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

Convolutions for the Fourier transforms with geometric variables and applications Bui Thi Giang1, Nguyen Van Mau2, and Nguyen Minh Tuan∗3 1 Dept.. of Edu, Ha Noi National Univ., G7 Build

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Convolutions for the Fourier transforms with geometric variables and applications

Bui Thi Giang1, Nguyen Van Mau2, and Nguyen Minh Tuan∗3

1 Dept of Basic Science, Institute of Cryptography Science, 141 Chien Thang str., Thanh Tri dist., Hanoi, Vietnam

2 Dept of Mathematical Analysis, University of Hanoi, 334 Nguyen Trai str., Thanh Xuan dist., Hanoi, Vietnam

3 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 onL1(Rn) are constructed, and explicit solutions of integral equations of convolution type are obtained

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 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, gen-eralized 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 geo-metric 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 convolu-tions of the Fourier-cosine and Fourier-sine transforms As usual, there exist different convoluconvolu-tions 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: tuannm@hus.edu.vn, nguyentuan@vnu.edu.vn, Phone: +84 4 37548092, Fax: +84 4 37548092

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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 T c , T s 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, U3be linear spaces on the field of scalarsK, 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, U3to V, respectively Let δ denote an element in the algebra V.

Definition 2.1 (See also [5, 19, 20]) A bilinear map∗ : U1× U2 :ư→ 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

K3,K1,K2g If δ is the unit of V, we say briefly the convolution for K3, K1, K2. If U1= U2= U3 and K1= K2= K3, the convolution is denoted simply f ∗ δ

K1g, and f ∗

K1g if δ is the unit of V As the notation

K3,K1,K2g already defines the factorization identity K3



f ∗ δ

K1g



= 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 onRn For

x, y ∈ R n ,letx, y denote the scalar product, and |x|2= x, x.

2.2 Convolutions for the Fourier transforms with geometric variables

2.2.1 Convolutions for the Fourier transform with shift

Let h ∈ R n be fixed Let F denote the Fourier and inverse transforms as

(F f)(x) := 1

(2π) n2



Rn

e ưix,y f (y) dy, (F ư1 f )(x) := 1

(2π) n2



Rn

e ix,y f (y) dy.

Consider the following integral transforms

(F h f )(x) := 1

(2π) n

2



Rn

e ưix+h,y f (y) dy, (F ư1

h f )(x) := 1

(2π) n

2



Rn

e ix,y+h f (y)dy.

We call F h the Fourier transform with shift, and F h ư1 its inverse transform The inversion formula of F his proved

by Lemma 3.7 Put γ1(x) = e ư1|x|2

.

Theorem 2.2 If f, g ∈ L1(Rn ), then each of the integral expressions (2.1), (2.2) below defines a convolution:



F h g



(x) := 1 (2π) n

2



Rn



f F γ ∗1

h

g



(x) := (2π)1 n



Rn



Rn

f (u)g(v)e ư1|xưuưv|2+ih,xưuưv du dv. (2.2)

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 ∈ R nthe following formulae hold

1

(2π) n

2



Rn

e ưix,y±u±vư1|y±u±v|2

dy = e ư |x|22 ;



Rn

e ư1|xưuưv|2

dx = (2π) n2 (2.2a) (see [24, Lemma 7.6]) We then have



Rn



f γ ∗1

F h g



(x)

 dx ≤ (2π)1 n



Rn



Rn



Rn |f(u)| |g(v)|e ư1|xưuưv|2 du dv dx < +∞.

It implies that f γ ∗1

F h g ∈ L1(Rn ) We prove the factorization identity Using formula (2.2a), we obtain

γ1(x)(F h f )(x)(F h g )(x)

(2π) 3n

2



Rn



Rn



Rn

f (u)g(v)e ư|yưuưv|22 ưix,y+i<h,ưuưv du dv dy

(2π) n

2



Rn

e ưix+h,y

1

(2π) n



Rn



Rn

f (u)g(v)e ư|yưuưv|22 +ih,yưuưv du dv

dy

= F h



f γ ∗1

F h g



(x).

The proof is complete

2.2.2 Convolutions for the Fourier transform with similarity

Let α = (α1, , α n ) ∈ R n

+ (α i > 0 ∀i = 1, , n) be fixed Write α · x = (α1x1, , α n x n) for any

x ∈ R n Consider the following integral transforms

(F α f )(x) := |α|

(2π) n

2



Rn

e ưiα·x,y f (y) dy,

F ư1

α :=

n



j=1

α j

|α|(2π) n

2



Rn

e iα·x,y (F α f )(y) dy.

We call F α the Fourier transform with similarity, and F ư1

α 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 α g



(x) := |α|

(2π) n

2



Rn



f γ ∗1

F α g



(x) :=

n j=1 α j |α|

(2π) n



Rn



Rn

f (u)g(v)e ư |α·(xưuưv)|22 du dv. (2.4)

P r o o f It is easy to prove the convolution (2.3) The fact f γ ∗1

F α g ∈ L1(Rn) for the convolution (2.4) is proved immediately We prove the factorization identity of the convolution (2.4) From (2.2a) we get

(n j=1 α j)

(2π) n

2



Rn

e ưix,α·(yưuưv)ư1|α·(yưuưv)|2

dy = e ư |x|2

We then have

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 2010 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim www.mn-journal.com

γ1(x)(F α f )(x)(F α g )(x)

= |α|2

n

j=1 α j

(2π) 3n

2



Rn



Rn

f (u)g(v)



Rn

e ưix,α·(yưuưv)ư1|α·(yưuưv)|2

dy



e ưiα·x,u e ưiα·x,v du dv

=

|α|2(n

j=1

α j)

(2π) 3n

2



Rn



Rn



Rn

f (u)g(v)e ư1|α·(yưuưv)|2

e ưiα·x,y du dv dy

= |α|

(2π) n2



Rn

e ưiα·x,y

⎡

⎣|α|

n j=1 α j

(2π) n



Rn



Rn

f (u)g(v)e ư1|α·(yưuưv)|2

du dv

⎤

⎦ dy

= F α



f F γ ∗1

α

g



(x).

The theorem is proved

2.2.3 Convolutions for the Fourier transform with inversion

x:=1

x1, , 1

x n



.Consider the integral transform

(F v f )(x) :=

⎧

⎨

⎩

1

(2π) n

2



Rn

e ưiy,1

x  f (y) dy if x i

We call F vthe Fourier transform with inversion Consider the following function

γ2(x) =



e ư1|1

if x i

It is worth saying that the function γ2(x) is bounded and infinite differentiable on R n

Theorem 2.4 If f, g ∈ L1(Rn ), then each of the integral transforms (2.5), (2.6) below defines a generalized

convolution:



f F ∗

v

g



(x) = (2π)1 n

2



Rn



f γ ∗2

F v g



(x) = (2π)1 n



Rn



Rn

f (u)g(v)e ư1|xưuưv|2

P r o o f The convolution (2.5) is proved immediately For the convolution (2.6) the fact that f γ ∗2

F v g ∈ L1(Rn)

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

F v



f γ ∗2

F v g



(x) = γ2(x)(F v f )(x)(F v g )(x) Indeed, if at least one of the x i is zero, this identity is clear

Consider x i

1

(2π) n

2



Rn

e ưiyưuưv,1

x ư1|yưuưv|2

dy = e ư1|1

We then have

γ2(x)(F v f )(x)(F v g )(x)

(2π) 3n

2



Rn



Rn

f (u)g(v)



Rn

e ưiyưuưv,1

x ư1|yưuưv|2

dy



e ưiu,1

x ưiv,1

x  du dv

(2π) 3n

2



Rn



Rn



Rn

f (u)g(v)e ư1|yưuưv|2ưiy,1

x  du dv dy=

= 1

(2π) n

2



Rn

e ưiy,1

1

(2π) n



Rn



Rn

f (u)g(v)e ư1|yưuưv|2

du dv



dy

= F v



f γ ∗2

F v g



(x).

The theorem is proved

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

For x, y, z ∈ R n ,we writecos xy := cosx, y, sin xy := sinx, y, cos x(y±z) := cosx, y±z, sin x(y±z) := sinx, y ± z as there is no danger of confusion The Fourier-cosine and Fourier-sine transforms are defined by

(T c f )(x) = 1

(2π) n

2



Rn cos xyf(y) dy, and (T s f )(x) = 1

(2π) n

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 γ1for the transforms T c , T s :



f γ ∗1

T c g



(x) = 1

4(2π) n



Rn



Rn

f (u)g(v)



2



du dv.

(2.7)



f γ ∗1

T c ,T s ,T s g



(x) = 4(2π)1 n



Rn



Rn

f (u)g(v)



2

du dv.

(2.8)



f γ ∗1

T s ,T c ,T s g



(x) = 4(2π)1 n



Rn



Rn

f (u)g(v)





du dv.

(2.9)



f γ ∗1

T s ,T s ,T c g



(x) = 1

4(2π) n



Rn



Rn

f (u)g(v)



2



du dv.

(2.10)

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)(T c f )(x)(T c g )(x) = γ (2π)1(x) n



Rn



Rn

f (u)g(v) cos xu cos xv du dv

= γ1(x)

4(2π) n



Rn



Rn

f (u)g(v) cos x(u + v) du dv

+4(2π) γ1(x) n



Rn



Rn

f (u)g(v) cos x(u ư v) du dv

+4(2π) γ1(x) n



Rn



Rn

f (u)g(v) cos x(u ư v) du dv

+4(2π) γ1(x) n



Rn



Rn

f (u)g(v) cos x(u + v) du dv.

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 2010 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim www.mn-journal.com

Using formula (2.2a), we have

γ1(x)

4(2π) n



Rn



Rn

f (u)g(v) cos x(u + v) du dv

=8(2π) γ1(x) n



Rn



Rn

f (u)g(v)e ix,u+v + e ưix,u+v

du dv

=8(2π)13n/2



Rn



Rn

f (u)g(v)



Rn

e ưiy,x e ư |y+u+v|2

8(2π) 3n/2



Rn



Rn

f (u)g(v)



Rn

e iy,x e ư |y+u+v|2

=4(2π)13n/2



Rn cos xy



Rn



Rn

f (u)g(v)e ư |y+u+v|2

(2.11)

Similarly,

γ1(x)

4(2π) n



Rn



Rn

f (u)g(v) cos x(u ư v) du dv

4(2π) 3n

2



Rn cos xy



Rn



Rn

f (u)g(v)e ư|y+uưv|22 dy du dv,

(2.12)

γ1(x)

4(2π) n



Rn



Rn

f (u)g(v) cos x(u ư v) du dv

4(2π) 3n

2



Rn cos xy



Rn



Rn

f (u)g(v)e ư |yưu+v|22 dy du dv, (2.13)

γ1(x)

4(2π) n



Rn



Rn

f (u)g(v) cos x(u + v) du dv

4(2π) 3n

2



Rn cos xy



Rn



Rn

f (u)g(v)e ư |yưuưv|22 dy du dv. (2.14)

We thus have

γ1(x)(T c f )(x)(T c g )(x)

= 4(2π)13n/2



Rn cos xy



Rn



Rn

f (u)g(v)



2

2



du dv dy

= T c



f γ ∗1

T c g



(x).

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

3 Applications

3.1 Normed rings onL1 (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 = (2π) |α| n

2



Rn |f(x)| dx, f = (2π)1 n

2



Rn |f(x)| dx,

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

(2π) n

2



Rn |f ∗ g|(x) dx

(2π) 3n2



Rn



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



Rn



2 +ih,xưuưv

 dxdudv

= (2π)1 n



Rn



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 implyT j f = γ0T k f T  e , where γ0 is the common symbol for the weight functions

1, γ1, γ2, andT j , T k , T  ∈ {F h , F α , F v , T c , T s } (note that it may be T j = T k = T  = T c ,etc.)

i) For the convolution (2.8), the factorization identity is γ1(T s f )(T s e ) = T c f We now choose f0(x) = e ư |x|2

2 .

Note that F = T c ư iT s on X Obviously, f0∈ X, and T s f0= 0 We then have T c f0= 0 which contradicts to

the following fact: T c f0= T c f0ư iT s f0= F f0= f0

ii) For the convolutions (2.9), (2.10), their factorization identities give(T s f )(γ1T c e ư 1) = 0 By choosing

it fails becauselimx→∞ γ1(x)(T c e )(x) = 0 (see [30, Theorem 1], [4, Theorem 31], or [24, Theorem 7.5]).

iii) For the other convolutions, the factorization identity isT k f (γ0T  e ư 1) = 0 By (2.2a), (2.4a), (2.7a)

in the respective case, we can choose f ∈ X for example, f (x) = e ư |x|2

2 so that (T k f

γ0(x)(T  e )(x) = 1 for every x ∈ R n But it fails becauselimx→∞ γ0(x)(T  e )(x) = 0 Therefore, the

convolu-tion multiplicaconvolu-tions have no unit We now prove the last conclusion

1) It is clear that X is commutative.

2) Note that T s is regarded as a linear operator from the linear spaces X to the linear space of all functions

(real-valued or complex-valued) defined onRn We may choose f ∈ ker T s \ ker T c , and g s(see [3, 4])

For the convolution (2.9) it follows f T γ ∗1

s ,T c ,T s g s , but g T γ ∗1

s ,T c ,T s f ∈ ker T s Again, by (2.10) we have

f γ ∗1

T s ,T c ,T s g ∈ ker T s , but g γ ∗1

T s ,T c ,T s f s The theorem is proved

3.2 Integral equations of convolution type

Consider the Fredholm integral equation

λϕ (x) +



Rn

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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 2010 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim www.mn-journal.com

the function identity f (x) = g(x) means that it is valid for almost every x ∈ R n However, if f (x) and g(x) are

continuous onRn , then the identity f (x) = g(x) means that it holds for every x ∈ R n

3.2.1 First equation

Consider the integral equation of convolution type

λϕ (x)

(2π) n

2



Rn



Rn





k (v)ϕ(u) du dv

= q(x),

(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 := ưa1ư a2 + a3+ a4, β1 := ưa1+ a2+ a3ư a4,

β2:= ưa1+ a2ư a3+ a4, and

D T c ,T s (x) := λ2+ 2λ(a3+ a4)γ1(x)(T c k )(x) + α1β2γ1(x)T c



k γ ∗1

T c k



(x) + α2β1γ1(x)T c



k γ ∗1

T c ,T s ,T s k



(x),

D T c (x) := T c



λq + β2k

γ1

∗

T c q ư β1k

γ1

∗

T c ,T s ,T s q



(x),

(3.3)

D T s (x) := T s



λq + α1k

γ1

∗

T s ,T c ,T s q ư α2k

γ1

∗

T s ,T s ,T c q



(3.2) has a solution in L1(Rn ) if and only if F ư1 D

given by

ϕ (x) = F ư1

D T c ư iD T s

D T c ,T s



(x).

P r o o f By using the factorization identities of the convolutions (2.7), (2.8), we obtain

T c



1

2(2π) n2



Rn



Rn



f (u)g(v) du dv



(x)

= γ1(x)[(T c f )(x)(T c g )(x) ư (T s f )(x)(T s g )(x)],

T c

2(2π) n

2



Rn



2 ]f(u)g(v) du dv (x)

= γ1(x)[(T c f )(x)(T c g )(x) + (T s f )(x)(T s g )(x)].

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

T c



Rn



Rn

f (u)g(v)e ư |y+u+v|22 du dv

(x) = T c



Rn



Rn

f (u)g(v)e ư |yưuưv|22 du dv

(x),

T c



Rn



Rn

f (u)g(v)e ư |y+uưv|22 du dv



(x) = T c



Rn



Rn

f (u)g(v)e ư |yưu+v|22 du dv



(x).

We then have

T c



1

(2π) n

2



Rn



Rn

2 f (u)g(v) du dv



(x)

= γ1(x)[(T c f )(x)(T c g )(x) − (T s f )(x)(T s g )(x)],

(3.5)

T c



1

(2π) n

2



Rn



Rn

2 f (u)g(v) du dv



(x)

= γ1(x)[(T c f )(x)(T c g )(x) + (T s f )(x)(T s g )(x)],

(3.6)

T c



1

(2π) n

2



Rn



Rn

2 f (u)g(v) du dv



(x)

= γ1(x)[(T c f )(x)(T c g )(x) + (T s f )(x)(T s g )(x)],

(3.7)

T c



1

(2π) n

2



Rn



Rn

2 f (u)g(v) du dv



(x)

= γ1(x)[(T c f )(x)(T c g )(x) − (T s f )(x)(T s g )(x)].

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

T s



1

(2π) n

2



Rn



Rn

2 f (u)g(v) du dv



(x)

= −γ1(x)[(T c f )(x)(T s g )(x) + (T s f )(x)(T c g )(x)],

(3.9)

T s



1

(2π) n

2



Rn



Rn



(x)

= γ1(x)[(T c f )(x)(T s g )(x) − (T s f )(x)(T c g )(x)],

(3.10)

T s



1

(2π) n

2



Rn



Rn

2 f (u)g(v) du dv



(x)

= γ1(x)[(T s f )(x)(T c g )(x) − (T c f )(x)(T s g )(x)],

(3.11)

T s



1

(2π) n

2



Rn



Rn

2 f (u)g(v) du dv



(x)

= γ1(x)[(T c f )(x)(T s g )(x) + (T s f )(x)(T c g )(x)].

(3.12)

Necessity Suppose that ϕ ∈ L1(Rn ) is a solution of (3.2) Applying T c , T sto 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 + α1γ1(x)(T c k )(x)(T c ϕ )(x) +β1γ1(x)(T s k )(x)(T s ϕ )(x) = (T c q )(x),



α2γ1(x)(T s k )(x)(T c ϕ )(x) +λ + β2γ1(x)(T c k )(x)(T s ϕ )(x) = (T s q )(x), (3.13)

where(T c ϕ )(x), (T s ϕ )(x) are unknown functions The determinants of the system (3.13): D T c ,T s (x), D T c (x),

D T s (x) have been defined in (3.3), (3.4) Since D T c ,T s n ,we find(T c ϕ )(x), (T s ϕ )(x) Unfortunately, T c and T shave no inverse transforms Now, we use the inversion formula of the Fourier transform

to obtain the function ϕ(x) Since D T c ,T s n ,

(T c ϕ )(x) = D T c (x)

D T c ,T s (x) , (T s ϕ )(x) =

D T s (x)

D T c ,T s (x) .

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

(F ϕ)(x) = D T c (x) − iD T s (x)

D T c ,T s (x) .

c

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