On the polyconvolution for the Fourier cosine, Fourier sine, and Kontorovich–Lebedev integral transforms
The polyconvolution $∗_1(f,g,h)(x)$ of three functions $f, g$ and $h$ is constructed for the Fourier cosine $(F_c)$, Fourier sine $(F_s)$, and Kontorovich–Lebedev $(K_{iy})$ integral transforms whose factorization equality has the form $$F_c(∗_1(f,g,h))(y)=(F_sf)(y).(F_sg)(y).(K_{iy}h)\;\;∀y>...
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| author | Thao, N. X. Virchenko, N. A. Тао, Н. Х. Вірченко, Н. О. |
| author_facet | Thao, N. X. Virchenko, N. A. Тао, Н. Х. Вірченко, Н. О. |
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| datestamp_date | 2020-03-18T19:41:19Z |
| description | The polyconvolution $∗_1(f,g,h)(x)$ of three functions $f, g$ and $h$ is constructed for the Fourier cosine $(F_c)$, Fourier sine $(F_s)$, and Kontorovich–Lebedev $(K_{iy})$ integral transforms whose factorization equality has the form
$$F_c(∗_1(f,g,h))(y)=(F_sf)(y).(F_sg)(y).(K_{iy}h)\;\;∀y>0.$$
The relationships between this polyconvolution, the Fourier convolution, and the Fourier cosine convolution are established. In addition, we also establish the relationships between the product of the new polyconvolution and the products of the other known types of convolutions. As an application, we consider a class of integral equations with Toeplitz and Hankel kernels whose solutions can be obtained with the help of the new polyconvolution in the closed form. We also present the applications to the solution of systems of integral equations. |
| first_indexed | 2026-03-24T02:33:38Z |
| format | Article |
| fulltext |
UDC 517.581
N. X. Thao (Hanoi Univ. Technol., Vietnam),
N. O. Virchenko (Nat. Techn. Univ. Ukraine “KPI”, Kyiv)
ON THE POLYCONVOLUTION FOR THE FOURIER COSINE,
FOURIER SINE, AND THE KONTOROVICH – LEBEDEV
INTEGRAL TRANSFORMS*
ПРО ПОЛIЗГОРТКУ ДЛЯ КОСИНУС-ФУР’Є, СИНУС-ФУР’Є
ТА КОНТОРОВИЧА – ЛЕБЕДЄВА IНТЕГРАЛЬНИХ
ПЕРЕТВОРЕНЬ
The polyconvolution ∗
1
(f, g, h)(x) of three functions f, g, h is constructed for the Fourier cosine (Fc) integral
transform, the Fourier sine (Fs) integral transform, and the Kontorovich – Lebedev (Kiy) integral transform,
whose factorization equality is of the form
Fc(∗
1
(f, g, h))(y) = (Fsf)(y).(Fsg)(y).(Kiyh) ∀y > 0.
The relations of this polyconvolution to the Fourier convolution and the Fourier cosine convolution are ob-
tained. In addition, the relations between the new polyconvolution product and other known convolution
products are established. As application, we consider a class of integral equations with the Toeplitz kernel and
the Hankel kernel, whose solutions in closed form can be obtained with the help of the new polyconvolution.
Application in solving systems of integral equations is also presented.
Побудовано полiзгортку ∗
1
(f, g, h)(x) трьох функцiй f, g, h для косинус-Фур’є (Fc), синус-Фур’є (Fs)
i Конторовича – Лебедєва (Kiy) iнтегральних перетворень з рiвнiстю факторизацiї у формi
Fc(∗
1
(f, g, h))(y) = (Fsf)(y).(Fsg)(y).(Kiyh) ∀y > 0.
Одержано спiввiдношення цiєї полiзгортки iз згорткою Фур’є i косинус-Фур’є згорткою. Також вста-
новлено спiввiдношення мiж добутком нової полiзгортки та добутками iнших вiдомих згорток. Як
застосування, розглянуто клас iнтегральних рiвнянь з ядрами Теплiца i Ганкеля, розв’язки цих рiвнянь
за допомогою нової полiзгортки можна одержати у замкненiй формi. Наведено також застосування до
розв’язання систем iнтегральних рiвнянь.
Introduction. The convolution of two functions f and g for the Fourier transform is
well-known [1]:
(f ∗
F
g)(x) =
1√
2π
∞∫
−∞
f(x− y)g(y)dy, x ∈ R. (0.1)
This convolution has the factorization equality as belows
F (f ∗
F
g)(y) = (Ff)(y)(Fg)(y) ∀y ∈ R,
here F denotes the Fourier transform [1]
(Ff)(y) =
1√
2π
∞∫
−∞
e−ixyf(x)dx.
The convolution of f and g for the Kontorovich – Lebedev integral transform has been
*This research is supported partially by NAFOSTED of Vietnam, grant 101.01.21.09.
c© N. X. THAO, N. O. VIRCHENKO, 2010
1388 ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
ON THE POLYCONVOLUTION FOR THE FOURIER COSINE, FOURIER SINE AND. . . 1389
studied in [2]
(f ∗
K−L
g)(x) =
1
2x
∞∫
0
∞∫
0
exp
[
−1
2
(xu
v
+
xv
u
+
uv
x
)]
f(u)g(v)dudv, x > 0, (0.2)
for which the factorization identity holds
Kiy(f ∗
K−L
g) = (Kiyf).(Kiyg) ∀y > 0.
Here Kiy is the Kontorovich – Lebedev transform [2]
Kix[f ] =
∞∫
0
Kix(t)f(t)dt,
and Kix(t) is the Macdonald function [3].
The convolution of two function f and g for the Fourier cosine is of the form [1]
(f ∗
1
g)(x) =
1√
2π
∞∫
0
f(y)[g(|x− y|) + g(x+ y)]dy, x > 0, (0.3)
which satisfied the factorization equality
Fc(f ∗
1
g)(y) = (Fcf)(y)(Fcg)(y) ∀y > 0.
Here the Fourier cosine transform is of the form [1]
(Fcf)(y) =
√
2
π
∞∫
0
cos yx.f(x)dx, y > 0.
The convolution with a weight function γ(x) = sinx of two functions f and g for the
Fourier sine transform has introduced in [4]
(f
γ
∗g)(x) =
1
2
√
2π
+∞∫
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, (0.4)
and the factorization identity holds
Fs(f
γ
∗g)(y) = sin y(Fsf)(y)(Fsg)(y) ∀y > 0.
Here the Fourier sine is of the form [1]
(Fsf)(y) =
√
2
π
∞∫
0
sin yx.f(x)dx, y > 0.
The generalized convolution of two functions f, g for the Fourier sine and Fourier cosine
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
1390 N. X. THAO, N. O. VIRCHENKO
transforms has studied in [1]
(f ∗
2
g)(x) =
1√
2π
∞∫
0
f(u)[g(|x− u|)− g(x+ u)]du, x > 0, (0.5)
and the respectively factorization identity is [1]
Fs(f ∗
2
g)(y) = (Fsf)(y).(Fcg)(y) ∀y > 0.
The generalized convolution of two functions f and g for the Fourier cosine and the
Fourier sine transforms is defined by [5]
(f ∗
3
g)(x) =
1√
2π
∞∫
0
f(u)[sign(u− x)g(|u− x|) + g(u+ x)]du, x > 0. (0.6)
For this generalized convolution the factorization equality holds [5]
Fc(f ∗
3
g)(y) = (Fsf)(y)(Fsg)(y) ∀y > 0.
The generalized convolution with the weight function γ(x) = sinx for the Fourier cosine
and the Fourier sine transforms of f and g has introduced in [6]
(f
γ
∗
1
g)(x) =
1
2
√
2π
∞∫
0
f(u)[g(|x+ u− 1|) + g(|x− u+ 1|)− g(x+ u+ 1)−
−g(|x− u− 1|)]du, x > 0. (0.7)
It satisfies the factorization property [6]
Fc(f
γ
∗
1
g)(y) = sin y (Fsf)(y)(Fcg)(y) ∀y > 0.
The generalized convolution with the weight function γ(x) = sinx of f and g for the
Fourier sine and Fourier cosine has studied in [7]
(f
γ
∗
2
g)(x) =
1
2
√
2π
∞∫
0
f(u)[g(|x+ u− 1|) + g(|x− u− 1|)− g(x+ u+ 1)−
−g(|x− u+ 1|)]du, x > 0, (0.8)
and satisfy the factorization identity
Fs(f
γ
∗
2
g)(y) = sin y (Fcf)(y)(Fcg)(y) ∀y > 0.
In 1997, Kakichev V. A. introduced a constructive method for defining a polyconvolution
γ
∗(f1, f2, . . . , fn)(x) of functions f1, f2, . . . , fn with a weight function γ for the integral
transforms K,K1,K2, . . . ,Kn, for which the factorization property holds [8]
K[
γ
∗(f1, f2, . . . , fn)](y) = γ(y)
n∏
i=1
(Kifi)(y), n ≥ 3.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
ON THE POLYCONVOLUTION FOR THE FOURIER COSINE, FOURIER SINE AND. . . 1391
Polyconvolutions for the Hilbert, Stieltjes, Fourier cosine and Fourier sine integral
transforms has been studied in.
The polyconvolution of f , g and h for the Fourier cosine and the Fourier sine
transforms has the form [9]
∗(f, g, h)(x) =
1
2π
∞∫
0
∞∫
0
f(u)g(v)[h(|x+ u− v|) + h(x− u+ v|)−
−h(|x− u− v|)− h(x+ u+ v)]dudv, x > 0, (0.9)
which satisfies the following factorization property:
Fc(∗(f, g, h))(y) = (Fsf)(y).(Fsg)(y).(Fch)(y) ∀y > 0.
Recent years, many sciences interested in the theory of convolution for the integral
transforms and gave several interesting application (see [10]). Specially, the integral
equations with the Toeplitz plus Hankel kernel
f(x) +
∞∫
0
[k1(x+ y) + k2(x− y)]f(y)dy = g(x), x > 0, (0.10)
where k1, k2, g are known functions, and f is unknow 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 cosine, Fourier sine 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.
1. Polyconvolution.
Definition 1. The polyconvolution of functions f, g and h for the Fourier cosine,
Fourier sine and the Kontorovich – Lebedev integral transforms is defined as follows
∗
1
(f, g, h)(x) =
∞∫
0
∞∫
0
∞∫
0
θ(x, u, v, w)f(u)g(v)h(w)dudvdw, x > 0, (1.1)
where
θ(x, u, v, w) =
1
2
√
2π
[e−w cosh(x+u−v)+
+e−w cosh(x−u+v) − e−w cosh(x+u+v) − e−w cosh(x−u−v)].
Theorem 1. Let f, g be functions in L1(R+), and let h be function in L1
( 1√
w
,
R+
)
, then the polyconvolution (1.1) belongs to L1(R+) and satisfies the factorization
property
Fc(∗
1
(f, g, h))(y) = (Fsf)(y).(Fsg)(y).(Kiyh) ∀y > 0. (1.2)
Proof. Since |e−w cosh(x+u−v)−e−w cosh(x+u+v)| 6 1√
w
, for sufficient large w > 0,
we have
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
1392 N. X. THAO, N. O. VIRCHENKO
| ∗
1
(f, g, h)(x)| 6 1
2
√
2π
∞∫
0
∞∫
0
∞∫
0
|f(u)||g(v)||h(w)||θ(x, u, v, w)|dudvdw 6
6
√
1
2π
∞∫
0
|f(u)|du.
∞∫
0
|g(v)|dv.
∞∫
0
1√
w
|h(w)|dw < +∞.
On the other hand, note that cosh(x+ u− v) >
(x+ u− v)2
2
, we have
e−w cosh(x+u−v) 6 e−w
(x+u−v)2
2 ∀w > 0.
Then we have
∞∫
0
e−w cosh(x+u−v)dx 6
√
2
w
∞∫
0
e−(
√
w
2 (x+u−v))
2
d
(√
w
2
(x+ u− v)
)
6
6 2
√
2
w
∞∫
0
e−s
2
ds =
√
2π
w
.
Using this estimation we obtain
∞∫
0
∞∫
0
∞∫
0
∞∫
0
e−w cosh(x+u−v))|f(u)||g(v)||h(w)|dudvdwdx 6
6
∞∫
0
∞∫
0
∞∫
0
√
2π
w
|h(w)||f(u)||g(v)|dudvdw =
=
√
2π
∞∫
0
|f(u)|du.
∞∫
0
|g(v)|dv.
∞∫
0
1√
w
|h(w)|dw < +∞. (1.3)
The following estimation can be obtained by similar way
∞∫
0
∞∫
0
∞∫
0
∞∫
0
e−w cosh(x−u+v))|f(u)||g(v)||h(w)|dudvdwdx < +∞, (1.4)
∞∫
0
∞∫
0
∞∫
0
∞∫
0
e−w cosh(x+u+v))|f(u)||g(v)||h(w)|dudvdwdx < +∞, (1.5)
∞∫
0
∞∫
0
∞∫
0
∞∫
0
e−w cosh(x−u−v))|f(u)||g(v)||h(w)|dudvdwdx < +∞. (1.6)
From formulas (1.1), (1.3) – (1.6) we have
∞∫
0
| ∗
1
(f, g, h)(x)|dx < +∞.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
ON THE POLYCONVOLUTION FOR THE FOURIER COSINE, FOURIER SINE AND. . . 1393
It shows that the polyconvolution (1.1) belonging to L1(R+).
We now prove the factorization equality (1.2). We get
(Fsf)(y)(Fsg)(y)(Kiyh) =
=
2
π
∞∫
0
∞∫
0
∞∫
0
sin(yu) sin(yv)Kiy(w)f(u)g(v)h(w)dudvdw =
=
1
π
∞∫
0
∞∫
0
∞∫
0
∞∫
0
cos(yα)e−w coshα(cos y(u− v)−
− cos y(u+ v))f(u)g(v)h(w)dudvdwdα =
=
1
2π
∞∫
0
∞∫
0
∞∫
0
∞∫
0
[cos y(α+ u− v) + cos y(α− u+ v)− cos y(α+ u+ v)−
− cos y(α− u− v)]e−w coshαf(u)g(v)h(w)dudvdwdα. (1.7)
Changing variables we have
∞∫
0
[cos y(α+ u− v)− cos y(α+ u+ v)]e−w coshαdα =
=
∞∫
0
cosxy[e−w cosh(x−u+v) − e−w cosh(x−u−v)]dx. (1.8)
Similar,
∞∫
0
[cos y(α− u+ v)− cos y(α− u− v)]e−w coshαdα =
=
∞∫
0
cosxy[e−w cosh(x+u−v) − e−w cosh(x+u+v)]dx. (1.9)
From formulaes (1.7) – (1.9) we have
(Fsf)(y)(Fsg)(y)(Kiyh) = Fc(∗(f, g, h))(y).
Theorem 1 is proved.
Proposition. Let f, g ∈ L1(R+), and let h ∈ L1
(
1√
w
,R+
)
, then the identity holds
∗
1
(f, g, h) =
√
π
2
∞∫
0
h(w)
(
(g ∗
1
e−w cosh t) ∗
F
(f(|t|))
)
(x)dw. (1.10)
Proof. From the definition (1.1) of the polyconvolution and the convolution (0.3)
we have
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
1394 N. X. THAO, N. O. VIRCHENKO
∗
1
(f, g, h)(x) =
1
2
∞∫
0
∞∫
0
f(u)h(w)[(g ∗
1
e−w cosh t)(x+u)+(g ∗
1
e−w cosh t)(x−u)]dudw.
(1.11)
Therefore, in view of formula (0.1) we obtain
∗
1
(f, g, h) =
√
π
2
∞∫
0
h(w)
(
(g ∗
1
e−w cosh t) ∗
F
(f(|t|))
)
(x)dw.
Theorem 2. Let f, g, h be functions in L1(R+), and let k be functions in
L
(
1√
w
,R+
)
, then the following properties hold
a) ∗
1
(f ∗
2
g, h, k) = ∗
1
(f, h ∗
2
g, k);
b) ∗
1
(f
γ
∗ g, h, k) = ∗
1
(f, g
γ
∗ h, k).
Proof. We only need to prove the assertion a), since the second one can be obtained
similarly. From Theorem 1 and (0.5) we have
Fc(∗
1
(f, h ∗
2
g, k))(y) = Fs(f ∗
2
g)(y)(Fsh)(y)(Kiyk) =
= (Fsf)(y)(Fcg)(y)(Fsh)(y)(Kiyk) =
= (Fsf)(y)Fs(h ∗
2
g)(y)(Kiyk) = Fs(∗
1
(f ∗
2
g, h, k))(y).
Then we obtain assertion a).
Definition 2. Let f be a function in L1(R+) and g be a function in L1(β,R+),
β(v) =
2√
v
. Then their norm are defined as follows
‖f‖L1(R+) =
∞∫
0
|f(x)|dx, and ‖g‖L1(β,R+) =
∞∫
0
β(v)|f(v)|dv.
Theorem 3. Let f, g be functions in L1(R+), and let h be function in L1(β,R+),
then the estimation holds
‖ ∗
1
(f, g, h)‖L1(R+) 6 ‖f‖L1(R+)‖g‖L1(R+)‖h‖L1(β,R+).
Proof. From formulas (1.1), (1.3) – (1.6) we have
∫
| ∗
1
(f, g, h)(x)|dx 6
∞∫
0
2√
w
|h(w)dw.
∞∫
0
|f(u)|du.
∞∫
0
|g(v)|dv.
Therefore,
‖ ∗
1
(f, g, h)‖L1(R+) 6 ‖f‖L1(R+)‖g‖L1(R+)‖h‖L1(β,R+).
2. Applications. Consider the integral equation
f(x) +
∞∫
0
θ1(x, u)f(u)du+
∞∫
0
θ2(x, u)f(u)du+
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
ON THE POLYCONVOLUTION FOR THE FOURIER COSINE, FOURIER SINE AND. . . 1395
+
∞∫
0
∞∫
0
∞∫
0
θ(x, u, v, w)f(u)l(v)k(w)dudvdw = p(x), x > 0. (2.1)
Here θ(x, u, v, w) is given by the Definition 1, and θ1(x, u) and θ2(x, u) are defined by
θ1(x, u) =
1√
2π
[g(|x− u|)− g(x+ u)],
θ2(x, u) =
1
2
√
2π
[h(|x+ u− 1|) sign(x+ u− 1) + h(|x− u+ 1|) sign(x− u+ 1)−
−h(x+ u+ 1)− h(|x− u− 1|) sign(x− u− 1)].
Beside, g, h, l, k, p are known functions, f is unknow function.
Theorem 4. Let g, h1, h2, l, p1, p2 ∈ L1(R+), p = p1 + p2, and let k ∈
∈ L1
(
1√
w
,R+
)
, h = h1 ∗
2
h2 such that
1 + (Fcg)(y) + sin(Fsh)(y) 6= 0 ∀y > 0,
and
p2(x) = ∗
1
(p1, l, k)− l ∗
1
(∗
1
(p1, l, k))(x),
where l ∈ L1(R+) is defined uniquely by
(Fcl)(y) =
(Fcg)(y) + sin(Fsh)(y)
1 + (Fcg)(y) + sin(Fsh)(y)
.
Then the equation (2.1) has a unique solution in L1(R+) whose closed form is
f(x) = p1(x)− (p1 ∗
2
l)(x).
Proof. First, similarly to the proof of the Theorem 1, we obtain the following lemma.
Lemma 1. Let f, g ∈ L1(R+), then (f
γ
∗
3
g)(x) belongs to L1(R) the identity holds
F (f
γ
∗
3
g)(y) = −i sin y(Fsf)(y)(Fsg)(y),
where
(f
γ
∗
3
g)(x) =
1
2
√
2π
+∞∫
0
f(u)[g(|x+ u− 1|) sign(x+ u− 1)+
+g(|x− u+ 1|) sign(x− u+ 1)− g(|x+ u+ 1|) sign(x+ u+ 1)−
−g(|x− u− 1|) sign(x− u− 1)]du.
Lemma 2. Let f, g ∈ L1(R+), then (f
γ
∗
4
g)(x) belongs to L1(R) the identity holds
F (f
γ
∗
4
g)(y) = −i(Fsf)(y)(Fcg)(y),
where
(f
γ
∗
4
g)(x) =
1√
2π
+∞∫
0
f(u)[g(|x− u|)− g(|x+ u|)]du.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
1396 N. X. THAO, N. O. VIRCHENKO
We now prove the Theorem 4 with the help of the Fourier transform, Lemmas 1
and 2, Theorem 1, the generalized convolution (0.5) and the convolution (0.4). Extend
f, p1 oddly, and p2 evenly over whole real-line, we have
−i(Fsf)(y)− i(Fsf)(y).(Fcg)(y)− i(Fsf)(y).(Fsh)(y) sin y+
+(Fsf)(y).(Fsl)(y)(Kiyk) = −i(Fsp1)(y) + (Fcp2)(y). (2.2)
Note that the equation (2.2) is equivalent to the following system:
(Fsf)(y)(1 + (Fcg)(y) + sin y(Fsh)(y)) = (Fsp1)(y), (2.3)
(Fsf)(y).(Fsl)(y)(Kiyk) = (Fcp2)(y). (2.4)
From (2.3) and the given condition we have
(Fsf)(y) = (Fsp1)(y)
(
1− (Fcg)(y) + sin y(Fsh)(y)
1 + (Fcg)(y) + sin y(Fsh)(y)
)
. (2.5)
Since h = h1 ∗
2
h2 we have
sin y(Fsh)(y) = Fc(h1
γ
∗
1
h2)(y).
In virtue of the Wiener – Levy theorem [10], and the given condition, there exists a
function l ∈ L1(R+) such that
(Fcl)(y) =
(Fcg)(y) + Fc(h1
γ
∗
1
h2)(y)
1 + (Fcg)(y) + Fc(h1
γ
∗
1
h2)(y)
. (2.6)
From (2.4) – (2.6) we have
(Fsf)(y) = (1− (Fcl)(y))(Fsp1)(y).
Therefore,
f(x) = p1(x)− (p1 ∗
2
l)(x). (2.7)
Substitute (2.7) into (2.4) we obtain
(Fcp2)(y) = (1− (Fcl)(y))(Fsp1)(y)(Fsl)(y)(Kiyk).
Hence, using formula (0.5) and Theorem 1 we have
p2(x) = ∗
1
(p1, l, k)(x)− (l ∗
1
(∗
1
(p1, l, k)))(x), x > 0. (2.8)
From (2.3), (2.4), (2.7), (2.8), the solution of equation (2.1) has a closed form in L1(R+)
as
f(x) = p1(x)− (p1 ∗
2
l)(x).
Remark. The integral equation (2.1) is a special case of the integral equation with
the Toeplitz plus Hankel kernel (0.10) with
k1(t) = − 1√
2π
g(t)− 1
2
√
2π
[h(t+ 1)− h(|t− 1|) sign(t− 1)]−
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
ON THE POLYCONVOLUTION FOR THE FOURIER COSINE, FOURIER SINE AND. . . 1397
− 1
2
√
2π
∞∫
0
∞∫
0
l(v)k(w)[e−w cosh(t+v) − e−w cosh(t−v)]dvdw,
k2(t) =
1√
2π
g(|t|) +
1
2
√
2π
[h(|t+ 1|) sign(t+ 1)− h(|t− 1|) sign(t− 1)]+
+
1
2
√
2π
∞∫
0
∞∫
0
l(v)k(w)[e−w cosh(t+v) − e−w cosh(t−v)]dvdw.
Next, we consider the following system of two integral equations for x > 0:
f(x) +
∞∫
0
θ3(x, u)g(u)du+
∞∫
0
θ4(x, u)g(u)du+
+
∞∫
0
∞∫
0
∞∫
0
θ(x, u, v, w)h(u)g(v)h(w)dudvdw = p(x),
∞∫
0
θ5(x, u)f(u)du+
∞∫
0
θ6(x, u)f(u)du+ g(x) = q(x).
(2.9)
Here θ(x, u, v, w) is defined by (1.1), and
θ3(x, u) =
1√
2π
[h(u+ x) + h(|u− x|) sign(u− x)],
θ4(x, u) =
1
2
√
2π
[k(|x+ u− 1|) + k(|x− u+ 1|)− k(x+ u+ 1)− k(|x− u− 1|)],
θ5(x, u) =
1√
2π
[ψ(|x− u|) sign(x− u) + ψ(x+ u)],
θ6(x, u) =
1
2
√
2π
[ξ(|x+ u− 1|) + ξ(|x− u− 1|)− ξ(x+ u+ 1)− ξ(|x− u+ 1|)],
h, k, l, ϕ, ψ, ξ, p, q are known functions, f, g are unknown functions.
Theorem 5. Given that h, k, l, ψ, ξ1, ξ2, p, q ∈ L1(R+) and ϕ ∈ L1
(
1√
w
,R+
)
,
ξ = ξ1 ∗
3
ξ2 such that 1− (Fcr)(y) 6= 0 ∀y > 0, where
r(x) = (h ∗
3
ψ)(x) + (ψ
γ
∗
1
k)(x) + ∗
1
(ψ, l, ϕ)(x)+
+(h
γ
∗
1
ξ)(x) + (ξ1
γ
∗
1
(ξ2
γ
∗
1
k))(x) + ∗
1
(ξ1, ξ2
γ
∗ l, ϕ)(x).
Then the system (2.9) has a unique solution in L1(R+)×L1(R+) whose closed formed
as follows
f(x) = p(x)− (q ∗
3
h)(x)− (q
γ
∗
1
k)(x)− ∗
1
(q, l, ϕ)(x)+
+(η ∗
1
p)(x)− (η ∗
1
(q ∗
3
h))(x)− (η ∗
1
(q
γ
∗ k))(x)− (η ∗
1
(∗
1
(q, l, ϕ))(x),
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
1398 N. X. THAO, N. O. VIRCHENKO
g(x) = q(x)− (ψ ∗
2
p)(x)− (ξ
γ
∗
2
p)(x)+
+(q ∗
2
η)(x)− ((ψ ∗
2
p) ∗
2
η)(x)− ((ξ
γ
∗
2
p) ∗
2
η)(x).
Here, η ∈ L1(R+) is defined by
Fcη =
(Fcr)(y)
1− (Fcr)(y)
.
Proof. Using Theorem 1 and (0.5) – (0.8) we write the system (2.9) in the form
(Fcf)(y) + (Fsg)(y)[(Fsh)(y) + sin y(Fsk)(y) + (Fsl)(y)(Kiyϕ)] = (Fcp)(y),
(2.10)
(Fcf)(y)[(Fsψ)(y) + sin y(Fcξ)(y) + (Fsg)(y) = (Fsq)(y).
We obtain a system of two linear equations for (Fcf)(y) and (Fsg)(y). We have
∆ =
∣∣∣∣∣∣
1 (Fsh)(y) + sin y(Fck)(y) + (Fsl)(y)(Kiyϕ)
(Fsψ)(y) + sin y(Fcξ)(y) 1
∣∣∣∣∣∣ =
= 1− (Fcr)(y). (2.11)
In view of the Wiener – Levy theorem [10], by the given condition, there is a unique
function η ∈ L1(R+) such that
(Fcη)(y) =
(Fcr)(y)
1− (Fcr)(y)
. (2.12)
From (2.11) and (2.12) we have
1
∆
= 1 + (Fcη)(y). (2.13)
On the other hand,
∆1 =
∣∣∣∣∣∣
(Fcp)(y) (Fsh)(y) + sin y(Fck)(y) + (Fsl)(y)(Kiyϕ)
(Fsq)(y) 1
∣∣∣∣∣∣ =
= (Fcp)(y)− Fc(q ∗
3
h)(y)− Fc(q
γ
∗
1
k)(y)− Fc(∗(q, l, ϕ))(y). (2.14)
Hence, from (2.13), (2.14) we have
(Fcf)(y) =
∆1
∆
=
= [1+(Fcη)(y)][(Fcp)(y)−Fc(q ∗
3
h)(y)−Fc(q
γ
∗
1
k)(y)−Fc(∗
1
(q, l, ϕ))(y)] =
= (Fcp)(y)− Fc(q ∗
3
h)(y)− Fc(q
γ
∗
1
k)(y)− Fc(∗
1
(q, l, ϕ))(y) + Fc(η ∗
1
p)−
−Fc(η ∗
1
(q ∗
3
h)(y)− Fc(η ∗
1
(q
γ
∗
1
k)(y)− Fc(η ∗
1
(∗
1
(q, l, ϕ)))(y).
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
ON THE POLYCONVOLUTION FOR THE FOURIER COSINE, FOURIER SINE AND. . . 1399
It follows
f(x) = p(x)− (q ∗
3
h)(x)− (q
γ
∗
1
k)(x)− ∗
1
(q, l, ϕ)(x) + (η ∗
1
p)(x)−
−(η ∗
1
(q ∗
3
h))(x)− (η ∗
1
(q
γ
∗ k))(x)− (η ∗
1
(∗
1
(q, l, ϕ))(x). (2.15)
Similarly,
∆2 =
∣∣∣∣∣∣
1 (Fcp)(y)
(Fsψ)(y) + sin y(Fcξ)(y) (Fsq)(y)
∣∣∣∣∣∣ =
= (Fsq)(y)− Fs(ψ ∗
2
p)(y)− Fs(ξ
γ
∗
2
p)(y). (2.16)
Using formula (2.13) and (2.16) we have
(Fsg)(y) =
∆2
∆
=
= [1 + (Fcη)(y)][(Fsq)(y)− Fs(ψ ∗
2
p)(y)− Fs(ξ
γ
∗
2
p)(y)] =
= (Fsq)(y)− Fs(ψ ∗
2
p)(y)− Fs(ξ
γ
∗
2
p)(y)+
+Fs(q ∗
2
η)(y)− Fs((ψ ∗
2
p) ∗
2
η)(y)− Fs((ξ
γ
∗
2
p) ∗
2
η)(y).
It shows that
g(x) = q(x)−(ψ∗
2
p)(x)−(ξ
γ
∗
2
p)(x)+(q∗
2
η)(x)−((ψ∗
2
p)∗
2
η)(x)−((ξ
γ
∗
2
p)∗
2
η)(x). (2.17)
From (2.10), (2.15), (2.17), system (2.9) has a solution (f, g) in L1(R+)× L1(R+).
Theorem 5 is proved.
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integral transform // Dokl. Akad. Nauk BSSR. – 1987. – 31. – P. 101 – 103.
3. Erdely A. et al. Higher transcendental functions. – New York: McGraw-Hill, 1953. – 1. – 299 p.
4. Kakichev V. A. On the convolution for integral transforms // Izv. Vysh. Uchebn. Zaved. Mat. – 1967. –
№ 2. – S. 53 – 62.
5. Nguyen Xuan Thao, Kakichev V. A., Vu Kim Tuan. On the generalized convolution for Fourier cosine
and sine transforms // East-West J. Math. – 1998. – 1. – P. 85 – 90.
6. Nguyen Xuan Thao, Vu Kim Tuan, Nguyen Minh Khoa. On the generalized convolution with a weight-
function for the Fourier cosine and sine transforms // Frac. Cal. and Appl. Anal. – 2004. – 7, № 3. –
P. 323 – 337.
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Received 10.12.09
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
|
| id | umjimathkievua-article-2963 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:33:38Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/12/027b2e332926ac8b1bab3e2601c2bb12.pdf |
| spelling | umjimathkievua-article-29632020-03-18T19:41:19Z On the polyconvolution for the Fourier cosine, Fourier sine, and Kontorovich–Lebedev integral transforms Про полізгортку для косинус-Фур'є, синус-Фур'є та Конторовича - Лебедєва інтегральних перетворень Thao, N. X. Virchenko, N. A. Тао, Н. Х. Вірченко, Н. О. The polyconvolution $∗_1(f,g,h)(x)$ of three functions $f, g$ and $h$ is constructed for the Fourier cosine $(F_c)$, Fourier sine $(F_s)$, and Kontorovich–Lebedev $(K_{iy})$ integral transforms whose factorization equality has the form $$F_c(∗_1(f,g,h))(y)=(F_sf)(y).(F_sg)(y).(K_{iy}h)\;\;∀y>0.$$ The relationships between this polyconvolution, the Fourier convolution, and the Fourier cosine convolution are established. In addition, we also establish the relationships between the product of the new polyconvolution and the products of the other known types of convolutions. As an application, we consider a class of integral equations with Toeplitz and Hankel kernels whose solutions can be obtained with the help of the new polyconvolution in the closed form. We also present the applications to the solution of systems of integral equations. Побудовано полізгортку $∗_1(f,g,h)(x)$ трьох функцій $f, g, h$ для косинус-Фур'є $(F_c)$, синус-Фур'є $(F_s)$ і Комторовича-Лебедєва $(K_{iy})$ інтегральних перетворень з рівністю факторизації у формі $$F_c(∗_1(f,g,h))(y)=(F_sf)(y).(F_sg)(y).(K_{iy}h)\;\;∀y>0.$$ Одержано співвідношення цієї полізгортки із згорткою Фур'є і косинус-Фур'є згорткою. Також вста- новлено співвідношення між добутком нової полізгортки та добутками інших відомих згорток. Як застосування, розглянуто клас інтегральних рівнянь з ядрами Тепліца і Ганкеля, розв'язки цих рівнянь за допомогою нової полізгортки можна одержати у замкненій формі. Наведено також застосування до розв'язання систем інтегральних рівнянь. Institute of Mathematics, NAS of Ukraine 2010-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2963 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 10 (2010); 1388–1399 Український математичний журнал; Том 62 № 10 (2010); 1388–1399 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2963/2676 https://umj.imath.kiev.ua/index.php/umj/article/view/2963/2677 Copyright (c) 2010 Thao N. X.; Virchenko N. A. |
| spellingShingle | Thao, N. X. Virchenko, N. A. Тао, Н. Х. Вірченко, Н. О. On the polyconvolution for the Fourier cosine, Fourier sine, and Kontorovich–Lebedev integral transforms |
| title | On the polyconvolution for the Fourier cosine, Fourier sine, and Kontorovich–Lebedev integral transforms |
| title_alt | Про полізгортку для косинус-Фур'є, синус-Фур'є та Конторовича - Лебедєва інтегральних перетворень |
| title_full | On the polyconvolution for the Fourier cosine, Fourier sine, and Kontorovich–Lebedev integral transforms |
| title_fullStr | On the polyconvolution for the Fourier cosine, Fourier sine, and Kontorovich–Lebedev integral transforms |
| title_full_unstemmed | On the polyconvolution for the Fourier cosine, Fourier sine, and Kontorovich–Lebedev integral transforms |
| title_short | On the polyconvolution for the Fourier cosine, Fourier sine, and Kontorovich–Lebedev integral transforms |
| title_sort | on the polyconvolution for the fourier cosine, fourier sine, and kontorovich–lebedev integral transforms |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2963 |
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