Integral Transforms and Special Functions

ISSN: 1065-2469 (Print) 1476-8291 (Online) Journal homepage: http://www.tandfonline.com/loi/gitr20

On Fourier – Kontorovich–Lebedev generalized

convolution transforms

Nguyen Minh Khoa, Nguyen Thanh Hong & Vu Kim Tuan

To cite this article: Nguyen Minh Khoa, Nguyen Thanh Hong & Vu Kim Tuan (2017): On Fourier

– Kontorovich–Lebedev generalized convolution transforms, Integral Transforms and Special

Functions, DOI: 10.1080/10652469.2017.1281922

To link to this article: http://dx.doi.org/10.1080/10652469.2017.1281922

Published online: 29 Jan 2017.

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Date: 30 January 2017, At: 17:40

INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS, 2017

http://dx.doi.org/10.1080/10652469.2017.1281922

Research Article

On Fourier – Kontorovich–Lebedev generalized convolution

transforms

Nguyen Minh Khoaa , Nguyen Thanh Hongb and Vu Kim Tuanc

a Department of Mathematics, Electric Power University, Hanoi, Vietnam; b School for Gifted Students, Hanoi

National University of Education, Hanoi, Vietnam; c Department of Mathematics, University of West Georgia,

Carrollton, GA, USA

ABSTRACT

ARTICLE HISTORY

We deal with several classes of integral transformations of the form

Received 24 October 2016

Accepted 9 January 2017

f (x) → D

R2+

(e−v cosh(x−u) ± e−v cosh(x+u) )f (u)h(v) du dv,

KEYWORDS

where D is a special differential operator. In case D is the identical

operator, we prove a Young type theorem for a generalized convolution operator related to the Fourier and the Kontorovich–Lebedev

integral transforms. In case D is a certain finite order differential operator, we study the unitary property of these transforms and apply

them to solve a class of integro-differential equations. Several classes

of system of integral equations are also considered.

Convolution; Young’s

theorem; Watson’s theorem;

Fourier;

Kontorovich–Lebedev

transform;

integro-diﬀerential equation;

system of integral equations

AMS SUBJECT

CLASSIFICATION

33C10; 44A35; 45E10; 45J05;

47A30; 47B15

1. Introduction

The Fourier cosine and the Fourier sine transforms are defined as follows

(Fc f )(y) ≡ Fc [f ](y) =

2

π

(Fs f )(y) ≡ Fs [f ](y) =

2

π

∞

0

∞

0

f (x) cos xy dx =

2 d

π dy

f (x) sin xy dx =

2 d

π dy

∞

0

∞

0

f (x)

f (x)

sin xy

dx,

x

(1.1)

1 − cos xy

dx. (1.2)

x

The first integrals in (1.1) and (1.2) are well-defined for f ∈ L1 (R+ ), while the second

integrals in (1.1) and (1.2) are also well-defined for f ∈ L2 (R+ ) (see [1,2]).

The Kontorovich–Lebedev integral transform and its inverse have the form (see [3–5])

(Kf )(y) = K[f ](y) = g(y) =

f (x) = K −1 [g](x) =

2

π 2x

CONTACT Nguyen Minh Khoa

khoanm@epu.edu.vn

235 Hoang Quoc Viet, Hanoi, Vietnam

© 2017 Informa UK Limited, trading as Taylor & Francis Group

∞

0

∞

0

Kiy (x)f (x) dx,

(1.3)

y sinh(π y)Kiy (x)g(y) dy,

(1.4)

Department of Mathematics, Electric Power University,

2

N. M. KHOA ET AL.

here the integrals are understood in the mean-square norms with weights, if necessary. For

more properties of the Kontorovich–Lebedev transform and its convolution transforms,

we refer the reader to the references [6–8].

In [9], the following non-commutative convolutions for the Fourier and Kontorovich–Lebedev integral transforms were introduced

(f ∗ h){ c } (x) =

s

1

2

∞

0

∞

0

[e−v cosh(x−u) ± e−v cosh(x+u) ]f (u)h(v) du dv,

x > 0. (1.5)

The existence of the generalized convolutions (1.5) for two functions in L1 (R+ ) with a

weight and their application to solve integral equations of convolution type were studied

in [9]. Specially, in certain function spaces, the generalized convolutions (1.5) satisfy the

following factorization equalities

F{ c } [(f ∗ h){ c } ](y) = (F{ c } f )(y)(Kh)(y),

s

s

s

∀y > 0.

(1.6)

In any convolution (f ∗ h)(x) of two functions f and h, if we fix a function h and let f

vary in a certain function space, then one can study convolution transforms of the type f →

D[f ∗ h], where D is a certain differential operator. The most famous integral transform

constructed by that way is the Watson transform that is related to the Mellin convolution

and the Mellin transform (see [2])

∞

f (x) −→ g(x) =

0

k(xy)f (y) dy.

Recently, several authors have been investigated convolution transforms of this type

(see [10–13]). However, none of these references considered any application to integrodifferential equations. In this paper we are interested in transforms related to the generalized convolutions (1.5), namely, the transforms of the form given in the abstract. For the

case D is the identity operator, in Section 2 we study Young’s theorem and Young’s inequality for these generalized convolutions. In Section 3, for a certain finite order differential

operators D, we obtain a necessary and sufficient condition such that the respective transforms are unitary on L2 (R+ ), and derive the inverse transforms. In Sections 4 and 5, we

solve in closed form a class of integro-differential equations as well as several classes of

systems of two integral equations with the help of these integral transforms.

2. A Young type theorem

In this section, we will prove several norm properties of the generalized convolutions

(1.5). Throughout the paper, we are interested in the following two-parameter family of

Lebesgue’s spaces.

Definition 2.1 (see [9]): For α ∈ R, 0 < β

of all functions f (x) defined in R+ such that

∞

0

α,β

1, p ≥ 1, we denote by Lp (R+ ) the space

|f (x)|p K0 (βx)xα dx < ∞.

(2.1)

INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS

3

The norm of a function in this space is defined by

f

α,β

L p (R + )

=

∞

|f (x)|p K0 (βx)xα dx

0

1/p

.

Using the asymptotic of the Macdonald function Kν (z) (see [4])

π 1/2 −z

e [1 + O(1/z)],

2z

K0 (z) = − log z + O(1), z → 0,

Kν (z) =

z → ∞,

(2.2)

(2.3)

formula (2.1) can be expressed in an equivalent form

1

0

|f (x)|p | log x|xα dx +

∞

1

|f (x)|p xα−1/2 e−βx dx < ∞.

For the Fourier convolution (see [1])

1

(f ∗ h)(x) = √

F

2π

∞

−∞

h(x − y)f (y) dy,

(2.4)

Young’s theorem and its corollary, the so-called Young inequality are fundamental (see

[14]). We will prove now Young’s type theorem for the generalized convolutions (1.5).

Theorem 2.2 (Young’s Type Theorem): Let p,q,r be real numbers in (1; ∞) such that

0,β

1/p + 1/q + 1/r = 2 and let f ∈ Lp (R+ ), g ∈ Lq (R+ ), 0 < β 1, h ∈ Lr (R+ ), then

∞

0

(f ∗ g){ c } (x)h(x) dx

s

2−1/q f

Lp (R + )

g

0,β

Lq (R + )

h

Lr (R + ) .

Proof: Let p1 , q1 , r1 be the conjugate exponents of p,q,r, respectively, it means

1

1

1

1

1

1

+

= +

= +

= 1.

p p1

q q1

r

r1

Then it is obvious that 1/p1 + 1/q1 + 1/r1 = 1. Put

F(x, u, v) = |g(v)|q/p1 |h(x)|r/p1 |[e−v cosh(x−u) ± e−v cosh(x+u) ]|1/p1 ,

G(x, u, v) = |h(x)|r/q1 |f (u)|p/q1 |[e−v cosh(x−u) ± e−v cosh(x+u) ]|1/q1 ,

H(x, u, v) = |f (u)|p/r1 |g(v)|q/r1 |[e−v cosh(x−u) ± e−v cosh(x+u) ]|1/r1 .

(2.5)

4

N. M. KHOA ET AL.

We have

F(x, u, v)G(x, u, v)H(x, u, v) = |f (u)g(v)h(x)[e−v cosh(x−u) ± e−v cosh(x+u) ]|.

(2.6)

Using the following integral representation for the Macdonald function with a pure

imaginary index (see [4,5,15])

∞

Kiy (x) =

e−x cosh u cos yu du,

0

x > 0,

(2.7)

we easily see that

Hence, K0 (v)

F

p1

Lp1 (R3+ )

=

∞

1

2

K0 (v) =

0

(e−v cosh(x−u) + e−v cosh(x+u) ) du.

K0 (βv) for 0 < β

∞

∞

0

∞

0

0

∞

∞

0

0

1, and

|g(v)|q |h(x)|r |[e−v cosh(x−u) ± e−v cosh(x+u) ]| du dv dx

|g(v)|q K0 (βv)|h(x)|r dv dx

q

= g

(2.8)

r

Lr (R + ) ,

h

0,β

Lq (R + )

(2.9)

and similarly,

H

r1

Lr1 (R3+ )

∞

=

0

∞

∞

0

∞

0

0

∞

0

|f (u)|p |g(v)|q K0 (βv) du dv

p

Lp (R + )

= f

|f (u)|p |g(v)|q [e−v cosh(x−u) ± e−v cosh(x+u) ] du dv dx

q

g

0,β

Lq (R + )

.

(2.10)

On the other hand, in the space Lq1 (R3+ ), using formula (2.8) we have

G

q1

Lq1 (R3+ )

∞

=

∞

0

∞

0

∞

0

∞

∞

2

0

=2 f

0

p

Lp (R + )

|h(x)|r |f (u)|p |[e−v cosh(x−u) ± e−v cosh(x+u) ]| du dv dx

0

h

|f (u)|p |h(x)|r e−v du dv dx

r

Lr (R + ) .

(2.11)

From (2.11), (2.9), and (2.10) we obtain

F

Lp1 (R3+ )

G

Lq1 (R3+ )

H

Lr1 (R3+ )

2(q−1)/q f

Lp (R + )

g

0,β

Lq (R + )

h

Lr (R + ) .

From (2.6) and (2.12), by three-function form of Hölder inequality [14] we have

∞

0

(f ∗ g)(x)h(x) dx

(2.12)

INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS

5

1 ∞ ∞ ∞

|f (u) g(v) h(x)|[e−v cosh(x+u) + e−v cosh(x−u) ] du dv dx

2 0

0

0

1 ∞ ∞ ∞

=

F(x, u, v)G(x, u, v)H(x, u, v) du dv dx

2 0

0

0

1

F Lp (R 3 ) G Lq (R 3 ) H Lr (R 3 )

+

+

+

1

1

1

2

−1/q

2

f Lp (R+ ) g L0,β (R ) h Lr (R+ ) .

+

q

The proof is complete.

1/(r−1)

Putting h = (f ∗ g){ c }

and replacing r1 by r in (2.5) we obtain the following Young

s

type inequality

Corollary 2.3 (Young’s Type Inequality): Let 1 < p, q, r < ∞, be such that 1/p + 1/q =

0,β

1 + 1/r and let f ∈ Lp (R+ ), g ∈ Lq (R+ ), 0 < β 1, then the generalized convolutions

(1.5) are well-defined in Lr (R+ ), moreover, the following norm estimates hold

(f ∗ g){ c }

s

2−1/q f

Lr (R + )

Lp (R + )

g

0,β

Lq (R + )

.

(2.13)

3. A Watson type theorem

An important branch of the integral transform theory is to study unitary transforms. In this

section, for a special differential operator of finite order D, we give a condition of the kernel

h such that the convolution transformations (3.3) define unitary operators in L2 (R+ ), and

derive the inverse transformations.

0,β

Lemma 3.1 ([9]): Let h ∈ L2 (R+ ), 0 < β 1, and f ∈ L2 (R+ ), then the generalized

convolutions (1.5) satisfy the factorization equalities (1.6). Furthermore, the following generalized Parseval identities hold

2

π

(f ∗ h){ c } (x) =

s

∞

0

(F{ c } f )(y)K −1 [h](y)

s

cos xy

sin xy

dy,

(3.1)

where the integrals are understood in L2 (R+ ) norm, if necessary.

0,β

Theorem 3.2: Let h ∈ L2 (R+ ), 0 < β

1, then the condition

|K −1 [h](y)| =

1

1 + y2

a.e.,

(3.2)

is necessary and sufficient to ensure that the transformations Kh : f → g given by the

following formulas

(Kh f )(x) ≡ g(x) = 1 −

d2

dx2

∞

0

∞

0

(e−v cosh(x−u) ± e−v cosh(x+u) )f (u)h(v) du dv,

(3.3)

6

N. M. KHOA ET AL.

are unitary in L2 (R+ ). Moreover, the inverse transformations can be written in the conjugate

form

f (x) = 1 −

∞

d2

dx2

∞

0

0

¯

(e−v cosh(x−u) ± e−v cosh(x+u) )h(v)g(u)

du dv.

(3.4)

Proof: Sufficiency: Suppose that the function h satisfies the condition (3.2). Applying

Lemma 3.1, it is easy to see that the generalized convolution transforms (3.3) can be written

in the form

g(x) =

2

π

1−

∞

d2

dx2

(F{ c } f )(y)(Kh)(y)

s

0

cos xy

sin xy

dy.

(3.5)

It is well-known that h(y), yh(y), y2 h(y) ∈ L2 (R) if and only if (Fh)(x), (d/dx)(Fh)(x),

∈ L2 (R) (Theorem 68, page 92, [2]), where F is the Fourier transform,

and moreover,

(d2 /dx2 )(Fh)(x)

d2

(Fh)(x) = −F[y2 h(y)](x).

dx2

Therefore, if (1 + y2 )h(y) ∈ L2 (R+ ) then the following formula holds

1−

d2

dx2

(F{ c } h)(x) = F{ c } [(1 + y2 )h(y)](x).

s

s

(3.6)

From the condition (3.2), (1 + y2 )(Kh)(y)(F{ c } f )(y) ∈ L2 (R+ ), and the formula (3.6)

s

yields

g(x) = F{ c } [(1 + y2 )(F{ c } f )(y)(Kh)(y)](x) ∈ L2 (R+ ).

s

s

Applying the Fourier cosine (sine) transform to both sides of the above relation, we have

(F{ c } g)(y) = (1 + y2 )(Kh)(y)(F{ c } f )(y).

s

s

From the condition (3.2), it is easy to see that |(F{ c } g)(y)| = |(F{ c } f )(y)|, a.e., and

s

s

hence, F{ c } f L2 (R+ ) = F{ c } g L2 (R+ ) . Parseval’s equality for the Fourier cosine (sine)

s

s

transform F{ c } f L2 (R+ ) = f L2 (R+ ) shows that f L2 (R+ ) = g L2 (R+ ) , which implies

s

that the transforms (3.3) are unitary. Again from the condition (3.2) we obtain

¯

(1 + y2 )K[h](y)(F

{ c } g)(y) = (F{ c } f )(y),

s

s

a.e.

Thus, in the same manner as above it corresponds to (3.4) and the inversion formulas

of the transforms (3.3) follow.

Necessity: Suppose that the transforms (3.3) are unitary on L2 (R+ ) and the inversion

formulas are defined by (3.4). Then using Parseval’s type identities (3.1), and Parseval’s

identity for the Fourier cosine (sine) transform, we obtain

g

L2 (R + )

= (1 + y2 )(F{ c } f )(y)(Kh)(y)

s

= F{ c } f

s

L2 (R + )

= f

L2 (R + ) .

L2 (R + )

INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS

7

Therefore, the multiplication operator Mθ [·] with θ (y) = (1 + y2 )(Kh)(y) is unitary on

L2 (R+ ). This is equivalent to |θ(y)| = 1 on R+ a.e., namely

|(Kh)(y)| =

1

1 + y2

a.e.

It shows that h satisfies the condition (3.2). This completes the proof of the theorem.

4. A class of integro-differential equations

Inspite of having many useful applications (see [16]), not many integro-differential equations can be solved in closed form. No application of convolution type transforms for

solving integro-differential equations was considered in recent investigations [10–13]. In

this section, we apply the generalized convolution transforms (3.3) to solve a class of

integro-differential equations, which seems to be difficult to solve in closed form by using

other techniques. We consider the following integro-differential equation

f (x) + (Kh f )(x) = g(x),

f (0) = 0,

(4.1)

lim f (x) = 0.

x→∞

0,β

Here, h(x) = ((1/cosh τ ) ∗ h1 (τ )){ c } (x), and h1 ∈ L1 (R+ ), g ∈ L1 (R+ ) are given,

s

and f is an unknown function.

In order to give a solution of the above problem, note that, for h ∈ L1 (R+ ) such that

h (0) = 0, lim h(x) = 0, the Fourier sine and Fourier cosine transforms of h exist, and

x→∞

(Fs h )(y) =

2

π

=

2

π

∞

0

h (x) sin xy dx

∞

h(x) sin xy

0

−y

∞

h(x) cos xy dx

0

= −y(Fc h)(y),

(4.2)

and similarly,

(Fc h )(y) = y(Fs h)(y).

(4.3)

The convolution for the Fourier cosine transform [1]

1

(f ∗ g)(x) = √

Fc

2π

∞

0

f (u)[g(|x − u|) + g(x + u)] du,

x > 0,

(4.4)

satisfies the following factorization property

Fc [f ∗ g](y) = (Fc f )(y)(Fc g)(y),

Fc

∀y > 0, f , g ∈ L1 (R+ ).

(4.5)

The generalized convolution for the Fourier sine and Fourier cosine transforms [1]

1

(f ∗ g)(x) = √

1

2π

∞

0

f (u)[g(|x − u|) − g(x + u)] du,

x > 0,

(4.6)

8

N. M. KHOA ET AL.

satisfies the factorization equality [1]

∀y > 0, f , g ∈ L1 (R+ ).

Fs [f ∗ g](y) = (Fs f )(y)(Fc g)(y),

1

(4.7)

Theorem 4.1: Suppose the following condition holds

1

1 − 2Fc

cosh3 τ

∗ h1 (τ )

(y) = 0,

{c}

∀y > 0.

(4.8)

Then the problems (4.1) have the unique solutions in L1 (R+ ) that can be written as follows

f (x) = g(x) + (g ∗

{ Fc }

1

)(x),

(4.9)

here, ∈ L1 (R+ ) is defined by

(Fc )(y) =

1

∗ h1 (τ )){c} ](y)

cosh3 τ

.

1 − 2Fc [( 1 3 ∗ h1 (τ )){c} ](y)

cosh τ

2Fc [(

Proof: The Equations (4.1) can be rewritten in the form

f (x) + 1 −

d2

dx2

(f ∗ h){ c } (x) = g(x).

(4.10)

s

Applying the Fourier cosine (sine) transform to both side of (4.10), using the conditions (4.1) and the factorization equalities (1.6) and formulas (4.2), (4.3), we obtain

(F{ c } f )(y) + (y2 + 1)(Kh)(y)(F{ c } f )(y) = (F{ c } g)(y).

s

s

s

Using the formula (see relation (1.9.1) in [3])

1

(y) =

cosh t

Fc

1

π

,

2 cosh( π2y )

we have

(F{ c } f )(y) − (1 + y2 )(F{ c } f )(y)(Kh1 )(y)

s

s

1

π

= (F{ c } g)(y).

s

2 cosh( π2y )

In view of the formula (1.9.4) in [3] we have

(F{ c } f )(y) − 2(F{ c } f )(y)(Kh1 )(y)Fc

s

s

1

cosh3 t

(y) = (F{ c } g)(y),

s

or equivalently,

(F{ c } f )(y) 1 − 2Fc

s

1

cosh3 τ

∗ h1 (τ )

{c}

(y) = (F{ c } g)(y).

s

(4.11)

INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS

9

From condition (4.8) we get

(F{ c } f )(y) = 1 +

s

1

∗ h1 (τ )){c} ](y)

cosh3 τ

1

1 − 2Fc [(

∗ h1 (τ )){c} ](y)

cosh3 τ

2Fc [(

(F{ c } g)(y).

s

(4.12)

Recall that the Wiener–Levy theorem [17] states that if f is the Fourier transform of

an L1 (R) function, and ϕ is analytic in a neighbourhood of the origin that contains the

domain {f (y), ∀y ∈ R}, and ϕ(0) = 0, then ϕ(f ) is also the Fourier transform of an L1 (R)

function. For the Fourier cosine transform it means that if f is the Fourier cosine transform

of an L1 (R+ ) function, and ϕ is analytic in a neighbourhood of the origin that contains the

domain {f (y), ∀y ∈ R+ }, and ϕ(0) = 0, then ϕ(f ) is also the Fourier cosine transform of

an L1 (R+ ) function.

By the given condition (4.8) the function ϕ(z) = 2z/(1 − 2z) satisfies conditions of the

Wiener–Levy theorem, and therefore, there exists a unique function ∈ L1 (R+ ) such that

(Fc )(y) =

1

∗ h1 (τ )](y)

cosh3 τ

.

1

1 − 2Fc [

∗ h1 (τ )](y)

cosh3 τ

2Fc [

Therefore Equations (4.12) become

(F{ c } f )(y) = (1 + (Fc )(y))(F{ c } g)(y) = F{ c } (g + g ∗

s

s

that imply f (x) = g(x) + (g ∗

{ Fc }

1

s

{ Fc }

1

)(y),

)(x). The proof is complete.

Remark: We can obtain similar results for a general differential operator D =

n

n

k

2k

2k

2k

k=0 (−1) ak (d /dx ) such that the polynomial P(x) =

k=0 ak x has no real zero.

5. Systems of integral equations

(a) Consider the following system of integral equations

λ1 ∞ ∞ −v cosh(x−u)

[e

+ e−v cosh(x+u) ]g(u)h(v) du dv = p(x)

2 0

0

∞

λ2

g(x) + √

[k(u + x) + k(|u − x|)]f (u) du = q(x),

2π 0

f (x) +

(5.1)

where λ1 , λ2 are complex constants, h,k,p,q are given functions in L1 (R+ ), and f,g are

unknown functions.

10

N. M. KHOA ET AL.

Theorem 5.1: If the following condition satisfies

1 − λ1 λ2 Fc [(k ∗ h){c} ](y) = 0,

∀y > 0,

(5.2)

then there exists a unique solution in L1 (R+ ) × L1 (R+ ) of the system (5.1), which is defined

by

f (x) = p(x) − λ1 (q ∗ h){c} (x) + ( ∗ p)(x) − λ1 [(q ∗ h){c} ∗ ](x)

Fc

Fc

(5.3)

g(x) = q(x) − λ2 (k ∗ p)(x) + ( ∗ q)(x) − λ2 [(k ∗ p) ∗ ](x).

Fc

Fc

Fc

Fc

Here is given by (5.5).

Proof: Using (1.5) and (4.4), the system (5.1) can be rewritten in the form

f (x) + λ1 (g ∗ h){c} (x) = p(x),

(5.4)

λ2 (f ∗ k)(x) + g(x) = q(x).

Fc

With the help of the factorization properties (1.6), (4.5), we obtain the linear system of

algebraic equations

(Fc f )(y) + λ1 (Fc g)(y)(Kh)(y) = (Fc p)(y),

λ2 (Fc f )(y)(Fc k)(y) + (Fc g)(y) = (Fc q)(y).

The inverse of the determinant of this system has the form

1

=1+

λ1 λ2 Fc [(k ∗ h){c} ](y)

.

1 − λ1 λ2 Fc [(k ∗ h){c} ](y)

According to the Wiener–Levy theorem [17], there exists a function

that

λ1 λ2 Fc [(k ∗ h)(y){c} ]

(Fc )(y) =

, ∀y > 0.

1 − λ1 λ2 Fc [(k ∗ h)(y){c} ]

Hence 1/

∈ L1 (R+ ) such

(5.5)

= 1 + (Fc )(y). Moreover,

=

1

(Fc p)(y)

(Fc q)(y)

λ1 (Kh)(y)

= (Fc p)(y) − λ1 Fc [(q ∗ h){c} ](y),

1

and

2

=

1

λ2 (Fc k)(y)

(Fc p)(y)

= (Fc q)(y) − λ2 Fc (k ∗ p)(y).

(Fc q)(y)

Fc

Again, using the formulas (1.6), (4.5) we get

(Fc f )(y) =

1

= [1 + (Fc )(y)][(Fc p)(y) − λ1 Fc [(q ∗ h){c} ](y)]

= (Fc p)(y) − λ1 Fc [(q ∗ h){c} ](y) + Fc ( ∗ p)(y) − λ1 Fc [(q ∗ h){c} ∗ ](y).

Fc

Fc

INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS

11

It follows that

f (x) = p(x) − λ1 (q ∗ h){c} (x) + ( ∗ p)(x) − λ1 ((q ∗ h){c} ∗ )(x) ∈ L1 (R+ ).

Fc

Fc

Similarly,

2

(Fc g)(y) =

= [1 + (Fc )(y)][(Fc q)(y) − λ2 Fc (k ∗ p)(y)]

Fc

= (Fc q)(y) − λ2 Fc (k ∗ p)(y) + Fc ( ∗ q)(y) − λ2 Fc [(k ∗ p) ∗ ](y).

Fc

Fc

Fc

Fc

Consequently,

g(x) = q(x) − λ2 (k ∗ p)(x) + ( ∗ q)(x) − λ2 [(k ∗ p) ∗ ](x) ∈ L1 (R+ ).

Fc

Fc

Fc

Fc

The proof is complete.

(b) Finally, we consider the following system of integral equations

f (x) +

λ2

√

2 2π

λ1

2

∞

0

∞

0

∞

0

[e−v cosh(x−u) + e−v cosh(x+u) ]g(u)h(v) du dv = p(x),

(5.6)

k(y)[f (|x + y − 1|) + f (|x − y + 1|) − f (x + y + 1)

− f (|x − y − 1|)] dy + g(x) = q(x),

where λ1 , λ2 are complex constants, h,p,q are given functions in L1 (R+ ); f,g are unknown

functions; and k(x) = (k1 ∗ k2 )(x), here the generalized convolution · ∗ · is defined

1

1

by (4.6), and k1 , k2 are given functions in L1 (R+ ).

To solve this system of integral equations, we recall the generalized convolution with

the weight function γ (y) = sin y for the Fourier cosine and sine transforms (see [18])

∞

γ

1

(f ∗ g)(x) = √

2

2 2π

0

f (y)[g(|x + y − 1|) + g(|x − y + 1|) − g(x + y + 1)

− g(|x − y − 1|)] dy,

x > 0.

(5.7)

This generalized convolution satisfies the following factorization equality for f , g ∈

L1 (R+ )

γ

Fc (f ∗ g)(y) = sin y(Fs f )(y)(Fc g)(y),

y > 0.

(5.8)

∀y > 0,

(5.9)

2

Theorem 5.2: If the following condition satisfies

γ

1 − λ1 λ2 Fc (k1 ∗ (k2 ∗ h){c} )(y) = 0,

2

then there exists a unique solution in L1 (R+ ) of the system (5.6), which is defined by

f (x) = p(x) − λ1 (q ∗ h){c} (x) + ( ∗ p)(x) − λ1 [ ∗ (q ∗ h){c} ](x),

Fc

γ

Fc

γ

g(x) = q(x) − λ2 (k ∗ p)(x) + ( ∗ p)(x) − λ2 [(k ∗ p) ∗ )](x).

2

Here is given by (5.11).

Fc

2

Fc

(5.10)

12

N. M. KHOA ET AL.

Proof: Using formulas (1.5) and (5.7), the system (5.6) can be rewritten in the form

f (x) + λ1 (g ∗ h){c} (x) = p(x),

γ

λ2 (k ∗ f )(x) + g(x) = q(x).

2

With the help of the factorization properties (1.6), (5.8), we obtain the linear system of

algebraic equations

(Fc f )(y) + λ1 (Fc g)(y)(Kh)(y) = (Fc p)(y),

λ2 sin y(Fs k)(y)(Fc f )(y) + (Fc g)(y) = (Fc q)(y).

We have

λ1 (Kh)(y)

= 1 − λ1 λ2 sin y(Fs k)(y)(Kh)(y)

1

1

λ2 sin y(Fs k)(y)

=

= 1 − λ1 λ2 sin y(Fs k1 )(y)(Fc k2 )(y)(Kh)(y)

γ

= 1 − λ1 λ2 Fc [k1 ∗ (k2 ∗ h){c} ](y).

2

Therefore,

γ

1

=1+

λ1 λ2 Fc (k1 ∗ (k2 ∗ h){c} )(y)

2

γ

1 − λ1 λ2 Fc (k1 ∗ (k2 ∗ h){c} )(y)

.

2

According to the Wiener–Levy theorem [17], there exists a function

that

∈ L1 (R+ ) such

γ

(Fc )(y) =

λ1 λ2 Fc (k1 ∗ (k2 ∗ h){c} )(y)

2

γ

1 − λ1 λ2 Fc (k1 ∗ (k2 ∗ h){c} )(y)

∀y > 0.

,

(5.11)

2

Hence, 1/

= 1 + (Fc )(y). Moreover,

1

=

(Fc p)(y)

(Fc q)(y)

λ1 (Kh)(y)

= (Fc p)(y) − λ1 Fc (q ∗ h){c} (y),

1

and

2

=

1

λ2 sin y(Fs k)(y)

γ

(Fc p)(y)

= (Fc q)(y) − λ2 Fc (k ∗ p)(y).

(Fc q)(y)

2

Hence,

f (x) = p(x) − λ1 (q ∗ h){c} (x) + ( ∗ p)(x) − λ1 [ ∗ (q ∗ h){c} ](x),

Fc

γ

Fc

γ

g(x) = q(x) − λ2 (k ∗ p)(x) + ( ∗ p)(x) − λ2 [(k ∗ p) ∗ )](x).

2

Fc

2

Fc

It is clearly that the solution f,g defined as above are functions in L1 (R+ ). The proof is

complete.

INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS

13

Disclosure statement

No potential conflict of interest was reported by the authors.

References

[1] Sneddon IN. Fourier transforms. New York: McGray-Hill; 1951.

[2] Titchmarsh EC. Introduction to the theory of Fourier integrals. 3rd ed. New York: Chelsea

Publishing Co.; 1986.

[3] Bateman H, Erdelyi A. Table of integral transforms, Vol. I. New York: McGraw-Hill; 1954.

[4] Glaeske HJ, Prudnikov AP, Skornik KA. Operational calculus and related topics. Raton:

Chapman & Hall/CRC; 2006.

[5] Wimp J. A class of integral transforms. Proc Edinb Math Soc. 1964;14:33–40.

[6] Yakubovich SB, Luchko Yu. The hypergeometric approach to integral transforms and convolutions. Mathematics and its applications, Vol. 287. Dordrecht: Kluwer Academic Publishing;

1994.

[7] Yakubovich SB. On the Kontorovich-Lebedev transformation. J Integral Equations Appl.

2003;15(1):95–112.

[8] Yakubovich SB. Boundedness and inversion properties of certain convolution transforms. J

Korean Math Soc. 2003;40(6):999–1014.

[9] Yakubovich SB, Britvina LE. Convolutions related to the Fourier and Kontorovich-Lebedev

transforms revisited. Integral Transforms Spec Funct. 2010;21(4):259–276.

[10] Al-Musallam F, Tuan VK. Integral transforms related to a generalized convolution. Results

Math. 2000;38:197–208.

[11] Britvina LE. A class of integral transforms related to the Fourier cosine convolution. Integral

Transforms Spec Funct. 2005;16(5–6):379–389.

[12] Tuan VK. Integral transforms of Fourier cosine convolution type. J Math Anal Appl.

1999;229:519–529.

[13] Yakubovich SB. Integral transforms of the Kontorovich-Lebedev convolution type. Collect

Math. 2003;54(2):99–110.

[14] Adams RA, Fournier JJF. Sobolev spaces. 2nd ed. New York: Academic Press; 2003.

[15] Abramowitz M, Stegun IA. Handbook of mathematical functions, with formulas, graphs and

mathematical tables. National Bureau of Standards Applied Mathematics Series. Washington

(DC): United States of America Dover Publications, Inc.; 1964.

[16] Grigoriev YN, Ibragimov NH, Kovalev VF, Meleshko SV. Symmetries of integro-differential

equations with applications in mechanics and plasma physics. Lect Notes Phys. Dordrecht:

Springer; 2010.

[17] Paley REAC, Wiener N. Fourier transforms in the complex domain. New York: AMS; 1934.

[18] Thao NX, Tuan VK, Khoa NM. A generalized convolution with a weight function for the

Fourier cosine and sine transforms. Fract Calc Appl Anal. 2004;7(3):323–337.

ISSN: 1065-2469 (Print) 1476-8291 (Online) Journal homepage: http://www.tandfonline.com/loi/gitr20

On Fourier – Kontorovich–Lebedev generalized

convolution transforms

Nguyen Minh Khoa, Nguyen Thanh Hong & Vu Kim Tuan

To cite this article: Nguyen Minh Khoa, Nguyen Thanh Hong & Vu Kim Tuan (2017): On Fourier

– Kontorovich–Lebedev generalized convolution transforms, Integral Transforms and Special

Functions, DOI: 10.1080/10652469.2017.1281922

To link to this article: http://dx.doi.org/10.1080/10652469.2017.1281922

Published online: 29 Jan 2017.

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Date: 30 January 2017, At: 17:40

INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS, 2017

http://dx.doi.org/10.1080/10652469.2017.1281922

Research Article

On Fourier – Kontorovich–Lebedev generalized convolution

transforms

Nguyen Minh Khoaa , Nguyen Thanh Hongb and Vu Kim Tuanc

a Department of Mathematics, Electric Power University, Hanoi, Vietnam; b School for Gifted Students, Hanoi

National University of Education, Hanoi, Vietnam; c Department of Mathematics, University of West Georgia,

Carrollton, GA, USA

ABSTRACT

ARTICLE HISTORY

We deal with several classes of integral transformations of the form

Received 24 October 2016

Accepted 9 January 2017

f (x) → D

R2+

(e−v cosh(x−u) ± e−v cosh(x+u) )f (u)h(v) du dv,

KEYWORDS

where D is a special differential operator. In case D is the identical

operator, we prove a Young type theorem for a generalized convolution operator related to the Fourier and the Kontorovich–Lebedev

integral transforms. In case D is a certain finite order differential operator, we study the unitary property of these transforms and apply

them to solve a class of integro-differential equations. Several classes

of system of integral equations are also considered.

Convolution; Young’s

theorem; Watson’s theorem;

Fourier;

Kontorovich–Lebedev

transform;

integro-diﬀerential equation;

system of integral equations

AMS SUBJECT

CLASSIFICATION

33C10; 44A35; 45E10; 45J05;

47A30; 47B15

1. Introduction

The Fourier cosine and the Fourier sine transforms are defined as follows

(Fc f )(y) ≡ Fc [f ](y) =

2

π

(Fs f )(y) ≡ Fs [f ](y) =

2

π

∞

0

∞

0

f (x) cos xy dx =

2 d

π dy

f (x) sin xy dx =

2 d

π dy

∞

0

∞

0

f (x)

f (x)

sin xy

dx,

x

(1.1)

1 − cos xy

dx. (1.2)

x

The first integrals in (1.1) and (1.2) are well-defined for f ∈ L1 (R+ ), while the second

integrals in (1.1) and (1.2) are also well-defined for f ∈ L2 (R+ ) (see [1,2]).

The Kontorovich–Lebedev integral transform and its inverse have the form (see [3–5])

(Kf )(y) = K[f ](y) = g(y) =

f (x) = K −1 [g](x) =

2

π 2x

CONTACT Nguyen Minh Khoa

khoanm@epu.edu.vn

235 Hoang Quoc Viet, Hanoi, Vietnam

© 2017 Informa UK Limited, trading as Taylor & Francis Group

∞

0

∞

0

Kiy (x)f (x) dx,

(1.3)

y sinh(π y)Kiy (x)g(y) dy,

(1.4)

Department of Mathematics, Electric Power University,

2

N. M. KHOA ET AL.

here the integrals are understood in the mean-square norms with weights, if necessary. For

more properties of the Kontorovich–Lebedev transform and its convolution transforms,

we refer the reader to the references [6–8].

In [9], the following non-commutative convolutions for the Fourier and Kontorovich–Lebedev integral transforms were introduced

(f ∗ h){ c } (x) =

s

1

2

∞

0

∞

0

[e−v cosh(x−u) ± e−v cosh(x+u) ]f (u)h(v) du dv,

x > 0. (1.5)

The existence of the generalized convolutions (1.5) for two functions in L1 (R+ ) with a

weight and their application to solve integral equations of convolution type were studied

in [9]. Specially, in certain function spaces, the generalized convolutions (1.5) satisfy the

following factorization equalities

F{ c } [(f ∗ h){ c } ](y) = (F{ c } f )(y)(Kh)(y),

s

s

s

∀y > 0.

(1.6)

In any convolution (f ∗ h)(x) of two functions f and h, if we fix a function h and let f

vary in a certain function space, then one can study convolution transforms of the type f →

D[f ∗ h], where D is a certain differential operator. The most famous integral transform

constructed by that way is the Watson transform that is related to the Mellin convolution

and the Mellin transform (see [2])

∞

f (x) −→ g(x) =

0

k(xy)f (y) dy.

Recently, several authors have been investigated convolution transforms of this type

(see [10–13]). However, none of these references considered any application to integrodifferential equations. In this paper we are interested in transforms related to the generalized convolutions (1.5), namely, the transforms of the form given in the abstract. For the

case D is the identity operator, in Section 2 we study Young’s theorem and Young’s inequality for these generalized convolutions. In Section 3, for a certain finite order differential

operators D, we obtain a necessary and sufficient condition such that the respective transforms are unitary on L2 (R+ ), and derive the inverse transforms. In Sections 4 and 5, we

solve in closed form a class of integro-differential equations as well as several classes of

systems of two integral equations with the help of these integral transforms.

2. A Young type theorem

In this section, we will prove several norm properties of the generalized convolutions

(1.5). Throughout the paper, we are interested in the following two-parameter family of

Lebesgue’s spaces.

Definition 2.1 (see [9]): For α ∈ R, 0 < β

of all functions f (x) defined in R+ such that

∞

0

α,β

1, p ≥ 1, we denote by Lp (R+ ) the space

|f (x)|p K0 (βx)xα dx < ∞.

(2.1)

INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS

3

The norm of a function in this space is defined by

f

α,β

L p (R + )

=

∞

|f (x)|p K0 (βx)xα dx

0

1/p

.

Using the asymptotic of the Macdonald function Kν (z) (see [4])

π 1/2 −z

e [1 + O(1/z)],

2z

K0 (z) = − log z + O(1), z → 0,

Kν (z) =

z → ∞,

(2.2)

(2.3)

formula (2.1) can be expressed in an equivalent form

1

0

|f (x)|p | log x|xα dx +

∞

1

|f (x)|p xα−1/2 e−βx dx < ∞.

For the Fourier convolution (see [1])

1

(f ∗ h)(x) = √

F

2π

∞

−∞

h(x − y)f (y) dy,

(2.4)

Young’s theorem and its corollary, the so-called Young inequality are fundamental (see

[14]). We will prove now Young’s type theorem for the generalized convolutions (1.5).

Theorem 2.2 (Young’s Type Theorem): Let p,q,r be real numbers in (1; ∞) such that

0,β

1/p + 1/q + 1/r = 2 and let f ∈ Lp (R+ ), g ∈ Lq (R+ ), 0 < β 1, h ∈ Lr (R+ ), then

∞

0

(f ∗ g){ c } (x)h(x) dx

s

2−1/q f

Lp (R + )

g

0,β

Lq (R + )

h

Lr (R + ) .

Proof: Let p1 , q1 , r1 be the conjugate exponents of p,q,r, respectively, it means

1

1

1

1

1

1

+

= +

= +

= 1.

p p1

q q1

r

r1

Then it is obvious that 1/p1 + 1/q1 + 1/r1 = 1. Put

F(x, u, v) = |g(v)|q/p1 |h(x)|r/p1 |[e−v cosh(x−u) ± e−v cosh(x+u) ]|1/p1 ,

G(x, u, v) = |h(x)|r/q1 |f (u)|p/q1 |[e−v cosh(x−u) ± e−v cosh(x+u) ]|1/q1 ,

H(x, u, v) = |f (u)|p/r1 |g(v)|q/r1 |[e−v cosh(x−u) ± e−v cosh(x+u) ]|1/r1 .

(2.5)

4

N. M. KHOA ET AL.

We have

F(x, u, v)G(x, u, v)H(x, u, v) = |f (u)g(v)h(x)[e−v cosh(x−u) ± e−v cosh(x+u) ]|.

(2.6)

Using the following integral representation for the Macdonald function with a pure

imaginary index (see [4,5,15])

∞

Kiy (x) =

e−x cosh u cos yu du,

0

x > 0,

(2.7)

we easily see that

Hence, K0 (v)

F

p1

Lp1 (R3+ )

=

∞

1

2

K0 (v) =

0

(e−v cosh(x−u) + e−v cosh(x+u) ) du.

K0 (βv) for 0 < β

∞

∞

0

∞

0

0

∞

∞

0

0

1, and

|g(v)|q |h(x)|r |[e−v cosh(x−u) ± e−v cosh(x+u) ]| du dv dx

|g(v)|q K0 (βv)|h(x)|r dv dx

q

= g

(2.8)

r

Lr (R + ) ,

h

0,β

Lq (R + )

(2.9)

and similarly,

H

r1

Lr1 (R3+ )

∞

=

0

∞

∞

0

∞

0

0

∞

0

|f (u)|p |g(v)|q K0 (βv) du dv

p

Lp (R + )

= f

|f (u)|p |g(v)|q [e−v cosh(x−u) ± e−v cosh(x+u) ] du dv dx

q

g

0,β

Lq (R + )

.

(2.10)

On the other hand, in the space Lq1 (R3+ ), using formula (2.8) we have

G

q1

Lq1 (R3+ )

∞

=

∞

0

∞

0

∞

0

∞

∞

2

0

=2 f

0

p

Lp (R + )

|h(x)|r |f (u)|p |[e−v cosh(x−u) ± e−v cosh(x+u) ]| du dv dx

0

h

|f (u)|p |h(x)|r e−v du dv dx

r

Lr (R + ) .

(2.11)

From (2.11), (2.9), and (2.10) we obtain

F

Lp1 (R3+ )

G

Lq1 (R3+ )

H

Lr1 (R3+ )

2(q−1)/q f

Lp (R + )

g

0,β

Lq (R + )

h

Lr (R + ) .

From (2.6) and (2.12), by three-function form of Hölder inequality [14] we have

∞

0

(f ∗ g)(x)h(x) dx

(2.12)

INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS

5

1 ∞ ∞ ∞

|f (u) g(v) h(x)|[e−v cosh(x+u) + e−v cosh(x−u) ] du dv dx

2 0

0

0

1 ∞ ∞ ∞

=

F(x, u, v)G(x, u, v)H(x, u, v) du dv dx

2 0

0

0

1

F Lp (R 3 ) G Lq (R 3 ) H Lr (R 3 )

+

+

+

1

1

1

2

−1/q

2

f Lp (R+ ) g L0,β (R ) h Lr (R+ ) .

+

q

The proof is complete.

1/(r−1)

Putting h = (f ∗ g){ c }

and replacing r1 by r in (2.5) we obtain the following Young

s

type inequality

Corollary 2.3 (Young’s Type Inequality): Let 1 < p, q, r < ∞, be such that 1/p + 1/q =

0,β

1 + 1/r and let f ∈ Lp (R+ ), g ∈ Lq (R+ ), 0 < β 1, then the generalized convolutions

(1.5) are well-defined in Lr (R+ ), moreover, the following norm estimates hold

(f ∗ g){ c }

s

2−1/q f

Lr (R + )

Lp (R + )

g

0,β

Lq (R + )

.

(2.13)

3. A Watson type theorem

An important branch of the integral transform theory is to study unitary transforms. In this

section, for a special differential operator of finite order D, we give a condition of the kernel

h such that the convolution transformations (3.3) define unitary operators in L2 (R+ ), and

derive the inverse transformations.

0,β

Lemma 3.1 ([9]): Let h ∈ L2 (R+ ), 0 < β 1, and f ∈ L2 (R+ ), then the generalized

convolutions (1.5) satisfy the factorization equalities (1.6). Furthermore, the following generalized Parseval identities hold

2

π

(f ∗ h){ c } (x) =

s

∞

0

(F{ c } f )(y)K −1 [h](y)

s

cos xy

sin xy

dy,

(3.1)

where the integrals are understood in L2 (R+ ) norm, if necessary.

0,β

Theorem 3.2: Let h ∈ L2 (R+ ), 0 < β

1, then the condition

|K −1 [h](y)| =

1

1 + y2

a.e.,

(3.2)

is necessary and sufficient to ensure that the transformations Kh : f → g given by the

following formulas

(Kh f )(x) ≡ g(x) = 1 −

d2

dx2

∞

0

∞

0

(e−v cosh(x−u) ± e−v cosh(x+u) )f (u)h(v) du dv,

(3.3)

6

N. M. KHOA ET AL.

are unitary in L2 (R+ ). Moreover, the inverse transformations can be written in the conjugate

form

f (x) = 1 −

∞

d2

dx2

∞

0

0

¯

(e−v cosh(x−u) ± e−v cosh(x+u) )h(v)g(u)

du dv.

(3.4)

Proof: Sufficiency: Suppose that the function h satisfies the condition (3.2). Applying

Lemma 3.1, it is easy to see that the generalized convolution transforms (3.3) can be written

in the form

g(x) =

2

π

1−

∞

d2

dx2

(F{ c } f )(y)(Kh)(y)

s

0

cos xy

sin xy

dy.

(3.5)

It is well-known that h(y), yh(y), y2 h(y) ∈ L2 (R) if and only if (Fh)(x), (d/dx)(Fh)(x),

∈ L2 (R) (Theorem 68, page 92, [2]), where F is the Fourier transform,

and moreover,

(d2 /dx2 )(Fh)(x)

d2

(Fh)(x) = −F[y2 h(y)](x).

dx2

Therefore, if (1 + y2 )h(y) ∈ L2 (R+ ) then the following formula holds

1−

d2

dx2

(F{ c } h)(x) = F{ c } [(1 + y2 )h(y)](x).

s

s

(3.6)

From the condition (3.2), (1 + y2 )(Kh)(y)(F{ c } f )(y) ∈ L2 (R+ ), and the formula (3.6)

s

yields

g(x) = F{ c } [(1 + y2 )(F{ c } f )(y)(Kh)(y)](x) ∈ L2 (R+ ).

s

s

Applying the Fourier cosine (sine) transform to both sides of the above relation, we have

(F{ c } g)(y) = (1 + y2 )(Kh)(y)(F{ c } f )(y).

s

s

From the condition (3.2), it is easy to see that |(F{ c } g)(y)| = |(F{ c } f )(y)|, a.e., and

s

s

hence, F{ c } f L2 (R+ ) = F{ c } g L2 (R+ ) . Parseval’s equality for the Fourier cosine (sine)

s

s

transform F{ c } f L2 (R+ ) = f L2 (R+ ) shows that f L2 (R+ ) = g L2 (R+ ) , which implies

s

that the transforms (3.3) are unitary. Again from the condition (3.2) we obtain

¯

(1 + y2 )K[h](y)(F

{ c } g)(y) = (F{ c } f )(y),

s

s

a.e.

Thus, in the same manner as above it corresponds to (3.4) and the inversion formulas

of the transforms (3.3) follow.

Necessity: Suppose that the transforms (3.3) are unitary on L2 (R+ ) and the inversion

formulas are defined by (3.4). Then using Parseval’s type identities (3.1), and Parseval’s

identity for the Fourier cosine (sine) transform, we obtain

g

L2 (R + )

= (1 + y2 )(F{ c } f )(y)(Kh)(y)

s

= F{ c } f

s

L2 (R + )

= f

L2 (R + ) .

L2 (R + )

INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS

7

Therefore, the multiplication operator Mθ [·] with θ (y) = (1 + y2 )(Kh)(y) is unitary on

L2 (R+ ). This is equivalent to |θ(y)| = 1 on R+ a.e., namely

|(Kh)(y)| =

1

1 + y2

a.e.

It shows that h satisfies the condition (3.2). This completes the proof of the theorem.

4. A class of integro-differential equations

Inspite of having many useful applications (see [16]), not many integro-differential equations can be solved in closed form. No application of convolution type transforms for

solving integro-differential equations was considered in recent investigations [10–13]. In

this section, we apply the generalized convolution transforms (3.3) to solve a class of

integro-differential equations, which seems to be difficult to solve in closed form by using

other techniques. We consider the following integro-differential equation

f (x) + (Kh f )(x) = g(x),

f (0) = 0,

(4.1)

lim f (x) = 0.

x→∞

0,β

Here, h(x) = ((1/cosh τ ) ∗ h1 (τ )){ c } (x), and h1 ∈ L1 (R+ ), g ∈ L1 (R+ ) are given,

s

and f is an unknown function.

In order to give a solution of the above problem, note that, for h ∈ L1 (R+ ) such that

h (0) = 0, lim h(x) = 0, the Fourier sine and Fourier cosine transforms of h exist, and

x→∞

(Fs h )(y) =

2

π

=

2

π

∞

0

h (x) sin xy dx

∞

h(x) sin xy

0

−y

∞

h(x) cos xy dx

0

= −y(Fc h)(y),

(4.2)

and similarly,

(Fc h )(y) = y(Fs h)(y).

(4.3)

The convolution for the Fourier cosine transform [1]

1

(f ∗ g)(x) = √

Fc

2π

∞

0

f (u)[g(|x − u|) + g(x + u)] du,

x > 0,

(4.4)

satisfies the following factorization property

Fc [f ∗ g](y) = (Fc f )(y)(Fc g)(y),

Fc

∀y > 0, f , g ∈ L1 (R+ ).

(4.5)

The generalized convolution for the Fourier sine and Fourier cosine transforms [1]

1

(f ∗ g)(x) = √

1

2π

∞

0

f (u)[g(|x − u|) − g(x + u)] du,

x > 0,

(4.6)

8

N. M. KHOA ET AL.

satisfies the factorization equality [1]

∀y > 0, f , g ∈ L1 (R+ ).

Fs [f ∗ g](y) = (Fs f )(y)(Fc g)(y),

1

(4.7)

Theorem 4.1: Suppose the following condition holds

1

1 − 2Fc

cosh3 τ

∗ h1 (τ )

(y) = 0,

{c}

∀y > 0.

(4.8)

Then the problems (4.1) have the unique solutions in L1 (R+ ) that can be written as follows

f (x) = g(x) + (g ∗

{ Fc }

1

)(x),

(4.9)

here, ∈ L1 (R+ ) is defined by

(Fc )(y) =

1

∗ h1 (τ )){c} ](y)

cosh3 τ

.

1 − 2Fc [( 1 3 ∗ h1 (τ )){c} ](y)

cosh τ

2Fc [(

Proof: The Equations (4.1) can be rewritten in the form

f (x) + 1 −

d2

dx2

(f ∗ h){ c } (x) = g(x).

(4.10)

s

Applying the Fourier cosine (sine) transform to both side of (4.10), using the conditions (4.1) and the factorization equalities (1.6) and formulas (4.2), (4.3), we obtain

(F{ c } f )(y) + (y2 + 1)(Kh)(y)(F{ c } f )(y) = (F{ c } g)(y).

s

s

s

Using the formula (see relation (1.9.1) in [3])

1

(y) =

cosh t

Fc

1

π

,

2 cosh( π2y )

we have

(F{ c } f )(y) − (1 + y2 )(F{ c } f )(y)(Kh1 )(y)

s

s

1

π

= (F{ c } g)(y).

s

2 cosh( π2y )

In view of the formula (1.9.4) in [3] we have

(F{ c } f )(y) − 2(F{ c } f )(y)(Kh1 )(y)Fc

s

s

1

cosh3 t

(y) = (F{ c } g)(y),

s

or equivalently,

(F{ c } f )(y) 1 − 2Fc

s

1

cosh3 τ

∗ h1 (τ )

{c}

(y) = (F{ c } g)(y).

s

(4.11)

INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS

9

From condition (4.8) we get

(F{ c } f )(y) = 1 +

s

1

∗ h1 (τ )){c} ](y)

cosh3 τ

1

1 − 2Fc [(

∗ h1 (τ )){c} ](y)

cosh3 τ

2Fc [(

(F{ c } g)(y).

s

(4.12)

Recall that the Wiener–Levy theorem [17] states that if f is the Fourier transform of

an L1 (R) function, and ϕ is analytic in a neighbourhood of the origin that contains the

domain {f (y), ∀y ∈ R}, and ϕ(0) = 0, then ϕ(f ) is also the Fourier transform of an L1 (R)

function. For the Fourier cosine transform it means that if f is the Fourier cosine transform

of an L1 (R+ ) function, and ϕ is analytic in a neighbourhood of the origin that contains the

domain {f (y), ∀y ∈ R+ }, and ϕ(0) = 0, then ϕ(f ) is also the Fourier cosine transform of

an L1 (R+ ) function.

By the given condition (4.8) the function ϕ(z) = 2z/(1 − 2z) satisfies conditions of the

Wiener–Levy theorem, and therefore, there exists a unique function ∈ L1 (R+ ) such that

(Fc )(y) =

1

∗ h1 (τ )](y)

cosh3 τ

.

1

1 − 2Fc [

∗ h1 (τ )](y)

cosh3 τ

2Fc [

Therefore Equations (4.12) become

(F{ c } f )(y) = (1 + (Fc )(y))(F{ c } g)(y) = F{ c } (g + g ∗

s

s

that imply f (x) = g(x) + (g ∗

{ Fc }

1

s

{ Fc }

1

)(y),

)(x). The proof is complete.

Remark: We can obtain similar results for a general differential operator D =

n

n

k

2k

2k

2k

k=0 (−1) ak (d /dx ) such that the polynomial P(x) =

k=0 ak x has no real zero.

5. Systems of integral equations

(a) Consider the following system of integral equations

λ1 ∞ ∞ −v cosh(x−u)

[e

+ e−v cosh(x+u) ]g(u)h(v) du dv = p(x)

2 0

0

∞

λ2

g(x) + √

[k(u + x) + k(|u − x|)]f (u) du = q(x),

2π 0

f (x) +

(5.1)

where λ1 , λ2 are complex constants, h,k,p,q are given functions in L1 (R+ ), and f,g are

unknown functions.

10

N. M. KHOA ET AL.

Theorem 5.1: If the following condition satisfies

1 − λ1 λ2 Fc [(k ∗ h){c} ](y) = 0,

∀y > 0,

(5.2)

then there exists a unique solution in L1 (R+ ) × L1 (R+ ) of the system (5.1), which is defined

by

f (x) = p(x) − λ1 (q ∗ h){c} (x) + ( ∗ p)(x) − λ1 [(q ∗ h){c} ∗ ](x)

Fc

Fc

(5.3)

g(x) = q(x) − λ2 (k ∗ p)(x) + ( ∗ q)(x) − λ2 [(k ∗ p) ∗ ](x).

Fc

Fc

Fc

Fc

Here is given by (5.5).

Proof: Using (1.5) and (4.4), the system (5.1) can be rewritten in the form

f (x) + λ1 (g ∗ h){c} (x) = p(x),

(5.4)

λ2 (f ∗ k)(x) + g(x) = q(x).

Fc

With the help of the factorization properties (1.6), (4.5), we obtain the linear system of

algebraic equations

(Fc f )(y) + λ1 (Fc g)(y)(Kh)(y) = (Fc p)(y),

λ2 (Fc f )(y)(Fc k)(y) + (Fc g)(y) = (Fc q)(y).

The inverse of the determinant of this system has the form

1

=1+

λ1 λ2 Fc [(k ∗ h){c} ](y)

.

1 − λ1 λ2 Fc [(k ∗ h){c} ](y)

According to the Wiener–Levy theorem [17], there exists a function

that

λ1 λ2 Fc [(k ∗ h)(y){c} ]

(Fc )(y) =

, ∀y > 0.

1 − λ1 λ2 Fc [(k ∗ h)(y){c} ]

Hence 1/

∈ L1 (R+ ) such

(5.5)

= 1 + (Fc )(y). Moreover,

=

1

(Fc p)(y)

(Fc q)(y)

λ1 (Kh)(y)

= (Fc p)(y) − λ1 Fc [(q ∗ h){c} ](y),

1

and

2

=

1

λ2 (Fc k)(y)

(Fc p)(y)

= (Fc q)(y) − λ2 Fc (k ∗ p)(y).

(Fc q)(y)

Fc

Again, using the formulas (1.6), (4.5) we get

(Fc f )(y) =

1

= [1 + (Fc )(y)][(Fc p)(y) − λ1 Fc [(q ∗ h){c} ](y)]

= (Fc p)(y) − λ1 Fc [(q ∗ h){c} ](y) + Fc ( ∗ p)(y) − λ1 Fc [(q ∗ h){c} ∗ ](y).

Fc

Fc

INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS

11

It follows that

f (x) = p(x) − λ1 (q ∗ h){c} (x) + ( ∗ p)(x) − λ1 ((q ∗ h){c} ∗ )(x) ∈ L1 (R+ ).

Fc

Fc

Similarly,

2

(Fc g)(y) =

= [1 + (Fc )(y)][(Fc q)(y) − λ2 Fc (k ∗ p)(y)]

Fc

= (Fc q)(y) − λ2 Fc (k ∗ p)(y) + Fc ( ∗ q)(y) − λ2 Fc [(k ∗ p) ∗ ](y).

Fc

Fc

Fc

Fc

Consequently,

g(x) = q(x) − λ2 (k ∗ p)(x) + ( ∗ q)(x) − λ2 [(k ∗ p) ∗ ](x) ∈ L1 (R+ ).

Fc

Fc

Fc

Fc

The proof is complete.

(b) Finally, we consider the following system of integral equations

f (x) +

λ2

√

2 2π

λ1

2

∞

0

∞

0

∞

0

[e−v cosh(x−u) + e−v cosh(x+u) ]g(u)h(v) du dv = p(x),

(5.6)

k(y)[f (|x + y − 1|) + f (|x − y + 1|) − f (x + y + 1)

− f (|x − y − 1|)] dy + g(x) = q(x),

where λ1 , λ2 are complex constants, h,p,q are given functions in L1 (R+ ); f,g are unknown

functions; and k(x) = (k1 ∗ k2 )(x), here the generalized convolution · ∗ · is defined

1

1

by (4.6), and k1 , k2 are given functions in L1 (R+ ).

To solve this system of integral equations, we recall the generalized convolution with

the weight function γ (y) = sin y for the Fourier cosine and sine transforms (see [18])

∞

γ

1

(f ∗ g)(x) = √

2

2 2π

0

f (y)[g(|x + y − 1|) + g(|x − y + 1|) − g(x + y + 1)

− g(|x − y − 1|)] dy,

x > 0.

(5.7)

This generalized convolution satisfies the following factorization equality for f , g ∈

L1 (R+ )

γ

Fc (f ∗ g)(y) = sin y(Fs f )(y)(Fc g)(y),

y > 0.

(5.8)

∀y > 0,

(5.9)

2

Theorem 5.2: If the following condition satisfies

γ

1 − λ1 λ2 Fc (k1 ∗ (k2 ∗ h){c} )(y) = 0,

2

then there exists a unique solution in L1 (R+ ) of the system (5.6), which is defined by

f (x) = p(x) − λ1 (q ∗ h){c} (x) + ( ∗ p)(x) − λ1 [ ∗ (q ∗ h){c} ](x),

Fc

γ

Fc

γ

g(x) = q(x) − λ2 (k ∗ p)(x) + ( ∗ p)(x) − λ2 [(k ∗ p) ∗ )](x).

2

Here is given by (5.11).

Fc

2

Fc

(5.10)

12

N. M. KHOA ET AL.

Proof: Using formulas (1.5) and (5.7), the system (5.6) can be rewritten in the form

f (x) + λ1 (g ∗ h){c} (x) = p(x),

γ

λ2 (k ∗ f )(x) + g(x) = q(x).

2

With the help of the factorization properties (1.6), (5.8), we obtain the linear system of

algebraic equations

(Fc f )(y) + λ1 (Fc g)(y)(Kh)(y) = (Fc p)(y),

λ2 sin y(Fs k)(y)(Fc f )(y) + (Fc g)(y) = (Fc q)(y).

We have

λ1 (Kh)(y)

= 1 − λ1 λ2 sin y(Fs k)(y)(Kh)(y)

1

1

λ2 sin y(Fs k)(y)

=

= 1 − λ1 λ2 sin y(Fs k1 )(y)(Fc k2 )(y)(Kh)(y)

γ

= 1 − λ1 λ2 Fc [k1 ∗ (k2 ∗ h){c} ](y).

2

Therefore,

γ

1

=1+

λ1 λ2 Fc (k1 ∗ (k2 ∗ h){c} )(y)

2

γ

1 − λ1 λ2 Fc (k1 ∗ (k2 ∗ h){c} )(y)

.

2

According to the Wiener–Levy theorem [17], there exists a function

that

∈ L1 (R+ ) such

γ

(Fc )(y) =

λ1 λ2 Fc (k1 ∗ (k2 ∗ h){c} )(y)

2

γ

1 − λ1 λ2 Fc (k1 ∗ (k2 ∗ h){c} )(y)

∀y > 0.

,

(5.11)

2

Hence, 1/

= 1 + (Fc )(y). Moreover,

1

=

(Fc p)(y)

(Fc q)(y)

λ1 (Kh)(y)

= (Fc p)(y) − λ1 Fc (q ∗ h){c} (y),

1

and

2

=

1

λ2 sin y(Fs k)(y)

γ

(Fc p)(y)

= (Fc q)(y) − λ2 Fc (k ∗ p)(y).

(Fc q)(y)

2

Hence,

f (x) = p(x) − λ1 (q ∗ h){c} (x) + ( ∗ p)(x) − λ1 [ ∗ (q ∗ h){c} ](x),

Fc

γ

Fc

γ

g(x) = q(x) − λ2 (k ∗ p)(x) + ( ∗ p)(x) − λ2 [(k ∗ p) ∗ )](x).

2

Fc

2

Fc

It is clearly that the solution f,g defined as above are functions in L1 (R+ ). The proof is

complete.

INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS

13

Disclosure statement

No potential conflict of interest was reported by the authors.

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