution (2.1) is immediate and referred to the readers. We prove the convolution
(2.2). For any u, v ∈ Rn the following formulae hold
1
n
(2π) 2

e−i

x,y±u±v − 12 |y±u±v|2

Rn

dy = e−

|x|2
2

;

2

1

Rn

n

e− 2 |x−u−v| dx = (2π) 2

(2.2a)

(see [24, Lemma 7.6]). We then have
γ1

f ∗ g (x) dx ≤
Fh

Rn

1
(2π)n

1

Rn

Rn

Rn

|f (u)| |g(v)| e− 2 |x−u−v|

2

du dv dx < +∞.

γ1

It implies that f ∗ g ∈ L1 (Rn ). We prove the factorization identity. Using formula (2.2a), we obtain

Fh

γ1 (x)(Fh f )(x)(Fh g)(x)
−|y−u−v|2
1
−i x,y +i 2
=
f (u)g(v)e
du dv dy
3n
(2π) 2 Rn Rn Rn
−|y−u−v|2
1
1
+i h,y−u−v
2
e−i x+h,y
f
(u)g(v)e
du dv dy
=
n
(2π)n Rn Rn
(2π) 2 Rn
γ1

= Fh f ∗ g (x).
Fh

The proof is complete.
2.2.2

Convolutions for the Fourier transform with similarity

Let α = (α1 , . . . , αn ) ∈ Rn+ (αi > 0 ∀i = 1, . . . , n) be fixed. Write α · x = (α1 x1 , . . . , αn xn ) for any
x ∈ Rn . Consider the following integral transforms
(Fα f )(x) :=

|α|
n
(2π) 2

n

Fα−1

:=

j=1

e−i

α·x,y

Rn

f (y) dy,

αj

n

|α|(2π) 2

ei

α·x,y

Rn

(Fα f )(y) dy.

We call Fα the Fourier transform with similarity, and Fα−1 its inverse transform.
Theorem 2.3 If f, g ∈ L1 (Rn ), then each of the integral expressions (2.3), (2.4) below defines a convolution:
f ∗ g (x) :=
Fα

|α|
n
(2π) 2

γ1

f ∗ g (x) :=
Fα

n
j=1

Rn

f (x − y)g(y) dy,

αj |α|

(2π)n

Rn

Rn

(2.3)

f (u)g(v)e−

|α·(x−u−v)|2
2

du dv.

(2.4)

γ1

P r o o f. It is easy to prove the convolution (2.3). The fact f ∗ g ∈ L1 (Rn ) for the convolution (2.4) is proved
Fα

immediately. We prove the factorization identity of the convolution (2.4). From (2.2a) we get
(

n
j=1

(2π)

αj )
n
2

e−i

x,α·(y−u−v) − 12 |α·(y−u−v)|2

Rn

dy = e−

|x|2
2

.

(2.4a)

We then have
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Math. Nachr. 283, No. 12 (2010) / www.mn-journal.com

1761

γ1 (x)(Fα f )(x)(Fα g)(x)
=

n
j=1

|α|2

(2π)
|α|2 (

n

j=1

=

(2π)

|α|
=
n
(2π) 2

αj

3n
2

e−i

f (u)g(v)

Rn

Rn

x,α·(y−u−v) − 12 |α·(y−u−v)|2

dy e−i

αj )

3n
2

2

1

Rn

e−i

Rn

⎡
⎣

α·x,y

Rn

f (u)g(v)e− 2 |α·(y−u−v)| e−i
n
j=1

|α|

e−i

α·x,v

du dv

α·x,y

du dv dy
⎤

αj

− 12 |α·(y−u−v)|2

(2π)n

Rn

α·x,u

Rn

Rn

Rn

f (u)g(v)e

du dv ⎦ dy

γ1

= Fα f ∗ g (x).
Fα

The theorem is proved.
2.2.3

Convolutions for the Fourier transform with inversion

For any x ∈ Rn (xi = 0, ∀i = 1, . . . , n), let us write
(Fv f )(x) :=

⎧
⎨

1
n
(2π) 2
⎩
0

e−i

1
y, x

Rn

1
x

1
1
x1 , . . . , xn

:=

f (y) dy

. Consider the integral transform

if xi = 0 ∀i = 1, . . . , n,
if xi = 0.

We call Fv the Fourier transform with inversion. Consider the following function

1 2

1

e− 2 | x |
0

γ2 (x) =

if xi = 0 ∀i = 1, . . . , n,
if xi = 0.

It is worth saying that the function γ2 (x) is bounded and infinite differentiable on Rn .
Theorem 2.4 If f, g ∈ L1 (Rn ), then each of the integral transforms (2.5), (2.6) below defines a generalized
convolution:
1
n
Fv
(2π) 2
γ2
1
f ∗ g (x) =
Fv
(2π)n

f ∗ g (x) =

Rn

f (x − y)g(y) dy,

(2.5)
2

1

Rn

Rn

f (u)g(v)e− 2 |x−u−v| du dv.

(2.6)
γ2

P r o o f. The convolution (2.5) is proved immediately. For the convolution (2.6) the fact that f ∗ g ∈ L1 (Rn )
Fv

is proved similarly to the proof of (2.2). We shall prove the factorization identity of the convolution (2.6) as
γ2

Fv f ∗ g (x) = γ2 (x)(Fv f )(x)(Fv g)(x). Indeed, if at least one of the xi is zero, this identity is clear.
Fv

Consider xi = 0, ∀i = 1, 2, . . . , n. Formula (2.2a) gives
1
n
(2π) 2

e−i

1
y−u−v, x
− 12 |y−u−v|2

Rn

1

1 2

dy = e− 2 | x | .

(2.7a)

We then have
γ2 (x)(Fv f )(x)(Fv g)(x)
=
=
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1
(2π)
1

3n
2

(2π)

3n
2

Rn

Rn

f (u)g(v)

e−i

Rn

Rn

dy e−i

1
1
u, x
−i v, x

du dv

Rn
1

Rn

1

y−u−v, x
− 12 |y−u−v|2

f (u)g(v)e− 2 |y−u−v|

2

1
−i y, x

du dv dy =
c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


1762

Bui Thi Giang, Nguyen Van Mau, and Nguyen Minh Tuan: Convolutions for Fourier transforms

=

1
n
(2π) 2

e−i

1
y, x

Rn

1
(2π)n

2

1

Rn

Rn

f (u)g(v)e− 2 |y−u−v| du dv dy

γ2

= Fv f ∗ g (x).
Fv

The theorem is proved.
2.3

Convolutions for the Fourier-cosine transform and Fourier-sine transform

For x, y, z ∈ Rn , we write cos xy := cos x, y , sin xy := sin x, y , cos x(y±z) := cos x, y±z , sin x(y±z) :=
sin x, y ± z as there is no danger of confusion. The Fourier-cosine and Fourier-sine transforms are defined by
(Tc f )(x) =

1
n

(2π) 2

Rn

cos xyf (y) dy,

and (Ts f )(x) =

1
n
(2π) 2

Rn

sin xyf (y) dy.

Theorem 2.5 If f, g ∈ L1 (Rn ), then each of the integral transforms (2.7)–(2.10) below defines a generalized
convolution with weight-function γ1 for the transforms Tc , Ts :
γ1

f ∗ g (x) =
Tc

1
4(2π)n

Rn

Rn

f (u)g(v) e−

|x+u+v|2
2

+ e−
+ e−

f

γ1

∗

Tc ,Ts ,Ts

g (x) =

1
4(2π)n

Rn

Rn

f (u)g(v) −e−

|x+u−v|2
2

|x−u+v|2
2

|x+u+v|2
2

+ e−

γ1

∗

Ts ,Tc ,Ts

g (x) =

1
4(2π)n

Rn

Rn

f (u)g(v) −e−

|x+u+v|2
2

+ e−

γ1

∗

Ts ,Ts ,Tc

g (x) =

1
4(2π)n

Rn

Rn

f (u)g(v) −e−

|x+u+v|2
2

− e−
+ e−

(2.7)
du dv.

|x+u−v|2
2

− e−

|x−u−v|2
2

(2.8)
du dv.

|x+u−v|2
2

− e−
f

|x−u−v|2
2

|x−u+v|2
2

+ e−
f

+ e−

|x−u+v|2
2

+ e−

|x−u−v|2

2

(2.9)
du dv.

|x+u−v|2
2

|x−u+v|2
2

+ e−

|x−u−v|2
2

(2.10)
du dv.

P r o o f. The first part for the convolutions (2.7)–(2.10) can be proved in the same way as in that of the
convolution (2.2). Therefore, it suffices to prove the factorization identities of those convolutions.
P r o o f o f t h e c o n v o l u t i o n (2.7). We have
γ1 (x)
f (u)g(v) cos xu cos xv du dv
(2π)n Rn Rn
γ1 (x)
=
f (u)g(v) cos x(u + v) du dv
4(2π)n Rn Rn
γ1 (x)

+
f (u)g(v) cos x(u − v) du dv
4(2π)n Rn Rn
γ1 (x)
+
f (u)g(v) cos x(u − v) du dv
4(2π)n Rn Rn
γ1 (x)
+
f (u)g(v) cos x(u + v) du dv.
4(2π)n Rn Rn

γ1 (x)(Tc f )(x)(Tc g)(x) =

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Math. Nachr. 283, No. 12 (2010) / www.mn-journal.com

1763

Using formula (2.2a), we have
γ1 (x)
f (u)g(v) cos x(u + v) du dv
4(2π)n Rn Rn
γ1 (x)
=
f (u)g(v) ei x,u+v + e−i x,u+v du dv

8(2π)n Rn Rn
|y+u+v|2
1
−i y,x −
2
f
(u)g(v)
e
e
dy du dv
=
8(2π)3n/2 Rn Rn
Rn
|y+u+v|2
1
f (u)g(v)
ei y,x e− 2
dy du dv
+
3n/2
8(2π)
Rn Rn
Rn
|y+u+v|2
1
cos xy
f (u)g(v)e− 2
dy du dv.
=
3n/2

4(2π)
Rn
Rn Rn

(2.11)

Similarly,
γ1 (x)
4(2π)n
=

Rn

1

4(2π)

γ1 (x)
4(2π)n

3n
2

Rn

1

=

Rn

f (u)g(v) cos x(u − v) du dv

Rn

Rn

cos xy

Rn

Rn

f (u)g(v)e

(2.12)

−|y+u−v|2
2

dy du dv,

|y−u+v|2
2

dy du dv,

f (u)g(v) cos x(u − v) du dv
f (u)g(v)e−

cos xy

3n

(2.13)

4(2π) 2 Rn
Rn Rn
γ1 (x)
f (u)g(v) cos x(u + v) du dv
4(2π)n Rn Rn
|y−u−v|2
1
2
cos xy
f (u)g(v)e−
dy du dv.
=
3n
4(2π) 2 Rn
Rn Rn

(2.14)

We thus have
γ1 (x)(Tc f )(x)(Tc g)(x)
=

1
4(2π)3n/2

Rn

cos xy

Rn

Rn

f (u)g(v) e−
+e−

|y+u+v|2
2

|y+u−v|2
2

+ e−

|y−u+v|2
2

+ e−

|y−u−v|2
2

du dv dy

γ1

= Tc f ∗ g (x).
Tc

The convolutions (2.8)–(2.10) can be proved similarly to the proof of (2.7). The theorem is proved.

3
3.1

Applications
Normed rings on L1 (Rn )

Definition 3.1 (See [23].) A vector space V with a ring structure and a vector norm is called a normed ring if
vw ≤ v w , for all v, w ∈ V. If V has a multiplicative unit element e, it is also required that e = 1.
Let X denote the linear space L1 (Rn ). For the convolution (2.4), and for the others the norm of f ∈ X is
defined by
f =
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|α|
n
(2π) 2

Rn

|f (x)| dx,

f =

1
n
(2π) 2

Rn

|f (x)| dx,
c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


1764

Bui Thi Giang, Nguyen Van Mau, and Nguyen Minh Tuan: Convolutions for Fourier transforms

respectively. Theorem 3.2 deals with normed ring structures on L1 (Rn ) that could be used in the theories of
Banach algebra.
Theorem 3.2 X, equipped with each one of the above mentioned convolution multiplications, becomes a
normed ring having no unit. Moreover,
1) For the convolutions from (2.1) to (2.8), X is commutative.
2) For the convolutions (2.9), (2.10), X is non-commutative.
P r o o f. The proof for the first statement is divided into two steps.
Step 1. X has a normed ring structure. We will use the common symbols ∗ for the above convolutions. It is
clear that X has a ring structure. We now prove the multiplicative inequality for convolution (2.2), the proof for
the others is similar. Using formula (2.2a), we have
1
n
(2π) 2

Rn

≤

|f ∗ g|(x) dx

1
3n
2

(2π)
1
=
(2π)n

Rn

Rn

Rn

Rn

|f (u)| |g(v)|

e−

|x−u−v|2
2

+i h,x−u−v

dx du dv

Rn

|f (u)| |g(v)| du dv.

This implies that f ∗ g ≤ f g .
Step 2. X has no unit. Suppose that there exists an element e ∈ X such that f ∗ e = e ∗ f = f, for any f ∈ X.
The factorization identities imply Tj f = γ0 Tk f T e, where γ0 is the common symbol for the weight functions
1, γ1 , γ2 , and Tj , Tk , T ∈ {Fh , Fα , Fv , Tc , Ts } (note that it may be Tj = Tk = T = Tc , etc.).
|x|2

i) For the convolution (2.8), the factorization identity is γ1 (Ts f )(Ts e) = Tc f. We now choose f0 (x) = e− 2 .
Note that F = Tc − iTs on X. Obviously, f0 ∈ X, and Ts f0 = 0. We then have Tc f0 = 0 which contradicts to
the following fact: Tc f0 = Tc f0 − iTs f0 = F f0 = f0 = 0 (see [24, Theorem 7.5]).
ii) For the convolutions (2.9), (2.10), their factorization identities give (Ts f )(γ1 Tc e − 1) = 0. By choosing
f ∈ X such that (Ts f )(x) = 0 for every x ∈ Rn (see [3, 30]), we get γ1 (x)(Tc e)(x) = 1 for every x ∈ Rn . But
it fails because limx→∞ γ1 (x)(Tc e)(x) = 0 (see [30, Theorem 1], [4, Theorem 31], or [24, Theorem 7.5]).
iii) For the other convolutions, the factorization identity is Tk f (γ0 T e − 1) = 0. By (2.2a), (2.4a), (2.7a)
in the respective case, we can choose f ∈ X

for example, f (x) = e−

|x|2
2

so that (Tk f )(x) = 0 to get
γ0 (x)(T e)(x) = 1 for every x ∈ Rn . But it fails because limx→∞ γ0 (x)(T e)(x) = 0. Therefore, the convolution multiplications have no unit. We now prove the last conclusion.
1) It is clear that X is commutative.
2) Note that Ts is regarded as a linear operator from the linear spaces X to the linear space of all functions

(real-valued or complex-valued) defined on Rn . We may choose f ∈ ker Ts \ ker Tc , and g ∈ ker Ts (see [3, 4]).
γ1
γ1
∗
g ∈ ker Ts , but g
∗
f ∈ ker Ts . Again, by (2.10) we have
For the convolution (2.9) it follows f
f

γ1

∗

Ts ,Tc ,Ts

3.2

g ∈ ker Ts , but g

γ1

∗

Ts ,Tc ,Ts

Ts ,Tc ,Ts

Ts ,Tc ,Ts

f ∈ ker Ts . The theorem is proved.

Integral equations of convolution type

Consider the Fredholm integral equation
λϕ(x) +

Rn

K(x, y)ϕ(y) dy = h(x),

(3.1)

where λ ∈ C. Integral equations are important in many applications. Problems in which integral equations are
encountered include radiative energy transfer and the oscillation of a string, membrane, or axle. The equations of
the form (3.1) also occur while solving problems of synthesis of electrostatic and magnetic fields, and of digital
signal processing (see [2, 8, 10, 11]).
The purpose of this subsection is to solve Equation (3.1) in some cases of kernel K(x, y). In what follows,
the given functions are assumed to be in L1 (Rn ), and a unknown function will be determined there. Moreover,
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the function identity f (x) = g(x) means that it is valid for almost every x ∈ Rn . However, if f (x) and g(x) are
continuous on Rn , then the identity f (x) = g(x) means that it holds for every x ∈ Rn .

3.2.1

First equation

Consider the integral equation of convolution type
λϕ(x)
+

1
n
(2π) 2
= q(x),

Rn

Rn

a1 e−

|x+u+v|2
2

+ a2 e−

|x+u−v|2
2

+a3 e−

|x−u+v|2

2

+ a4 e−

|x−u−v|2
2

k(v)ϕ(u) du dv

(3.2)

where λ, a1 , a2 , a3 , a4 ∈ C are predetermined, k, q ∈ L1 (Rn ) are given, and ϕ(x) is to be determined. The
integral equations of convolution type with Gaussian kernels have applications in Physics, Medicine and Biology
(see [8, 10, 11]). Put α1 := a1 + a2 + a3 + a4 , α2 := −a1 − a2 + a3 + a4 , β1 := −a1 + a2 + a3 − a4 ,
β2 := −a1 + a2 − a3 + a4 , and
γ1

DTc ,Ts (x) := λ2 + 2λ(a3 + a4 )γ1 (x)(Tc k)(x) + α1 β2 γ1 (x)Tc k ∗ k (x)
Tc

+ α2 β1 γ1 (x)Tc k

γ1

∗

Tc ,Ts ,Ts

γ1

γ1

Tc

Tc ,Ts ,Ts

DTc (x) := Tc λq + β2 k ∗ q − β1 k
DTs (x) := Ts λq + α1 k

k (x),

γ1

∗

Ts ,Tc ,Ts

∗

q − α2 k

(3.3)

q (x),
γ1

∗

Ts ,Ts ,Tc

q (x).

Theorem 3.3 Assume that DTc ,Ts (x) = 0 for every x ∈ Rn , and
(3.2) has a solution in L1 (Rn ) if and only if F −1
given by
ϕ(x) = F −1

DTc − iDTs
DTc ,Ts

DTc −iDTs
DTc ,Ts

(3.4)

DT c
DTc ,Ts

D

s
, DT T,T
∈ L1 (Rn ). Then Equation
c

s

∈ L1 (Rn ). If this is in case, then the solution is

(x).

P r o o f. By using the factorization identities of the convolutions (2.7), (2.8), we obtain
|y+u+v|2
|y−u−v|2
1
2
+ e−
e− 2
f (u)g(v) du dv (x)
n
2(2π) 2 Rn Rn
= γ1 (x)[(Tc f )(x)(Tc g)(x) − (Ts f )(x)(Ts g)(x)],
|y+u−v|2
|y+u−v|2
1
Tc
[e− 2
+ e− 2
]f (u)g(v) du dv (x)
n
2(2π) 2 Rn Rn
= γ1 (x)[(Tc f )(x)(Tc g)(x) + (Ts f )(x)(Ts g)(x)].

Tc

By (2.11) and (2.14), (2.12), (2.13) it is easy to prove that
Tc
Tc

Rn

Rn

Rn

Rn

f (u)g(v)e−
f (u)g(v)e−

|y+u+v|2
2

|y+u−v|2
2

du dv (x) = Tc
du dv (x) = Tc

Rn

Rn

Rn

Rn

f (u)g(v)e−
f (u)g(v)e−

|y−u−v|2
2

|y−u+v|2
2

du dv (x),
du dv (x).

We then have
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Tc

Tc

Tc

Tc

|y+u+v|2
1
e− 2

f (u)g(v) du dv (x)
n
(2π) 2 Rn Rn
= γ1 (x)[(Tc f )(x)(Tc g)(x) − (Ts f )(x)(Ts g)(x)],

(3.5)

|y+u−v|2
1
e− 2
f (u)g(v) du dv (x)
n
(2π) 2 Rn Rn
= γ1 (x)[(Tc f )(x)(Tc g)(x) + (Ts f )(x)(Ts g)(x)],

(3.6)

|y−u+v|2
1
e− 2
f (u)g(v) du dv (x)
n
(2π) 2 Rn Rn
= γ1 (x)[(Tc f )(x)(Tc g)(x) + (Ts f )(x)(Ts g)(x)],

(3.7)

|y−u−v|2
1
2

e−
f (u)g(v) du dv (x)
n
(2π) 2 Rn Rn
= γ1 (x)[(Tc f )(x)(Tc g)(x) − (Ts f )(x)(Ts g)(x)].

(3.8)

Similarly, by using the convolutions (2.9), (2.10) we prove the following identities
Ts

Ts

Ts

Ts

|y+u+v|2
1
e− 2
f (u)g(v) du dv (x)
n
(2π) 2 Rn Rn
= −γ1 (x)[(Tc f )(x)(Ts g)(x) + (Ts f )(x)(Tc g)(x)],

(3.9)

|y+u−v|2
1
e− 2

f (u)g(v) du dv (x)
n
(2π) 2 Rn Rn
= γ1 (x)[(Tc f )(x)(Ts g)(x) − (Ts f )(x)(Tc g)(x)],

(3.10)

|y−u+v|2
1
e− 2
f (u)g(v) du dv (x)
n
2
(2π)
Rn Rn
= γ1 (x)[(Ts f )(x)(Tc g)(x) − (Tc f )(x)(Ts g)(x)],

(3.11)

|y−u−v|2
1
2
e−
f (u)g(v) du dv (x)
n
(2π) 2 Rn Rn
= γ1 (x)[(Tc f )(x)(Ts g)(x) + (Ts f )(x)(Tc g)(x)].

(3.12)

Necessity. Suppose that ϕ ∈ L1 (Rn ) is a solution of (3.2). Applying Tc , Ts to both sides of this equation
and using (3.5)–(3.8), and (3.9)–(3.12), we obtain a system of two linear equations
λ + α1 γ1 (x)(Tc k)(x) (Tc ϕ)(x) + β1 γ1 (x)(Ts k)(x) (Ts ϕ)(x) = (Tc q)(x),
α2 γ1 (x)(Ts k)(x) (Tc ϕ)(x) + λ + β2 γ1 (x)(Tc k)(x) (Ts ϕ)(x) = (Ts q)(x),

(3.13)

where (Tc ϕ)(x), (Ts ϕ)(x) are unknown functions. The determinants of the system (3.13): DTc ,Ts (x), DTc (x),
DTs (x) have been defined in (3.3), (3.4). Since DTc ,Ts (x) = 0 for every x ∈ Rn , we find (Tc ϕ)(x), (Ts ϕ)(x).
Unfortunately, Tc and Ts have no inverse transforms. Now, we use the inversion formula of the Fourier transform
to obtain the function ϕ(x). Since DTc ,Ts (x) = 0 for every x ∈ Rn ,
(Tc ϕ)(x) =

DTc (x)
,
DTc ,Ts (x)

(Ts ϕ)(x) =

DTs (x)
.
DTc ,Ts (x)

As L1 (Rn ) is the domain of F, F −1 , we get
(F ϕ)(x) =

DTc (x) − iDTs (x)
.
DTc ,Ts (x)

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By the Fourier inverse transform, we get
ϕ(x) = F −1

DTc (x) − iDTs
DTc ,Ts

(x).

The necessary condition is proved.
Sufficiency. Let us first prove the following claim.
Claim 3.4 Suppose that f1 , f2 ∈ L1 (Rn ). If f1 (x) = f1 (−x), and if f2 (x) = −f2 (−x) for every x ∈ Rn ,
then F −1 (f1 − if2 ) = F (f1 + if2 ).
P r o o f o f t h e c l a i m. Since L1 (Rn ) is considered as a domain, F = Tc − iTs , and F −1 = Tc + iTs .
Clearly, f1 − if2 , f1 + if2 ∈ L1 (Rn ), and Tc f2 = Ts f1 = 0. Therefore,
F −1 (f1 − if2 ) = (Tc + iTs )(f1 − if2 ) = Tc f1 − iTc f2 + iTs f1 + Ts f2 = Tc f1 + Ts f2 ,
F (f1 + if2 ) = (Tc − iTs )(f1 + if2 ) = Tc f1 + iTc f2 − iTs f1 + Ts f2 = Tc f1 + Ts f2 .
The claim is proved.
Put
ϕ(x) := F −1

DTc − iDTs

DTc ,Ts

(x).

Is is easy to check that
f1 (x) :=

DTc (x)
,
DTc ,Ts (x)

and f2 (x) :=

satisfy the conditions of the claim. Hence, ϕ(x) = F

DTs (x)
DTc ,Ts (x)
DTc +iDTs
DTc ,Ts

(x). It follows that (F ϕ)(x) =

DTc (x)−iDTs (x)
,
DTc ,Ts (x)

−1
−1
DTc (x)+iDTs (x)
D c (x)

. Since Tc = F +F
and Ts = F 2i−F , we get (Tc ϕ)(x) = DT T,T
, and
DTc ,Ts (x)
2
c s (x)
DTs (x)
DTc ,Ts (x) . Therefore, (Tc ϕ)(x), (Ts ϕ)(x) satisfy (3.13). In this system of the equations, multiplying

(F −1 ϕ)(x) =

(Ts ϕ)(x) =
the second equation by the unit imaginary number i and subtracting it from the first, and calculating without
difficulty we get
1
n
(2π) 2
= (F q)(x).

F λϕ(x) +

Rn Rn

[a1 e−

|x+u+v|2
2

+ a2 e−

|x+u−v|2
2

+ a3 e−

|x−u+v|2
2

+ a4 e−

|x−u−v|2
2

]k(v)ϕ(u) du dv

By the uniqueness theorem of the Fourier transform F , the function ϕ satisfies Equation (3.2) for almost every
x ∈ Rn (see [4, Theorem 34]). The theorem is proved.
It is known that (3.1) is called Fredholm integral equation of first the kind if λ = 0, and that of the second
kind if λ = 0. For the second kind, there exists a great class of functions satisfying the assumptions in item (b)
of Theorem 3.3.
Proposition 3.5 Assume that λ = 0. Then
(a) DTc ,Ts (x) = 0 for every x outside a ball with finite radius.
(b) Assume that DTc ,Ts (x) = 0 for every x ∈ Rn . If DTc , DTs ∈ L1 (Rn ), then
P r o o f. (a) Consider γ1 ∈ S, k ∗ k, k
Tc

γ1

∗

Tc ,Ts ,Ts

DT c
DTc ,Ts

D

s
, DT T,T
∈ L1 (Rn ).
c

s

k ∈ L1 (Rn ) and the Riemann-Lebesgue’s lemma for the

Fourier-cosine transform Tc , the function DTc ,Ts (x) is continuous on Rn , and lim|x|→∞ DTc ,Ts (x) = λ (see [24,
Theorem 7.5], or [30, Theorem 1]). Thus item (a) follows from λ = 0 and the continuity of DTc ,Ts (x).
(b) Due to the continuity of DTc ,Ts (x), and lim|x|→∞ DTc ,Ts (x) = λ = 0, there exist R > 0, ε1 > 0 so that
inf |x|>R |DTc ,Ts (x)| > ε1 .
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Bui Thi Giang, Nguyen Van Mau, and Nguyen Minh Tuan: Convolutions for Fourier transforms

Since DTc ,Ts (x) is continuous, not vanishing in the compact set S(0, R) = {x ∈ Rn : |x| ≤ R}, there

exists ε2 > 0 so that inf |x|≤R |DTc ,Ts (x)| > ε2 . We then have supx∈Rn |DT ,T1 (x)| ≤ max{ ε11 , ε12 } < ∞. As a
c

result, the function |DT ,T1 (x)| is continuous and bounded on Rn . Therefore,
c s
provided DTc , DTs ∈ L1 (Rn ). Item (b) is proved.
3.2.2

s

DT c
DT s
|DTc ,Ts (x)| , |DTc ,Ts (x)|

∈ L1 (Rn ),

Second equation

Consider the following integral equation of convolution type
λϕ(x) +

1
(2π)n

1

e− 2 |x−u−v|
Rn

2

+i h,x−u−v

Rn

k(v)ϕ(u) du dv = g(x),

(3.14)

where λ ∈ C, g, k are given functions from L1 (Rn ), ϕ is the function to be determined.
Theorem 3.6 (a) If the equation (3.14) has a solution, then (Fh g)(x) = 0 whenever λ + γ1 (x)(Fh k)(x) = 0.
(b) Assume that λ + γ1 (x)(Fh k)(x) = 0 for every x ∈ Rn , and λ+γF1h(Fg h k) ∈ L1 (Rn ). Then the equation
(3.14) has a solution in L1 (Rn ) if and only if Fh−1
of the equation (3.14) is given by ϕ(x) = Fh−1

Fh g
λ+γ1 (Fh k)

Fh g
λ+γ1 (Fh k)

∈ L1 (Rn ). If this is in case, then the solution

(x).

We need the following lemma.
Lemma 3.7 (Inversion Theorem) Assume that f, Fh f ∈ L1 (Rn ). Then
1
n
(2π) 2

f (x) =

ei

x,y+h

Rn

(Fh f )(y)dy := Fh−1 Fh f (y) (x)
for almost every x ∈ Rn .

P r o o f o f t h e l e m m a. We have
(Fh f )(x) =

1
n
(2π) 2

e−i

x,y

e−i

h,y

Rn

f (y) dy = F e−i

h,y

f (y) (x) ∈ L1 (Rn ).

Since the function e−i h,y is continuous and bounded on Rn , the function e−i h,y f (y) belongs to L1 (Rn ).
Using the inversion theorem of the Fourier transform (see [24, Theorem 7.7]), we obtain
f (x) =

1
n
(2π) 2

ei

x,y+h

Rn

(Fh f )(y)dy

for almost every

x ∈ Rn .

The lemma is proved.
First, we prove item (b) of the theorem. Suppose that ϕ ∈ L1 (Rn ) is a solution of (3.14). By (2.2), Equation
γ1

(3.14) can be rewritten in the form: λϕ(x) + ϕ ∗ k (x) = g(x). Applying Fh to both sides of this equation

Fh

and using the factorization identity of convolution (2.2), we get
λ + γ1 (x)(Fh k)(x) (Fh ϕ)(x) = (Fh g)(x).

(3.15)
(Fh g)(x)
λ+γ1 (x)(Fh k)(x) . Using Lemma 3.7, we
function ϕ = Fh−1 λ+γF1h(Fg h k) ∈ L1 (Rn ).

From the assumption λ + γ1 (x)(Fh k)(x) = 0 it follows that (Fh ϕ)(x) =
obtain ϕ = Fh−1

Fh g
λ+γ1 (Fh k)

By Lemma 3.7, (Fh ϕ)(x) =

∈ L1 (Rn ). Conversely, consider the
(Fh g)(x)
λ+γ1 (x)(Fh k)(x) .

This implies that λ + γ1 (x)(Fh k)(x) (Fh ϕ)(x) = (Fh g)(x). By

γ1

formula (2.2) Fh λϕ + ϕ ∗ k (x) = (Fh g)(x). Again by Lemma 3.7 ϕ(x) satisfies (3.14) for almost every
Fh

x ∈ Rn . Item (b) is proved.

Item (a) now is just a consequence of (3.15) and the fact that the functions on both sides of (3.15) are continuous on Rn . Theorem 3.6 is proved.
Analogously to Proposition 3.5 we can prove the following proposition.
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Proposition 3.8 Let λ = 0. Then λ + γ1 (x)(Fh k)(x) = 0 for every x outside a ball with finite radius.
Fh g
∈ L1 (Rn ).
Moreover, if Fh g ∈ L1 (Rn ) and if λ + γ1 (x)(Fh k)(x) = 0 for every x ∈ Rn , then λ+γ
1 Fh k
Remark 3.9 If λ ∈ C \ (−∞, 0] and k(x) is Gaussian, then λ + γ1 (x)(Fh k)(x) = 0 for every x ∈ Rn .

3.2.3

Third equation

Let α = (α1 , . . . , αn ) ∈ Rn+ (αi > 0, i = 1, . . . , n) be given. The notation α · x has been defined in
Sub-subsection 2.2.2. We consider the equation
n
j=1

λϕ(x) +

αi |α|

(2π)n

k(v)e−
Rn

|α·(x−u−v)|2
2

Rn

ϕ(u) du dv = g(x),

(3.16)

where λ ∈ C, g, k are given functions in L1 (Rn ), ϕ is to be determined.
Theorem 3.10 (a) If Equation (3.16) has a solution, then (Fα g)(x) = 0 whenever λ + γ1 (x)(Fα k)(x) = 0.
(Fα g)
(b) Assume that λ + γ1 (x)(Fα k)(x) = 0 for every x ∈ Rn , and λ+γ
∈ L1 (Rn ). Then the equation
1 (Fα k)
(3.16) has a solution in L1 (Rn ) if and only if Fα−1
of Equation (3.16) is given by ϕ(x) = Fα−1

Fα g
λ+γ1 (Fα k)

Fα g
λ+γ1 (Fα k)

∈ L1 (Rn ). If this is in case, then the solution

(x).

As Lemma 3.7 we can prove the following lemma.
Lemma 3.11 (Inversion Theorem) Assume that f, Fα f ∈ L1 (Rn ). Then
n

f (x) =

j=1

αj
n

|α|(2π) 2

ei
Rn

α·x,y

(Fα f )(y)dy := Fα−1 (Fα f ) (x) for almost every x ∈ Rn .

We first prove item (b). By (2.4) Equation (3.16) can be rewritten in the form of the following convolution
γ1

equation: λϕ(x) + f ∗ k (x) = g(x). Applying Fα to both sides of this equation and using (2.4), we get
Fα

λ + γ1 (x)(Fα k)(x) (Fα ϕ)(x) = (Fα g)(x).

(3.17)

α g)(x)
. Using Lemma 3.11, we obtain ϕ(x) = Fα−1 λ+γF1α(Fg α k) (x).
Hence (Fα ϕ)(x) = λ+γ(F
1 (x)(Fα k)(x)
The conversion is proved similarly as that of Theorem 3.6. Thus item (b) is proved. Item (a) is just a
consequence of (3.17) and the fact that the functions in this equation are continuous on Rn . Theorem 3.10 is
proved.
The proof of Proposition 3.12 below is similar to that of Proposition 3.5.

Proposition 3.12 Let λ = 0. Then λ + γ1 (x)(Fα k)(x) = 0 for every x outside a ball with finite radius.
Fα g
∈ L1 (Rn ).
Moreover, if Fα g ∈ L1 (Rn ), and if λ + γ1 (x)(Fα k)(x) = 0 for every x ∈ Rn , then λ+γ
1 Fα k
Remark 3.13 If λ ∈ C \ (−∞, 0] and g(x) is Gaussian, then λ + γ1 (x)(Fα k)(x) = 0 for every x ∈ Rn .

In the general theory of integral equations, each one of the requirements DTc ,Ts (x) = 0, λ + γ1 (x)(Fh k)(x)
= 0, and λ + γ1 (x)(Fα k)(x) = 0 for every x ∈ Rn as in Theorems 3.3, 3.6, 3.10 is the normally solvable
condition of the corresponding equation (see [2, 16, 30]).
Comparison In constructing convolutions for integral transforms, previously published works (for example,
see [1,16,25–28]) solved integral equations of convolution type in a manner that provided the suffcient conditions
for the solvability of equations and gave implicit solutions via the Wiener-L`evy theorem (see [16, 29]).
By means of normally solvable condition of integral equation, the generalized convolutions in Section 2 work
out the suffcient and necessary conditions for the solvability of the equations and their explicit solutions.
Acknowledgements The second named author was partially supported by the Central Project, grant No. QGTD-0809-VNU.

The third named author is partially supported by the Vietnam National Foundation for Science and Technology Development.
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