On the generalized convolution for $F_c$, $F_c$, and $K - L$ integral transforms
We study new generalized convolutions $f \overset{\gamma}{*} g$ with weight function $\gamma(y) = y$ for the Fourier cosine, Fourier sine, and Kontorovich-Lebedev integral transforms in weighted function spaces with two parameters $L(\mathbb{R}_{+}, x^{\alpha} e^{-\beta x} dx)$. These generalized...
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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. |
| author_sort | Thao, N. X. |
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| description | We study new generalized convolutions $f \overset{\gamma}{*} g$ with weight function $\gamma(y) = y$ for the Fourier cosine, Fourier sine, and Kontorovich-Lebedev
integral transforms in weighted function spaces with two parameters $L(\mathbb{R}_{+}, x^{\alpha} e^{-\beta x} dx)$.
These generalized convolutions satisfy the factorization equalities
$$F_{\left\{\frac SC\right\}} (f \overset{\gamma}{*} g)_{\left\{\frac 12\right\}}(y) = y (F_{\left\{\frac SC\right\}} f)(y) (K_{sy}g) \quad \forall y > 0$$
We establish a relationship between these generalized convolutions and known convolutions, and also relations that associate them with other convolution operators.
As an example, we use these new generalized convolutions for the solution of a class of integral equations with Toeplitz-plus-Hankel kernels and a class of systems of two integral equations with Toeplitz-plus-Hankel kernels. |
| first_indexed | 2026-03-24T02:25:45Z |
| format | Article |
| fulltext |
UDC 517.581
N. X. Thao (Hanoi Univ. Technol., Vietnam),
N. O. Virchenko (Nat. Techn. Univ. Ukraine “KPI”, Kyiv)
ON THE GENERALIZED CONVOLUTION
FOR Fc, Fs, AND K –L INTEGRAL TRANSFORMS*
ПРО УЗАГАЛЬНЕНУ ЗГОРТКУ
ДЛЯ Fc, Fs ТА K –L IНТЕГРАЛЬНИХ ПЕРЕТВОРЕНЬ
We study new generalized convolutions f
γ
∗ g with weight function γ(y) = y for the Fourier cosine, Fourier sine, and
Kontorovich – Lebedev integral transforms in weighted function spaces with two parameters L(R+, x
αe−βxdx). These
generalized convolutions satisfy the factorization equalities
F{ sc}(f
γ
∗ g){ 1
2
}(y) = y(F{ cs}f)(y)(Kiyg) ∀y > 0.
We establish a relationship between these generalized convolutions and known convolutions, and also relations that asso-
ciate them with other convolution operators. As an example, we use these new generalized convolutions for the solution
of a class of integral equations with Toeplitz-plus-Hankel kernels and a class of systems of two integral equations with
Toeplitz-plus-Hankel kernels.
Вивчаються новi узагальненi згортки f
γ
∗ g з ваговою функцiєю γ(y) = y для косинус-Фур’є, синус-Фур’є та
Конторовича – Лебедєва iнтегральних перетворень у вагових функцiональних просторах з двома параметрами
L(R+, x
αe−βxdx). Для цих узагальнених згорток справджуються функцiональнi рiвностi
F{ sc}(f
γ
∗ g){ 1
2
}(y) = y(F{ cs}f)(y)(Kiyg) ∀y > 0.
Одержано спiввiдношення мiж цими узагальненими згортками та вiдомими згортками, а також вiдповiднi спiввiд-
ношення з iншими операторами згорток. Як приклад, цi новi узагальненi згортки застосовано до класу iнтегральних
рiвнянь з сумою ядер Теплiца i Ганкеля, а також до класу системи двох iнтегральних рiвнянь з сумою ядер Теплiца
i Ганкеля.
Introduction. The commutative convolution of two functions f and g for the Fourier cosine trans-
form is well known [16]:
(f ∗ g)(x) =
1√
2π
∞∫
0
f(y)[g(x+ y) + g(|x− y|)]dy, x > 0. (0.1)
For f, g ∈ L1(R+), this convolution belongs to L1(R+), and the following identity holds:
Fc(f ∗ g)(y) = (Fcf)(y)(Fcg)(y) ∀y ∈ R, (0.2)
where Fc denotes the Fourier cosine transform [16]. In 1967, Kakichev gave a constructive method
for defining a convolution with weight function for an arbitrary integral transform (see [11]). On the
basis of this method, a convolution of two functions f and g with weight function γ(y) = sin y for
the Fourier sine transform was introduced in [11]:
(f
γ
∗
Fs
g)(x) =
1
2
√
2π
+∞∫
0
f(y)[sign(x+ y − 1)g(|x+ y − 1|) + sign(x− y + 1)g(|x− y + 1|)−
*This research was partially supported by Vietnam National Foundation for Science and Technology Development
(NAFOSTED) (grant No. 101.01.21.09).
c© N. X. THAO, N. O. VIRCHENKO, 2012
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 1 81
82 N. X. THAO, N. O. VIRCHENKO
−g(x+ y + 1)− sign(x− y − 1)g(|x− y − 1|)]dy, x > 0. (0.3)
For f, g ∈ L1(R+), the convolution f
γ
∗
Fc
g belongs to L1(R+) and the following factorization identity
holds:
Fs(f
γ
∗
Fs
g)(y) = sin y(Fsf)(y)(Fsg)(y) ∀y > 0. (0.4)
In 1998, Kakichev and Thao introduced a constructive method for defining a generalized convolution
with weight function for three arbitrary integral transforms (see [12]), which seems to be very impor-
tant in convolution theory. The following noncommutative generalized convolution of two functions
f and g for the Fourier sine and Fourier cosine transforms was studied in [16]:
(f ∗
1
g)(x) =
1√
2π
∞∫
0
f(u)[g(|x− u|)− g(x+ u)]du, x > 0. (0.5)
If f, g ∈ L1(R+), then (f ∗
1
g) belongs to L1(R+) and satisfies the following identity:
Fs(f ∗
1
g)(y) = (Fsf)(y)(Fcg)(y) ∀y > 0. (0.6)
Here, Fs is the Fourier sine transform [17]. The following commutative generalized convolution of
two functions f and g for the Fourier cosine and sine transforms was defined in [13]:
(f ∗
2
g)(x) =
1√
2π
∞∫
0
f(u)[sign(u− x)g(|u− x|) + g(u+ x)]du, x > 0. (0.7)
For f, g ∈ L1(R+), this generalized convolution belongs to L1(R+) and the following factorization
equality holds:
Fc(f ∗
2
g)(y) = (Fsf)(y)(Fsg)(y) ∀y > 0. (0.8)
The generalized convolution with weight function γ(y) = sin y for the Fourier cosine and sine
transforms of f and g has the form [14]
(f
γ
∗
3
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. (0.9)
For f, g ∈ L1(R+), (f
γ
∗
3
g)(x) also belongs to L1(R+) and
Fc(f
γ
∗
1
g)(y) = sin y (Fsf)(y)(Fcg)(y) ∀y > 0. (0.10)
The Kontorovich – Lebedev transform is of the form [17]
Kiy[f ] =
∞∫
0
Kiy(x)f(x)dx,
where Kix(t) is the modified Bessel function [2].
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 1
ON THE GENERALIZED CONVOLUTION FOR THE Fc, Fs, AND THE K − L . . . 83
Throughout this paper, we are interested in the following function spaces of two parameters:
Lα,βp (R+) ≡ Lp(R+;xαe−βxdx), α ∈ R, 0 < β 6 1.
The norm of a function f in this space is defined as follows:
‖f‖
Lα,βp (R+)
=
∞∫
0
|f(x)|pxαe−βxdx
1/p .
In recent years, there has been much interest in convolution theory for integral transforms, and
several interesting applications have been considered (see [4, 6, 7, 18, 20]), in particular, the integral
equations with Toeplitz-plus-Hankel kernel [10, 15, 19]
f(x) +
∞∫
0
[k1(x+ y) + k2(x− y)]f(y)dy = g(x), x > 0, (0.11)
where k1, k2, g are known functions and f is an unknown function. The problem of solving this
equation in a closed form in the general case of a Toeplitz kernel k1 and Hankel kernel k2 remains
open. Many partial cases of this equation can be solved in a closed form with the help of convolutions
and generalized convolutions and have interesting applications in biology and medicine (see [8, 9]).
In this paper, we construct and investigate two generalized convolutions for the Fourier cosine,
Fourier sine, and Kontorovich – Lebedev transforms in the function spaces Lα,βp (R+). Applications
to the solution, in a closed form, of a class of integral equations with Toeplitz-plus-Hankel kernels
and systems of two integral equations with Toeplitz-plus-Hankel kernels are considered.
1. Noncommutative generalized convolutions.
Definition 1.1. The generalized convolution of two functions f and g with weight function
γ(y) = y for the Fourier cosine, Fourier sine, and Kontorovich – Lebedev integral transforms is
defined as follows:
(f
γ
∗ g){1
2
}(x) =
1
2
∫
R2
+
v
[
sinh(x+ u)e−v cosh(x+u) ± sinh(x− u)e−v cosh(x−u)
]
f(u)g(v)dudv.
(1.1)
For convenience, throughout this paper we use the following notation:
θ{1
2
}(x, u, v) = sinh(x+ u)e−v cosh(x+u) ± sinh(x− u)e−v cosh(x−u), x > 0.
Theorem 1.1. Let f ∈ L1(R+), g ∈ L0,β
1 (R+), and 0 < β 6 1. Then the generalized
convolution (1.1) belongs to L1(R+) and satisfies the factorization equalities
F{ sc}(f
γ
∗ g){1
2
}(y) = y(F{ cs}f)(y)(Kiyg) ∀y > 0. (1.2)
Moreover, the following estimates are true:∥∥∥∥(f
γ
∗ g){1
2
}∥∥∥∥
L1(R+)
6 ‖f‖L1(R+)‖g‖L0,β
1 (R+)
. (1.3)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 1
84 N. X. THAO, N. O. VIRCHENKO
Furthermore, the generalized convolution (1.1) belongs to C0(R+), and the following Parseval-type
identities are true:
(f
γ
∗ g){1
2
}(x) =
√
2
π
∞∫
0
y(F{ cs}f)(y)(Kiyg)
{
sinxy
cosxy
}
dy. (1.4)
Proof. We have
∞∫
0
v| sinh(x+ u)−v cosh(x+u) ± sinh(x− u)−v cosh(x−u)|dx 6
6
∞∫
0
v
[
| sinh(x+ u)|−v cosh(x+u) + | sinh(x− u)|−v cosh(x+u)
]
dx =
=
∞∫
u
v sinh te−v cosh tdt+
∞∫
−u
v| sinh t|e−v cosh tdt = 2
∞∫
0
v sinh te−v cosh tdt = 2e−v. (1.5)
Using (1.5) and the Fubini theorem, we get
∞∫
0
|(f
γ
∗ g){1
2
}(x)|dx 6
∫
R2
+
e−v|f(u)||g(v)|dudv 6
6
∫
R2
+
e−βv|f(u)||g(v)|dudv = ‖f‖L1(R+)‖g‖L0,β
1 (R+)
. (1.6)
This implies the existence of the generalized convolution (1.1) in L1(R+) and the validity of relation
(1.3). Note that integral (1.1) is absolutely convergent. Indeed, we have
v sinh(x+ u)e−v(cosh(x+u)−1) = v sinh(x+ u)e−v sinh(x+u) tanh
x+u
2 6
6 v sinh(x+ u)
(
etanh
x+u
2
)−v sinh(x+u)
6 e− tanh x+u
2 6 1. (1.7)
Similarly,
|v sinh(x− u)e−v(cosh(x−u)−1)| 6 1.
Then relation (1.7) yields ∣∣∣θ{1
2
}(x, u, v)
∣∣∣ 6 2e−v. (1.8)
Therefore,
|(f
γ
∗ g){1
2
}(x)| 6 ‖f‖L1(R+)‖g‖L0,β
1 (R+)
, (1.9)
Further, using relation (12.1.1) from [3, p. 130] (Theorem 2) and the Fubini theorem, we get
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 1
ON THE GENERALIZED CONVOLUTION FOR THE Fc, Fs, AND THE K − L . . . 85
(f
γ
∗ g){1
2
}(x) =
2
π
∫
R3
+
yf(u)g(v)
{
sinxy cos yu
cosxy sin yu
}
Kiy(v)dudvdy =
=
√
2
π
∞∫
0
y(F{ cs}f)(y)(Kiyg)
{
sinxy
cosxy
}
dy.
Then the Parseval-type identity (1.4) holds for f ∈ L1(R+) and g ∈ L0,β
1 (R+), 0 < β 6 1.
Using the Parseval-type identity (1.4), relation (1.9), and the reverse formula of the Fourier cosine
and Fourier sine transforms, one can easily obtain the factorization property (1.2).
On the other hand, integral (1.1) is absolutely convergent in x, and it follows from the Riemann –
Lebesgue lemma that (f
γ
∗ g){1
2
} ∈ C0(R+).
Theorem 1.1 is proved.
An extension of Theorem 1.1 to the spaces Lp(R+) and L0,β
p (R+) is given as follows:
Theorem 1.2. Let f ∈ Lp(R+), g ∈ L0,β
q (R+),
1
p
+
1
q
= 1, p, q > 1, and 0 < β 6 1. Then
the generalized convolutions (1.1) exist for all x > 0, belong to Lα,γr (R+), α > −1, γ > 0, r > 1,
and satisfy the relation
‖(f
γ
∗ g){1
2
}‖Lα,γr (R+) 6 γ−(α+1)/rΓ1/r(α+ 1)‖f‖Lp(R+)‖g‖L0,β
q (R+)
. (1.10)
If, in addition, f ∈ L1(R+) ∩ Lp(R+), then the factorization equalities (1.2) are true. Moreover, the
generalized convolutions (1.1) belong to C0(R+), and the Parseval-type identities (1.3) are true.
Proof. Using the Hölder inequality and relations (1.5) and (1.8), we obtain
|(f
γ
∗ g){1
2
}(x)| 6 1
2
∫
R2
+
|f(u)|p2e−vdudv
1/p ∞∫
0
|g(v)|q2e−vdv
1/q =
=
∞∫
0
|f(u)|pdu
1/p ∞∫
0
|g(v)|qe−vdv
1/q = ‖f‖Lp(R+)‖g‖L0,β
q (R+)
. (1.11)
On the other hand, using relations (3.225.3) from [3, p. 165] and (1.11), we get
‖(f
γ
∗ g){1
2
}‖Lα,γr (R+) 6
∞∫
0
xαe−γxdx
1/r ‖f‖Lp(R+)‖g‖L0,β
q (R+)
=
=
(
γ−(α+1)Γ(α+ 1)
)1/r‖f‖Lp(R+)‖g‖L0,β
q (R+)
, α > −1, γ > 0.
Estimate (1.10) is proved.
Furthermore, since L0,β
q (R+) ⊂ L0,β
1 (R+), Theorem 1.1 shows that (f
γ
∗ g){1
2
} ∈ L1(R+).
Therefore, using (1.11), relation (12.1.1) from [3, p. 130] (Theorem 2), and the Fubini theorem, we
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 1
86 N. X. THAO, N. O. VIRCHENKO
obtain the Parseval-type identity (1.4), and, hence, the factorization identity (1.2) is true. Finally, it
follows from relation (1.11) and the Riemann – Lebesgue lemma that (f
γ
∗ g){1
2
} ∈ C0(R+).
Theorem 1.2 is proved.
Corollary 1.1. Under the same assumptions as in Theorem 1.2, the generalized convolutions
(1.1) exist for all positive x, are continuous, belong to Lp(R+), and satisfy the following estimate:
‖(f
γ
∗ g){1
2
}(x)‖Lp(R+) 6 ‖f‖Lp(R+)‖g‖L0,β
q (R+)
. (1.12)
In particular, for p = 2, we obtain a Parseval identity of the Fourier type:
∞∫
0
|(f
γ
∗ g){1
2
}(x)|2dx =
∞∫
0
|y(F{ cs}f)(y)(Kiyg)|2dy. (1.13)
Proof. Using relations (1.5), (1.7), and (1.8) and the Hölder inequality, we get
∞∫
0
|(f
γ
∗ g){1
2
}(x)|pdx 6
1
2p
∞∫
0
∫
R2
+
∣∣∣f(u)|pv| sinh(x+ u)e−v cosh(x+u)±
± sinh(x− u)e−v cosh(x−u)
∣∣∣dudv
∫
R2
+
|g(v)|qv| sinh(x+ u)e−v cosh(x+u)±
± sinh(x− u)e−v cosh(x−u)|dudv
p/q
dx 6
6
1
2p
∫
R2
+
|f(u)|p2e−vdudv
∞∫
0
|g(v)|q2e−vdv
p/q .
Therefore,
‖(f
γ
∗ g){1
2
}(x)‖Lp(R+) 6 ‖f‖Lp(R+)‖g‖L0,β
q (R+)
.
Estimate (1.12) is proved. Moreover, (f
γ
∗ g){1
2
}(x) is continuous and belongs to Lp(R+).
In the case p = 2, we obtain the Parseval identity
‖F{ cs}‖L2(R+) = ‖f‖L2(R+).
Therefore, the factorization identity (1.2) yields a Parseval identity of the Fourier type.
Corollary 1.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 1
ON THE GENERALIZED CONVOLUTION FOR THE Fc, Fs, AND THE K − L . . . 87
Corollary 1.2. 1. Let f ∈ L1(R+) ∩ L2(R+), g ∈ L0,β
2 (R+), and 0 < β 6 1. Then the
generalized convolutions (1.1) exist, are continuous, belong to Lα,γr (R+), r > 1, γ > 0, α > −1,
and satisfy the following estimate:
‖(f
γ
∗ g){1
2
}(x)‖Lα,γr (R+) 6 (γ−(α+1)Γ(α+ 1))1/r‖f‖L1(R+)‖g‖L0,β
2 (R+)
. (1.14)
Furthermore, these generalized convolutions satisfy the factorization identities (1.2) and Parseval-
type identities (1.4).
2. Let f, g ∈ L1(R+). Then the generalized convolutions (1.1) exist, belong to Lα,γr (R+), r > 1,
γ > 0, α > −1, and satisfy the following estimate:
‖(f
γ
∗ g){1
2
}(x)‖Lα,γr (R+) 6 (γ−(α+1)Γ(α+ 1))1/r‖f‖L1(R+)‖g‖L1(R+). (1.15)
Moreover, the factorization identities (1.2) and Parseval-type identities (1.4) are true.
Proof. Using the Schwarz inequality and relations (1.7), (1.8), and (1.5), we get
|(f
γ
∗ g){1
2
}(x)| 6 1
2
∫
R2
+
|f(u)|2e−vdv
1/2∫
R2
+
|f(u)||g(v)|22e−vdudv
1/2
6
6 ‖f‖L1(R+)‖g‖L0,β
2 (R+)
.
Therefore, using relation (3.225.3) from [5, p. 165], we obtain
‖(f
γ
∗ g){1
2
}‖Lα,γr (R+) 6 (γ−(α+1)Γ(α+ 1))1/r‖f‖L2(R+)‖g‖L1(R+).
This yields estimate (1.14). Moreover, by virtue of Theorem 1.2, the factorization identities (1.2) and
Parseval-type identities (1.4) are true.
On the other hand, using the Schwarz inequality and relations (1.7) and (1.8), we get
|(f
γ
∗ g){1
2
}(x)| 6 1
2
∞∫
0
|f(u)||g(v)|2e−vdudv
1/2
∫
R2
+
|f(u)||g(v)|2e−vdudv
1/2
6
6 ‖f‖L1(R+)‖g‖L1(R+).
Therefore, using relation (3.225.3) from [5, p. 165], we obtain (1.15). Furthermore, using Theo-
rem 1.1, we obtain the factorization identities (1.2) and Parseval-type identities (1.4).
Corollary 1.2 is proved.
Since L0,β
1 (R+) ⊂ L1(R+), by using relations (1.1), (0.7), and (0.5) one can easily prove the
following assertion:
Proposition 1.1. Let f ∈ L1(R+) and g ∈ L0,β
1 (R+). Then
(f ∗ g)1(x) =
√
π
2
∞∫
0
vg(v)
(
f(u) ∗
2
sinhu e−v coshu
)
(x)dv,
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 1
88 N. X. THAO, N. O. VIRCHENKO
(f ∗ g)2(x) = −
√
π
2
∞∫
0
vg(v)
(
f(u) ∗
1
sinhu e−v coshu
)
(x)dv.
Proof. Using (1.1) and (0.7), we obtain the representation for the convolution (f ∗ g)1(x). On
the other hand, one can easily prove the representation for the convolution (f ∗g)2(x) by using (0.5).
Using Theorem 1.1 and relations (0.6), (0.8), (0.4), and (0.10), we obtain the following proposi-
tion:
Proposition 1.2. Let f, g ∈ L1(R+), h ∈ L0,β
1 (R+), and 0 < β 6 1. Then the following
equalities are true:
(a) f ∗
1
(g ∗ h)1 = g ∗
1
(f ∗ h)1,
(b) f ∗
2
(g ∗ h)2 = ((f ∗
1
g) ∗ h)2,
(c) ((f ∗
2
g) ∗ h)2 = f ∗
1
(g ∗ h)1,
(d) f
γ
∗
3
(g ∗ h)1 = ((f
γ
∗
Fs
g) ∗ h)1.
2. Applications. Integral equations with Toeplitz-plus-Hankel kernels were studied in [15, 19].
In this section, we consider a partial class of these integrals, namely, the integral equations
f(x) +
∞∫
0
(ϕ ∗
1
f)(y)[k(x+ y)− k(x− y)]dy = h(x), (2.1)
where
k(t) =
1
2
∞∫
0
v sinh te−v cosh tg(v)dv
ϕ, g, and h are given, and f is unknown.
Theorem 2.1. Let ϕ, h ∈ L1(R+) and g ∈ L0,β
1 (R+) be such that 1 + y(Fsϕ)(y) ×
×(Kiyg) 6= 0 for all positive y. Then Eq. (2.1) has a unique solution in L1(R+), which is deter-
mined as follows:
f(x) = h(x)− (h ∗ l)(x),
where l ∈ L1(R+) is defined by the formula (Fcl)(y) =
y(Fsϕ)(y)(Kiyg)
1 + y(Fsϕ)(y)(Kiyg)
.
Proof. Using Theorem 1.1 and relation (0.6) and applying the Fourier cosine transform to both
sides of (2.1), we get
(Fcf)(y) + y(Fsϕ)(y)(Fcf)(y)(Kiyg) = (Fch)(y).
By virtue of the factorization equality (1.2), we obtain
(Fcf)(y) = (Fch)(y)
[
1− y(Fsϕ)(y)(Kiyg)
1 + y(Fsϕ)(y)(Kiyg)
]
. (2.2)
According to the Wiener – Levy theorem [1], there is a function l ∈ L1(R+) such that
(Fcl)(y) =
y(Fsϕ)(y)(Kiyg)
1 + y(Fsϕ)(y)(Kiyg)
. (2.3)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 1
ON THE GENERALIZED CONVOLUTION FOR THE Fc, Fs, AND THE K − L . . . 89
Using (2.2) and (2.3), we obtain the unique solution of (2.1) as follows:
f(x) = h(x)− (h ∗ l)(x).
Theorem 2.1 is proved.
Corollary 2.1. The necessary condition for the existence of a solution f ∈ L1(R+) of Eq. (2.1)
is as follows:
‖f‖L1(R+) >
‖h‖L1(R+)
1 + ‖ϕ‖L1(R+)‖g‖L0,β
1 (R+)
.
Consider the following system of two integral equations with Toeplitz-plus-Hankel kernels:
f(x) +
∞∫
0
g(y)[k1(x+ y) + k2(x− y)]dy = p(x), x > 0,
g(x) +
∞∫
0
f(y)[k3(x+ y) + k4(x− y)] = q(x),
(2.4)
where p, q, and ki, i = 1, 4, are given functions and f and g are unknown functions. The problem
of solving these systems in explicit form remains open. Here, we consider the class
f(x) +
∞∫
0
(g ∗ ξ)(y)[k1(x+ y) + k2(x− y)]dy = p(x), x > 0,
g(x) +
∞∫
0
(f ∗
1
η)(y)[k3(x+ y) + k4(x− y)]dy = q(x),
(2.5)
where
k1(t) =
1
2
∞∫
0
v sinh te−v cosh tϕ(v)dv = k2(t),
k3(t) =
1
2
∞∫
0
v sinh te−v cosh tψ(v)dv = −k4(t),
ϕ, ψ, p, q, ξ, and η are given functions, and f and g are unknown functions.
Theorem 2.2. Let ξ, η, p, q ∈ L1(R+) and ϕ,ψ ∈ L0,β
1 (R+) be such that
1− y2(Fcξ)(y)(Fcη)(y)(Kiyϕ)(Kiyψ) 6= 0 ∀y > 0.
Then system (2.5) has the following unique solution (f, g) ∈ L1(R+)× L1(R+):
f(x) = p(x) + (p ∗
2
l)(x)− ((ξ
γ
∗ ϕ)1 ∗
2
q)(x)− (((ξ
γ
∗ ϕ)1 ∗
2
q) ∗
2
l)(x),
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 1
90 N. X. THAO, N. O. VIRCHENKO
g(x) = l(x) + (l ∗ q)(x)− (p ∗
2
(η
γ
∗ ψ)1)(x)− (l ∗ (p ∗
2
(η
γ
∗ ψ)1))(x),
where l ∈ L1(R+) is defined by the formula
(Fcl)(y) =
y2(Fcξ)(y)(Fcη)(y)(Kiyϕ)(Kiyψ)
1 + y2(Fcξ)(y)(Fcη)(y)(Kiyϕ)(Kiyψ)
. (2.6)
Proof. Using the factorization identity (1.2) and relations (0.2) and (0.6) and applying the Fourier
sine transform and the Fourier cosine transform, respectively, to the two equations of system (2.5),
we obtain
(Fsf)(y) + y(Fcξ)(y)(Kiyϕ)(Fcg)(y) = (Fsp)(y),
y(Fcη)(y)(Kiyψ)(Fsf)(y) + (Fcg)(y) = (Fcq)(y).
(2.7)
Solving the above linear system with the use of the Cramer technique and relations (1.2) and (0.8),
we get
∆ = 1− Fs(ξ ∗ ϕ)1(y)Fs(η ∗ ψ)1(y) = 1− Fc((ξ ∗ ϕ)1 ∗
2
(η ∗ ψ)1(y).
By virtue of the Wiener – Levy theorem (see [1]), there exists a function l ∈ L1(R+) defined by (2.6),
whence
1
∆
= 1 + (Fcl)(y). (2.8)
Using (2.7), (2.8), (1.2), and (0.8), we obtain
(Fsf)(y) = [1 + (Fcl)(y)]
∣∣∣∣∣(Fsp)(y) Fs(ξ
γ
∗ ϕ)1(y)
(Fcq)(y) 1
∣∣∣∣∣ =
= (Fsp)(y) + Fs(p ∗
2
l)(y)− Fs((ξ
γ
∗ ϕ)1 ∗
2
q)(y)− Fs(((ξ
γ
∗ ϕ)1 ∗
2
q) ∗
2
l)(y).
Using inverse Fourier sine formula, we get
f(x) = p(x) + (p ∗
2
l)(x)− ((ξ
γ
∗ ϕ)1 ∗
2
q)(x)− (((ξ
γ
∗ ϕ)1 ∗
2
q) ∗
2
l)(x) ∈ L1(R+). (2.9)
By analogy, using (2.8), (1.2), (0.2), and (0.8), we obtain
(Fcg)(y) = (Fcl)(y) + Fc(l ∗ q)(y)− Fc(p ∗
2
(η
γ
∗ ψ)1)(y)− Fc(l ∗ (p ∗
2
(η
γ
∗ ψ)1))(y).
Hence,
g(x) = l(x) + (l ∗ q)(x)− (p ∗
2
(η
γ
∗ ψ)1)(x)− (l ∗ (p ∗
2
(η
γ
∗ ψ)1))(x) ∈ L1(R+). (2.10)
Using (2.9) and (2.10), we obtain the unique solution of system (2.5) in L1(R+)× L1(R+).
Theorem 2.2 is proved.
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ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 1
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| id | umjimathkievua-article-2557 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:25:45Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b3/cebbe374e2cd7d54ddedb2eca0c539b3.pdf |
| spelling | umjimathkievua-article-25572020-03-18T19:29:28Z On the generalized convolution for $F_c$, $F_c$, and $K - L$ integral transforms Про узагальнену згортку для $F_c$, $F_s$, та $K - L$ iнтегральних перетворень Thao, N. X. Virchenko, N. A. Тао, Н. Х. Вірченко, Н. О. We study new generalized convolutions $f \overset{\gamma}{*} g$ with weight function $\gamma(y) = y$ for the Fourier cosine, Fourier sine, and Kontorovich-Lebedev integral transforms in weighted function spaces with two parameters $L(\mathbb{R}_{+}, x^{\alpha} e^{-\beta x} dx)$. These generalized convolutions satisfy the factorization equalities $$F_{\left\{\frac SC\right\}} (f \overset{\gamma}{*} g)_{\left\{\frac 12\right\}}(y) = y (F_{\left\{\frac SC\right\}} f)(y) (K_{sy}g) \quad \forall y > 0$$ We establish a relationship between these generalized convolutions and known convolutions, and also relations that associate them with other convolution operators. As an example, we use these new generalized convolutions for the solution of a class of integral equations with Toeplitz-plus-Hankel kernels and a class of systems of two integral equations with Toeplitz-plus-Hankel kernels. Вивчаються новi узагальненi згортки $f \overset{\gamma}{*} g$ з ваговою функцiєю $\gamma(y) = y$ для косинус-Фур’є, синус-Фур’є та Конторовича – Лебедєва iнтегральних перетворень у вагових функцiональних просторах з двома параметрами $L(\mathbb{R}_{+}, x^{\alpha} e^{-\beta x} dx)$. Для цих узагальнених згорток справджуються функцiональнi рiвностi $$F_{\left\{\frac SC\right\}} (f \overset{\gamma}{*} g)_{\left\{\frac 12\right\}}(y) = y (F_{\left\{\frac SC\right\}} f)(y) (K_{sy}g) \quad \forall y > 0.$$ Одержано спiввiдношення мiж цими узагальненими згортками та вiдомими згортками, а також вiдповiднi спiввiдношення з iншими операторами згорток. Як приклад, цi новi узагальненi згортки застосовано до класу iнтегральних рiвнянь з сумою ядер Теплiца i Ганкеля, а також до класу системи двох iнтегральних рiвнянь з сумою ядер Теплiца i Ганкеля. Institute of Mathematics, NAS of Ukraine 2012-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2557 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 1 (2012); 81-91 Український математичний журнал; Том 64 № 1 (2012); 81-91 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2557/1873 https://umj.imath.kiev.ua/index.php/umj/article/view/2557/1874 Copyright (c) 2012 Thao N. X.; Virchenko N. A. |
| spellingShingle | Thao, N. X. Virchenko, N. A. Тао, Н. Х. Вірченко, Н. О. On the generalized convolution for $F_c$, $F_c$, and $K - L$ integral transforms |
| title | On the generalized convolution for $F_c$, $F_c$, and $K - L$ integral transforms |
| title_alt | Про узагальнену згортку для $F_c$, $F_s$, та $K - L$ iнтегральних перетворень |
| title_full | On the generalized convolution for $F_c$, $F_c$, and $K - L$ integral transforms |
| title_fullStr | On the generalized convolution for $F_c$, $F_c$, and $K - L$ integral transforms |
| title_full_unstemmed | On the generalized convolution for $F_c$, $F_c$, and $K - L$ integral transforms |
| title_short | On the generalized convolution for $F_c$, $F_c$, and $K - L$ integral transforms |
| title_sort | on the generalized convolution for $f_c$, $f_c$, and $k - l$ integral transforms |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2557 |
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