Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2010, Article ID 709607, 16 pages
doi:10.1155/2010/709607
Research Article
On the Polyconvolution with the Weight
Function for the Fourier Cosine, Fourier Sine, and
the Kontorovich-Lebedev Integral Transforms
Nguyen Xuan Thao
Faculty of Applied Mathematics and Informatics, Hanoi University of Technology,
No. 1, Dai Co Viet, Hanoi, Vietnam
Correspondence should be addressed to Nguyen Xuan Thao, thaonxbmai@yahoo.com
Received 26 December 2009; Revised 5 April 2010; Accepted 17 May 2010
Academic Editor: Slimane Adjerid
Copyright q 2010 Nguyen Xuan Thao. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The polyconvolution with the weight function γ of three functions f, g, and h for the integral
transforms Fourier sine Fs , Fourier cosine Fc , and Kontorovich-Lebedev Kiy , which is
γ
denoted by ∗f, g, h(x), has been constructed. This polyconvolution satisfies the following
γ
factorization property Fc∗f, g, hy sin yFs fy · Fc gy · Kiy hy, for all y > 0. The
relation of this polyconvolution to the Fourier convolution and the Fourier cosine convolution has
been obtained. Also, the relations between the polyconvolution product and others convolution
product have been established. In application, we consider a class of integral equations with
Toeplitz plus Hankel kernel whose solution in closed form can be obtained with the help of the
new polyconvolution. An application on solving systems of integral equations is also obtained.
1. Introduction
The convolution of two functions f and g for the Fourier transform is well known 1
∞
1
f ∗ g x √
f x − y g y dy,
F
2π −∞
x ∈ R.
1.1
This convolution has the factorization equality as below
F f ∗ g y Ff y Fg y ,
F
∀y ∈ R,
1.2
2
Mathematical Problems in Engineering
where F denotes the Fourier transform
Ff
1
y √
2π
∞
−∞
fxe−ixy dx.
1.3
The convolution of f and g for the Kontorovich-Lebedev integral transform has been studied
in 2
∞
1
1 xu xv uv fugvdu dv,
f ∗ g x exp −
K−L
2x 0
2 v
u
x
x > 0,
1.4
for which the following factorization identity holds:
Kiy f ∗ g Kiy f · Kiy g ,
K−L
∀y > 0.
1.5
Here Kiy is the Kontorovich-Lebedev transform 3
Kix f ∞
Kix tftdt,
1.6
0
and Kix t is the Macdonald function 4.
The convolution of two functions f and g for the Fourier cosine is of the form 1
∞
1
f ∗ g x √
f y g x − y g x y dy,
1
2π 0
x > 0,
1.7
which satisfied the following factorization equality:
Fc f ∗ g y Fc f y F c g y ,
1
∀y > 0.
1.8
y > 0.
1.9
Here the Fourier cosine transform is of the form
Fc f y 2
π
∞
cos yx · fxdx,
0
The convolution with a weight function γx sin x of two functions f and g for the Fourier
sine transform has been introduced in 5, 6
1
f ∗ g x √
2 2π
γ
∞
0
f y sign x y − 1 g x y − 1 sign x − y 1 g x − y 1
−g x y 1 − sign x − y − 1 g x − y − 1 dy,
x > 0,
1.10
Mathematical Problems in Engineering
3
and the following factorization identity holds:
γ
Fs f ∗ g y sin y Fs f y Fs g y ,
∀y > 0.
1.11
Here the Fourier sine is of the form
Fs f y 2
π
∞
sin yx · fxdx,
1.12
y > 0.
0
The generalized convolution of two functions f and g for the Fourier sine and Fourier cosine
transforms has been studied in 1
1
f ∗ g x √
2
2π
∞
fu g|x − u| − gx u du,
x > 0,
1.13
0
and proved the following factorization identity 1:
Fs f ∗ g y F s f y · F c g y ,
2
∀y > 0.
1.14
The generalized convolution of two functions f and g for the Fourier cosine and the Fourier
sine transforms is defined by 7
∞
1
fu signu − xg|u − x| gu x du,
f ∗ g x √
3
2π 0
x > 0.
1.15
For this generalized convolution, the following factorization equality holds:
Fc f ∗ g
3
y Fs f y Fs g y ,
∀y > 0.
1.16
The generalized convolution with the weight function γx sin x for the Fourier cosine and
the Fourier sine transforms of f and g has been introduced in 8
∞
γ
1
fu g|x u − 1| g|x − u 1|
f ∗ g x √
1
2 2π 0
−gx u 1 − g|x − u − 1| du,
1.17
x > 0.
It satisfies the factorization property
γ
Fc f ∗ g y sin y Fs f y Fc g y ,
1
∀y > 0.
1.18
4
Mathematical Problems in Engineering
The generalized convolution with the weight function γx sin x of f and g for the Fourier
sine and Fourier cosine has been studied in 9
1
f ∗ g x √
2
2 2π
γ
∞
fu g|x u − 1| g|x − u − 1|
0
−gx u 1 − g|x − u 1| du,
1.19
x > 0,
and satisfies the following factorization identity:
γ
Fs f ∗ g
2
y sin y Fc f y Fc g y ,
∀y > 0.
1.20
Recently, the following generalized convolutions for Fourier cosine, Kontorovich-Lebedev
and Fourier sine, Kontorovich-Lebedev are studied in 10 f. 21
1
f ∗ g {c} x s
2πx
R2
fugv e−x coshu−v ± e−x coshuv du dv,
1.21
x > 0.
The respective factorization equalities are 10
F{c} f ∗ g {c} y F{c} f y Kiy g ,
s
s
s
0,β
x > 0, f ∈ L1 R ∩ Lp R , g ∈ Lp , p > 1,
1.22
where
0,β
Lp ∞
ftp K0 βt dt < ∞, 0 < β 1 .
f:
1.23
0
In 1997, Kakichev introduced a constructive method for defining a polyconvolution with a weight function γ of functions f1 , f2 , . . . , fn for the integral transforms
γ
K, K1 , K2 , . . . , Kn , which are denoted by ∗f1 , f2 , . . . , fn x, such that the following factorization property holds 11:
n
γ
K ∗ f1 , f2 , . . . , fn
y γ y
Ki fi y ,
n 3.
1.24
i1
Polyconvolutions for the Hilbert, Stieltjes, Fourier cosine, and Fourier sine integral transforms
have been studied in 12.
The polyconvolution of f, g, and h for the Fourier cosine and the Fourier sine
transforms has the form 13
1
∗ f, g, h x 2π
∞
fugvh|x u − v| hx − u v |
1.25
0
−h|x − u − v| − hx u vdu dv,
x > 0,
Mathematical Problems in Engineering
5
which satisfies the following factorization property:
Fc ∗ f, g, h y Fs f y · Fs g y · Fc h y ,
∀y > 0.
1.26
In recent years, many sciences were interested in the theory of convolution for the integral
transforms and gave several interesting application see 3, 14–21, specially, the integral
equations with the Toeplitz plus Hankel kernel 22–24
fx ∞
k1 x y k2 x − y f y dy gx,
x > 0,
1.27
0
where k1 , k2 , and g are known functions, and f is an unknown function. Many partial cases of
this equation can be solved in closed form with the help of the convolutions and generalized
convolutions. In this paper, we construct and investigate the polyconvolution for the Fourier
sine, Fourier cosine, and the Kontorovich-Lebedev transforms. Several properties of this new
polyconvolution and its application on solving integral equation with Toeplitz plus Hankel
equation and systems of integral equations are obtained.
2. Polyconvolution
Definition 2.1. The polyconvolution with the weight function γ sin x of functions f, g, and
h for the Fourier cosine, Fourier sine, and the Kontorovich-Lebedev integral transforms is
defined as follows:
γ
∗ f, g, h x ∞
θx, u, v, wfugvhwdu dv dw,
2.1
0
where
1 −w coshxvu−1
θx, u, v, w √
e
e−w coshxv−u1 e−w coshx−vu−1 e−w coshx−v−u1
4 2π
−e−w coshxvu1 − e−w coshxv−u−1 − e−w coshx−vu1 − e−w coshx−v−u−1 .
2.2
√
Theorem 2.2. Let f and g be functions in L1 R , and let h be a function in L1 1/ w, R ; then
the polyconvolution 2.1 belongs to L1 R and satisfies the following factorization equality:
γ
Fc ∗ f, g, h
y sin y Fs f y · Fc g y · Kiy h ,
∀y > 0.
2.3
6
Mathematical Problems in Engineering
√
Proof. Since |e−w coshxuv−1 − e−w coshxuv−1 | 1/ w for sufficient large w > 0, we have
∞
γ ∗ f, g, h x √1
fugv|hw||θx, u, v, w|du dv dw
4 2π
0
∞
∞
∞
1
1
√
fu du ·
gv dv ·
√ |hw|dw < ∞.
w
2π 0
0
0
2.4
On the other hand, note that coshx u v − 1 x u v − 12 /2; we have
2
e−w coshxuv−1 e−wxuv−1 /2 ,
2.5
∀w > 0.
Using formula 3.321.3, page 321, in 4, we have
∞
e
−w coshxuv−1
2
w
dx 0
∞
2
2
w
e
−
√
2
w/2xuv−1
d
0
∞
0
w
x u v − 1
2
e−s ds 2
2.6
2π
.
w
It shows that
∞
0
e−w coshxuv−1 fugv|hw|du dv dw dx
∞ 2π
|hw|fugvdu dv dw
w
0
∞
∞
∞
√
1
fudu ·
gvdv < ∞.
2π
√ |hw|dw ·
w
0
0
0
2.7
By the same way, we obtain similar estimations for the 7 other terms. Therefore, from
formulas 2.1, 2.2, and 2.7, we have
∞
γ ∗ f, g, h xdx < ∞.
2.8
0
It shows that the polyconvolution 2.1 belongs to L1 R . We now prove the factorization
equality 2.3. Indeed, we have
sin y Fs f y Fc g y Kiy h
∞
2
sin y sin yu cos yv Kiy wfugvhwdu dv dw.
π
0
2.9
Mathematical Problems in Engineering
7
Using formula 2, page 130 in 4, we get
sin y Fs f y Fc g y Kiy h
∞
2
sin y sin yu cos yv cos yα e−w cosh α fugvhwdu dv dw dα
π
0
∞
1
e−w cosh α cos yu − 1 v α cos yu − 1 − v − α cos yu − 1 v − α
4π
0
cos yu − 1 − v α − cos yu 1 v α − cos yu 1 − v − α
− cos yu 1 v − α − cos yu 1 − v α
× fugvhwdu dv dw dα.
2.10
Interchanging variables, we have
∞
e−w cosh α cos yu − 1 v α − cos yu 1 v α dα
0
∞
cos yx e
−w coshx−u1−v
−e
−w coshx−u−1−v
2.11
dx.
0
Similarly,
∞
e−w cosh α cos yu − 1 − v α − cos yu 1 − v α dα
0
∞
cos yx e−w coshx−u1v − e−w coshx−u−1v dx;
0
e−w cosh α cos yu − 1 − v − α − cos yu 1 − v − α dα
0
∞
∞
∞
cos yx e
−w coshxu−1−v
−e
−w coshxu1−v
2.12
dx;
0
e−w cosh α cos yu − 1 v − α − cos yu 1 v − α dα
0
∞
cos yx e−w coshxu−1v − e−w coshxu1v dx.
0
From fomulae 2.10– 2.8, we have
γ
sin y Fs f y Fc g y Kiy h Fc ∗ f, g, h
y .
The proof is complete.
2.13
8
Mathematical Problems in Engineering
Definition 2.3. Let f be a function in L1 R and let h be a function in L1 β, R ; their norms
are defined as follows:
f L1 R ∞
fxdx,
0
hL1 β,R ∞
βv|hv|dv.
2.14
0
√
here βv 2/ v.
Theorem 2.4. Let f and g be functions in L1 R , and let h be function in L1 β, R ; then the
following estimation holds:
γ ∗ f, g, h L1 R f L1 R g L1 R hL1 β,R .
2.15
Proof. From formulas 2.1, 2.2, and 2.7, we have
∞
∞
∞
γ 1
∗ f, g, h xdx 2
gvdv.
fu du ·
√ |hw|dw ·
w
0
0
0
2.16
Therefore, by Definition 2.3,
γ ∗ f, g, h L1 R f L1 R g L1 R hL1 β,R .
2.17
√
Proposition 2.5. Let f, g ∈ L1 R , and let h ∈ L1 1/ w, R ; then the following identity holds:
γ
1
∗ f, g, h 2
π
2
∞
−w cosh t
hw
g∗e
∗ f|t| sign t x 1
1
0
−
F
g ∗ e−w cosh t
1
∗ f|t| sign t
F
x − 1 dw.
2.18
Proof. From the definition 2.1 of the polyconvolution and the convolution 1.7, we have
γ ∗ f, g, h x
1 ∞
fuhw g ∗ e−w cosh t x − u 1 g ∗ e−w cosh t x u − 1
1
1
4 0
− g ∗ e−w cosh t x u 1 − g ∗ e−w cosh t x − u − 1 du dw.
1
1
2.19
Mathematical Problems in Engineering
9
From 2.19 and calculation, we obtain
∗ f, g, h x 1
∞
π
2
hw
g ∗ e−w cosh t ∗ f|t| sign t x 1
1
0
−
g∗e
F
−w cosh t
1
∗ f|t| sign t x − 1 dw.
2.20
F
The proof is complete.
Theorem 2.6. Let f, g, h be functions in L1 R , γx sin x, and let l and k be functions in
√
L1/ w, R; then the following properties holds:
γ
γ
γ
γ
a ∗f, ∗g, h, k, l ∗g, ∗f, h, k, l;
γ
γ
b ∗f ∗ g, h, k ∗f, g ∗ h, k;
2
γ
γ
γ
γ
γ
γ
1
γ
c ∗f ∗ g, h, k ∗f, g ∗ h, k;
1
γ
d ∗f ∗ g, h, k ∗f ∗ h, g, k;
2
2
γ
γ
e ∗f, g ∗ h, k ∗g, f ∗ h, k.
3
3
Proof. First, we prove the assertion c. From Theorem 2.2 and the convolutions 1.17, 1.10,
we have
γ
γ
γ
y sin yFs f ∗ g y · Fc h y Kiy h
Fc ∗ f ∗ g, h, k
sin y sin y Fs f y Fs g y Fc h y Kiy h
γ
sin y · Fs f y · Fc g ∗ h y · Kiy h
2.21
1
γ
γ
Fc ∗ f, g ∗ h, k
.
1
Therefore, the part c holds. Other parts can be proved in a similar way.
3. Applications in Solving Integral Equations and
Systems of Integral Equations
Consider the integral equation
fx ∞
θx, u, v, wgufvhwdu dv dw
0
∞
0
θ1 x, ufudu ∞
0
3.1
θ2 x, ufudu ϕx,
x > 0,
10
Mathematical Problems in Engineering
where g, h, k, l, and ϕ are known functions, f is an unknown function, θx, u, v, w is given
by the formula 2.2, and
1 θ1 x, u √
kx u 1 − k|x u − 1| signx u − 1
2 2π
k|x − u 1| signx − u 1 − k|x − u − 1| signx − u − 1 ,
3.2
1
θ2 x, u √ lx u l|x − u|.
2π
√
Theorem 3.1. Suppose that g, l, ϕ, k1 , k2 ∈ L1 R , h ∈ L1 1/ v, R , k k1 ∗ k2 such that
2
1 sin y Fs g y Kiy h Fs k1 y Fc k2 y Fc l y / 0,
3.3
then 3.1 has a unique solution in L1 R whose closed form is
fx ϕx − ϕ ∗ ξ x.
3.4
1
Here ξ ∈ L1 R is defined uniquely by
Fc ξ y sin y Fs g y Kiy h sin yFs k1 y Fc k2 y Fc l y
.
1 sin y Fs g y Kiy h sin yFs k1 y Fc k2 y Fc l y
3.5
Proof. We obtain the following lemmas.
Lemma 3.2. For f, k ∈ L1 R , then the following operator also belongs to L1 R ∞
fuθ1 x, udu.
3.6
0
Moreover, the following factorization equality holds:
∞
Fc
fuθ1 x, udu
y sin y · Fs k y Fc f y ,
∀y > 0.
3.7
0
γ
√
Lemma 3.3. Let g ∈ L1 R , h ∈ L1 1/ v, R ; then the generalized convolution g ∗ hx belongs
3
to L1 R and the respectively factorization equality is
Fc
g ∗ h y sin y Fs g y Kiy h ,
γ
3
∀y > 0,
3.8
Mathematical Problems in Engineering
11
where
1 ∞ −v coshxu−1
e
g ∗ h x e−v coshx−u1 − e−v coshxu1 − e−v coshx−u−1
3
4 0
γ
× guhvdu dv,
3.9
x > 0.
We now prove Theorem 3.1 with the help of convolution 1.7, Lemmas 1, and 2. We
have
Fc f y sin y Fs g y · Fc f y · Kiy h
Fs k y · Fc f y sin y Fc l y · Fc f y Fc ϕ y .
3.10
Therefore, by the given condition,
sin y Fs g y Kiy h sin yFs k y Fc l y
.
1 sin y Fs g y Kiy h sin yFs k y Fc l y
Fc f y F c ϕ y 1 −
3.11
γ
By the hypothesis k k1 ∗ k2 , we see that sin yFs ky Fc k1 ∗ k2 y; using Lemma 3.3, we
2
1
get
γ
⎞
Fc g ∗ h y Fc k1 ∗ k2 y Fc l y
⎟
⎜
4
1
⎟.
Fc f y F c ϕ y ⎜
⎝1 −
γ
γ
⎠
1 Fc g ∗ h y Fc k1 ∗ k2 y Fc l y
⎛
γ
4
3.12
1
In virtue of the Wiener-Levy theorem 25, by the given condition, there exists a function
ξ ∈ L1 R such that
Fc ξ y sin y Fs g y Kiy h sin yFs k1 y Fc k2 y Fc l y
.
1 sin y Fs g y Kiy h sin yFs k1 y Fc k2 y Fc l y
3.13
From 3.12 and 3.13, we have
Fc f
y Fc ϕ y 1 − Fc ξ y .
3.14
Then the solution in L1 R of 3.1 has the form
fx ϕx − ϕ ∗ ξ x.
1
The proof is complete.
3.15
12
Mathematical Problems in Engineering
Remark 3.4. The integral equation 3.1 is a special case of the integral equation with the Toeplitz plus
Hankel kernel 1.27 for x > 0 and
k1 t
1 1
√
kt 1 − k|t − 1| signt − 1 √ lt
2 2π
2π
∞
1
√
guhw e−w coshtu−1 e−w cosht−u1 − e−w coshtu1 − e−w cosht−u−1 du dw
4 2π 0
k2 t
1 1
l|t|
√
k|t 1| signt 1 − k|t − 1| signt − 1 √
2 2π
2π
∞
1
√
guhw e−w coshtu−1 e−w cosht−u1 − e−w coshtu1 − e−w cosht−u−1 du dw.
4 2π 0
3.16
Next, we consider the following system of two integral equations:
fx ∞
θx, u, v, wguhvkwdu dv dw 0
∞
θ4 x, ufudu 0
∞
∞
θ3 x, ugudu px
0
3.17
θ5 x, ufudu gx qx.
0
Here θx, u, v, w is defined by 2.2, and
1 θ3 x, u √
lx u − l|x − u| signx − u ,
2π
1 θ4 x, u √
ξx u ξ|x − u| signx − u ,
2π
3.18
1 θ5 x, u √
η| x u − 1 η|x − u − 1| − ηx u 1 − ηx − u 1 ,
2 2π
h, k, l, ξ, η, p, q are known functions, and f and g are unknown functions.
Theorem 3.5. Given that p, q, h, l, ξ, η1 , η2 ∈ L1 R and k ∈ L1 β, R , η η1 ∗ η2 such that
3
0, where
1 − Fc ψy /
γ
γ
ψx ∗ξ, h, k x − ξ ∗ l x − ∗ η1 ∗ η2 , h, k x − l ∗ η x.
3
1
3.19
Mathematical Problems in Engineering
13
Then the system 3.17 has a unique solution in L1 R × L1 R whose closed form is as follows
γ
fx px − ∗ q, h, k x q ∗ l l ∗ p x
3
1
γ
∗ q, h, k ∗ l x q ∗ l ∗ l x,
3
1
1
γ
γ
gx qx − ξ ∗ p x − η ∗ p x q ∗ l x −
ξ∗p ∗l −
η ∗ p ∗ l x.
2
2
2
2
2
2
2
3.20
Here, l ∈ L1 R is defined by
Fc l y Fc ψ y
.
1 − Fc ψ y
3.21
Proof. We need the following lemma.
Lemma 3.6. Let ξ, f ∈ L1 R ; then
∞
1
Fs √
2π
∞
ξx u ξ|x − u| signx − u fudu ∈ L1 R ,
0
ξx u ξ|x − u| signx − u fudu y Fs ξ y Fc f y ,
0
∀y > 0.
3.22
Using Theorem 2.2, Lemma 3.6, and the generalized convolution 1.15, 1.19, we
have
y sin y Fs g y Fc h y Kiy k Fs l y Fs g y Fc p y ,
Fs ξ y Fc f y sin y Fc η y Fc f y Fs g y Fs q y .
Fc f
3.23
γ
On the other hand, from η η1 ∗ η2 we have sin yFc ηy Fs η1 ∗ η2 y. Therefore, using
3
Theorem 2.2 and the generalized convolution 1.15, 1.17, we have
1
sin yFc h y · Kiy k Fs l y Δ Fs ξ y sin y Fc η y
1
γ
γ
γ
1 − Fc ∗ξ, h, k y − Fc ξ ∗ l y − Fc ∗ η1 ∗ η2 , h, k
y − Fc l ∗ η x.
3
1
3.24
14
Mathematical Problems in Engineering
Hence, in view of the Wiener-Levy theorem 25, by the given condition, there is a unique
function l ∈ L1 R such that
1
1 Fc l y ,
Δ
3.25
where
γ
γ
γ
Fc l ∗ η
Fc ∗ξ, h, k Fc ξ ∗ l Fc ∗ η1 ∗ η2 , h, k
3
1
.
Fc l γ
γ
γ
1 − Fc ∗ξ, h, k − Fc ξ ∗ l − Fc ∗ η1 ∗ η2 , h, k
− Fc l ∗ η
3
3.26
1
On the other hand, using Theorem 2.2 and the generalized convolution 1.15, we have
Fc p y
Δ1 Fs qy
sin yFc h y · Kiy k Fs l y 1
3.27
Fc p y − Fc ∗ q, h, k y − Fc q ∗ l y .
3
Hence, from 3.25, 3.27 we have
Fs f
y 1 Fc l y
Fc p y − Fc ∗ q, h, k y − Fc q ∗ l y
3
Fc p y − Fc ∗ q, h, k − Fc q ∗ l Fc
3
− Fc
q∗l ∗l
3
1
l ∗ p y − Fc ∗ q, h, k ∗ l y
1
1
y .
3.28
It shows that
fx px − ∗ q, h, k x − q ∗ l l ∗ p x − ∗ q, h, k ∗ l x −
q ∗ l ∗ l x.
3
1
1
3
1
3.29
Similarly, from the generalized convolutions 1.15, 1.19, we have
1
Fc p y Δ2 Fs ξy sin yFc ηy Fs qy
γ
Fs q y − Fs ξ ∗ p y − Fs η ∗ p y .
2
2
3.30
Mathematical Problems in Engineering
15
Using formulas 3.25, 3.30, we have
Fs g
γ
y 1 Fc l y
F s q y − Fs ξ ∗ p y − F s η ∗ p y
2
Fs q y − Fs
− Fs
γ
2
η∗p ∗l
2
2
2
γ
ξ ∗ p y − Fs η ∗ p y Fs q ∗ l − Fs
ξ∗p ∗l y
2
2
2
2
y .
3.31
It shows that
γ
γ
ξ∗p ∗l −
η ∗ p ∗ l x.
gx qx − ξ ∗ p x − η ∗ p x q ∗ l x −
2
2
2
2
2
2
2
3.32
Pair f, g defined by fomulae 3.29 and 3.32 is a solution in closed form in L1 R × L1 R of system 3.17. The proof is complete.
Acknowledgments
In the memory of professor V. A. Kakichev, the author wish to express his deep thanks to
him for all his encouragement to the author in this investigative direction. This research
is supported partially by Vietnam’s National Foundation for Science and Technology
Development, Grant no. 101.01.21.09.
References
1 I. N. Sneddon, Fourier Transform, McGraw–Hill, New York, NY, USA, 1951.
2 S. B. Yakubovich, “On the convolution for the Kontorovich-Lebedev transformation and its
applications to integral equations,” Doklady Akademii Nauk BSSR, vol. 31, no. 2, pp. 101–103, 1987
Russian.
3 I. N. Sneddon, The Use of Integral Transforms, McGraw–Hill, New York, NY, USA, 1972.
4 A. Ardelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Function, vol. 1, 2, 3,
McGraw–Hill, New York, NY, USA, 1953.
5 V. A. Kakichev, “On the convolution for integral transforms,” Izvestiya Vysshikh Uchebnykh Zavedenii.
Matematika, no. 2, pp. 53–62, 1967 Russian.
6 N. X. Thao and N. T. Hai, Convolution for Integral Transform and Their Application, Russian Academy,
Moscow, Russia, 1997.
7 N. X. Thao, V. A. Kakichev, and V. K. Tuan, “On the generalized convolutions for Fourier cosine and
sine transforms,” East-West Journal of Mathematics, vol. 1, no. 1, pp. 85–90, 1998.
8 N. X. Thao, V. K. Tuan, and N. M. Khoa, “A generalized convolution with a weight function for the
Fourier cosine and sine transforms,” Fractional Calculus & Applied Analysis, vol. 7, no. 3, pp. 323–337,
2004.
9 N. X. Thao and N. M. Khoa, “On the generalized convolution with a weight function for the Fourier
sine and cosine transforms,” Integral Transforms and Special Functions, vol. 17, no. 9, pp. 673–685, 2006.
10 S. B. Yakubovich and L. E. Britvina, “Convolutions related to the Fourier and Kontorovich-Lebedev
transforms revisited,” Integral Transforms and Special Functions, vol. 21, no. 4, pp. 259–276, 2010.
11 V. A. Kakichev, Polyconvolution, TPTU, Taganrog, Russia, 1997.
16
Mathematical Problems in Engineering
12 N. X. Thao, “On the polyconvolution for integral transforms,” Vestnik Novgorodskogo Gosudarstvennogo
Universiteta. Seriya Estestvennye i Tekhnicheskie Nauki, vol. 10, pp. 104–110, 1999.
13 N. X. Thao and N. D. Hau, “On the polyconvolution for the Fourier cosine and Fourier sine
transforms,” Acta Mathematica Vietnamica, vol. 33, no. 2, pp. 107–122, 2008.
14 J. J. Betancor and B. J. González, “Spaces of Lp -type and the Hankel convolution,” Proceedings of the
American Mathematical Society, vol. 129, no. 1, pp. 219–228, 2001.
15 J. J. Betancor, C. Jerez, S. M. Molina, and L. Rodrı́guez-Mesa, “Distributional convolutors for Fourier
transform,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 459–468, 2007.
16 J. J. Betancor, M. Linares, and J. M. R. Méndez, “A distributional convolution for a generalized finite
Fourier transformation,” Proceedings of the American Mathematical Society, vol. 128, no. 2, pp. 547–556,
2000.
17 H. M. Srivastava and V. K. Tuan, “A new convolution theorem for the stieltjes transform and its
application to a class of singular integral equations,” Archiv der Mathematik, vol. 64, no. 2, pp. 144–
149, 1995.
18 H. M. Titchmarsh, Introduction to the Theory of Fourier Integrals, Clarendon Press, Oxford, UK, 2nd
edition, 1967.
19 F. G. Tricomi, “On the finite Hilbert transformation,” The Quarterly Journal of Mathematics, vol. 2, pp.
199–211, 1951.
20 V. K. Tuan and M. Saigo, “Convolution of Hankel transform and its application to an integral
involving Bessel functions of first kind,” International Journal of Mathematics and Mathematical Sciences,
vol. 18, no. 3, pp. 545–549, 1995.
21 S. B. Yakubovich and A. I. Moshinskiı̆, “Integral equations and convolutions associated with
transformations of Kontorovich-Lebedev type,” Differentsial’nye Uravnenia, vol. 29, no. 7, pp. 1272–
1284, 1993.
22 H. H. Kagiwada and R. Kalaba, Integral Equations via Imbedding Methods, vol. 6 of Applied Mathematics
and Computation, Addison-Wesley, London, UK, 1974.
23 M. G. Kreı̆n, “On a new method of solution of linear integral equations of first and second kinds,”
Doklady Akademii Nauk SSSR, vol. 100, pp. 413–416, 1955 Russian.
24 J. N. Tsitsiklis and B. C. Levy, “Integral equations and resolvents of Toeplitz plus Hankel kernels,”
Tech. Rep. LIDS-P 1170, Laboratory for Information and Decision Systems, Massachusetts Institute of
Technology, Cambridge, Mass, USA, 1981.
25 N. I. Achiezer, Lectures on Approximation Theory, Science Publishing House, Moscow, Russia, 1965.