On the Fourier sine and Kontorovich–Lebedev generalized convolution transforms and applications
UDC 517.5 We study a generalized convolutions for the Fourier sine and Kontorovich - Lebedev transforms $ (h\underset{F_s,K}\ast f)(x)$ in a two-parameter function space $L_p^{\alpha, \beta}(\Bbb R_+)$. We obtain several estimates for the norms and prove a Young-type inequality for this generalized...
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2020
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| author | Tuan , T. Tuan , T. |
| author_facet | Tuan , T. Tuan , T. |
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We study a generalized convolutions for the Fourier sine and Kontorovich - Lebedev transforms $ (h\underset{F_s,K}\ast f)(x)$ in a two-parameter function space $L_p^{\alpha, \beta}(\Bbb R_+)$. We obtain several estimates for the norms and prove a Young-type inequality for this generalized convolution.
We impose necessary and sufficient conditions on the kernel $h$ to ensure that the generalized convolution transform$$D_h\colon f\mapsto D_{h}[f] = \left(1-\dfrac{d^2}{dx^2}\right)(h\underset{F_s,K} \ast f)(x)$$is a unitary operator in $L_2(\Bbb R_+)$ (Watson-type theorem) and derive its inverse formula. Finally, we apply these results to an integrodifferential equation and obtain an estimate for the solution in the $L_p$-norm. |
| first_indexed | 2026-03-24T02:22:29Z |
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UDC 517.5
T. Tuan (Electric Power Univ., Hanoi, Vietnam)
ON THE FOURIER SINE AND KONTOROVICH – LEBEDEV
GENERALIZED CONVOLUTION TRANSFORMS AND APPLICATIONS
ПРО СИНУС-ПЕРЕТВОРЕННЯ ФУР’Є I ПЕРЕТВОРЕННЯ
КОНТОРОВИЧА – ЛЕБЕДЄВА УЗАГАЛЬНЕНИХ ЗГОРТОК
TA ЇХ ЗАСТОСУВАННЯ
We study a generalized convolutions for the Fourier sine and Kontorovich – Lebedev transforms (h \ast
Fs,K
f)(x) in a two-
parameter function space L\alpha ,\beta
p (\BbbR +). We obtain several estimates for the norms and prove a Young-type inequality for this
generalized convolution.
We impose necessary and sufficient conditions on the kernel h to ensure that the generalized convolution transform
Dh : f \mapsto \rightarrow Dh[f ] =
\biggl(
1 - d2
dx2
\biggr)
(h \ast
Fs,K
f)(x)
is a unitary operator in L2(\BbbR +) (Watson-type theorem) and derive its inverse formula. Finally, we apply these results to
an integrodifferential equation and obtain an estimate for the solution in the Lp -norm.
Вивчається узагальнена згортка для синус-перетворення Фур’є i перетворення Конторовича – Лебедєва (h \ast
Fs,K
f)(x)
у двопараметричному просторi функцiй L\alpha ,\beta
p (\BbbR +). Отримано кiлька оцiнок для норм i встановлено нерiвнiсть типу
Юнга для цiєї узагальненoї згортки. Введено необхiднi та достатнi умови для ядра h, за яких перетворення
узагальненої згортки
Dh : f \mapsto \rightarrow Dh[f ] =
\biggl(
1 - d2
dx2
\biggr)
(h \ast
Fs,K
f)(x)
— це унiтарний оператор в L2(\BbbR +) (теорема типу Ватсона). Отримано формулу для оберненого перетворення. Крiм
того, цi результати застосовано до iнтегро-диференцiального рiвняння та отримано оцiнку для його розв’язку в
Lp -нормi.
1. Introduction. The Kontorovich – Lebedev integral transform was introduced by M. J. Kontorovich
and N. N. Lebedev during 1938 – 1939 (see [8, 14])
(Kf)(y) =
2
\pi 2
\infty \int
0
Kiy(x)f(x)
dx
x
, y > 0.
Here, the transform kernel contains the Macdonald function K\nu (x) (see [2]) of the pure imaginary in-
dex \nu = iy . There are several integral representations for the Macdonald function, and the following
one is very useful subsequently [2, 8, 17]:
Kiy(x) =
\infty \int
0
e - x coshu \mathrm{c}\mathrm{o}\mathrm{s} yu du, x > 0. (1.1)
The inverse Kontorovich – Lebedev transform is of the form [8, 14]
f(x) = K - 1[g](x) =
\infty \int
0
y \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\pi y)Kiy(x)g(y) dy.
c\bigcirc T. TUAN, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2 267
268 T. TUAN
Here, g(y) = (Kf)(y).
A generalized convolution for the Fourier sine and the Kontorovich – Lebedev integral transforms
has been studied in [12]:
\bigl(
h \ast
Fs,K
f
\bigr)
(x) =
1
\pi 2
\int
\BbbR 2
+
1
u
\Bigl[
e - u cosh(x - v) - e - u cosh(x+v)
\Bigr]
h(u)f(v) dudv, x > 0. (1.2)
Here, the Fourier sine integral transform is defined by [5, 11]
(Fsf)(y) =
\sqrt{}
2
\pi
\infty \int
0
f(x) \mathrm{s}\mathrm{i}\mathrm{n}xy dx.
The existence of the generalized convolution (1.2) for two functions in L1(\BbbR +) with weight and
its application to solving integral equations of generalized convolution type were studied in [12].
Namely, for h(x) \in L1
\Bigl(
\BbbR +, x
- 3/2 dx
\Bigr)
, f(x) \in L1(\BbbR +), the following factorization equality holds
(see [12]):
Fs(h \ast
Fs,K
f)(y) = (Kh)(y)(Fsf)(y) \forall y > 0. (1.3)
In any convolution h \ast f of two functions h and f, if we fix a function h and let f vary in a certain
function space, then we can define convolution transforms of the form f \rightarrow D(h \ast f), where D is a
certain (differential) operator. The most well-known integral transforms constructed by that way are
the Watson transforms that are related to the Mellin convolution and the Mellin transform [11]
f(x) \mapsto - \rightarrow g(x) =
\infty \int
0
k(xy)f(y) dy.
Recently, several authors have been interested in the convolution transforms of this type [3, 7,
13, 15]. In this paper, we will study the transforms f \rightarrow D(h \ast
Fs,K
f), where h \ast
Fs,K
f is the
generalized convolution (1.2). The case D is the identity operator is considered in Section 2, where
we study operator properties for the generalized convolution (1.2) in the two parameter Lebesgue
space L\alpha ,\beta
p (\BbbR +). In particular, we obtain the Young theorem and the Young inequality for this
generalized convolution. In Section 3, for the differential operators D = I - d2
dx2
, we derive a
necessary and sufficient condition such that the corresponding transforms are unitary on L2(\BbbR +),
and we draw the inverse transforms (a Watson-type theorem). Finally, in Section 4, we obtain the
solution in closed form of an integrodifferential equation related to the generalized convolution (1.2),
and an Lp-norm estimate of the solution with respect to the data.
2. Generalized convolution operator properties. In this section, we will prove several norm
properties of the generalized convolution (1.2). Throughout the paper, we are interested in the
following family of two parameter Lebesgue spaces.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
ON THE FOURIER SINE AND KONTOROVICH – LEBEDEV GENERALIZED CONVOLUTION . . . 269
Definition 2.1 [16]. For \alpha \in \BbbR , 0 < \beta \leq 1, we denote by L\alpha ,\beta
p (\BbbR +) the normed space of all
measurable functions f(x) on \BbbR + such that
\infty \int
0
| f(x)| pK0(\beta x)x
\alpha dx < \infty
with the norm
\| f\|
L\alpha ,\beta
p (\BbbR +)
=
\left( \infty \int
0
| f(x)| pK0(\beta x)x
\alpha dx
\right)
1
p
.
The boundedness of the generalized convolution (1.2) on the space L1(\BbbR +) is shown in the
following theorem.
Theorem 2.1. Let h \in L - 1,\beta
1 (\BbbR +) and f \in L1(\BbbR +), 0 < \beta < 1. Then the generalized
convolution (1.2) exists for almost all x > 0, belongs to L1(\BbbR +), and the following estimate holds:
\| (h \ast
Fs,K
f)\| L1(\BbbR +) \leq
2
\pi 2
\| h\|
L - 1,\beta
1 (\BbbR +)
\| f\| L1(\BbbR +).
Moreover, the factorization property (1.3) holds true. Furthermore, convolution (1.2) belongs to
C1
0 (\BbbR +), and the following Parseval-type equality takes place, for all x > 0:
(h \ast
Fs,K
f)(x) =
\sqrt{}
2
\pi
\infty \int
0
(Kh)(y)(Fsf)(y) \mathrm{s}\mathrm{i}\mathrm{n}xy dy. (2.1)
Proof. By using formula (1.1), we obtain
1
2
\infty \int
0
(e - u cosh(x+v) + e - u cosh(x - v)) dx = K0(u). (2.2)
Recalling that K0(u) \leq K0(\beta u), 0 < \beta \leq 1 [14], we have
\| (h \ast
Fs,K
f)\| L1(\BbbR +) \leq
2
\pi 2
\int
\BbbR 2
+
| h(u)|
u
K0(u)| f(v)| dudv \leq
\leq 2
\pi 2
\int
\BbbR 2
+
| h(u)|
u
K0(\beta u)| f(v)| dudv =
2
\pi 2
\| h\|
L - 1,\beta
1 (\BbbR +)
\| f\| L1(\BbbR +).
It shows that (h \ast
Fs,K
f)(x) belongs to L1(\BbbR +). We now prove the Parseval-type equality. By using
formula 2.16.48.19 in [9]
\infty \int
0
\mathrm{c}\mathrm{o}\mathrm{s} by Kiy(u)dy =
\pi
2
e - u cosh b,
we get
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
270 T. TUAN
(h \ast
Fs,K
f)(x) =
1
\pi 2
\int
\BbbR 2
+
1
u
[e - u cosh(x - v) - e - u cosh(x+v)]h(u)f(v) dudv =
=
1
\pi 2
\int
\BbbR 3
+
2
\pi
1
u
h(u)f(v)Kiy(u)[\mathrm{c}\mathrm{o}\mathrm{s}(x - v)y - \mathrm{c}\mathrm{o}\mathrm{s}(x+ v)y] dydudv =
=
4
\pi 3
\int
\BbbR 3
+
1
u
h(u)f(v)Kiy(u) \mathrm{s}\mathrm{i}\mathrm{n}xy \mathrm{s}\mathrm{i}\mathrm{n} vy dydudv.
By using the uniform estimate [14]
| Kiy(u)| \leq e - \delta yK0(u \mathrm{c}\mathrm{o}\mathrm{s} \delta ), 0 \leq \delta <
\pi
2
,
with \delta = \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\beta , we have\int
\BbbR 3
+
\bigm| \bigm| \bigm| \bigm| 1uh(u)f(v)Kiy(u) \mathrm{s}\mathrm{i}\mathrm{n}xy \mathrm{s}\mathrm{i}\mathrm{n} vy
\bigm| \bigm| \bigm| \bigm| dydudv \leq
\leq
\infty \int
0
1
u
| h(u)| K0(\beta u) du
\infty \int
0
| f(v)| dv
\infty \int
0
e - y arccos\beta dy =
=
1
\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\beta
\| h\|
L - 1,\beta
1 (\BbbR +)
\| f\| L1(\BbbR +) < \infty .
It means that we can apply Fubini’s theorem to obtain
(h \ast
Fs,K
f)(x) =
4
\pi 3
\int
\BbbR 3
+
1
u
h(u)f(v)Kiy(u) \mathrm{s}\mathrm{i}\mathrm{n}xy \mathrm{s}\mathrm{i}\mathrm{n} vy dydudv =
=
\sqrt{}
2
\pi
\infty \int
0
\left( 2
\pi 2
\infty \int
0
1
u
Kiy(u)h(u)du
\right) \left( \sqrt{}
2
\pi
\infty \int
0
f(v) \mathrm{s}\mathrm{i}\mathrm{n} vy dv
\right) \mathrm{s}\mathrm{i}\mathrm{n}xy dy =
=
\sqrt{}
2
\pi
\infty \int
0
(Kh)(y)(Fsf)(y) \mathrm{s}\mathrm{i}\mathrm{n}xy dy.
That is the Parseval identity (2.1). Since
| (Kh)(y)| \leq \| h\|
L - 1,\beta
1 (\BbbR +)
e - y arccos\beta , | (Fsf)(y)| \leq \| f\| L1(\BbbR +),
it follows that (1 + y)(Kh)(y)(Fsf)(y) \in L1(\BbbR +). Thus, the Parseval identity (2.1) shows that
(h \ast
Fs,K
f)(x) is the Fourier sine transform of a function from L1(\BbbR +), differentiable, and, therefore,
belongs to C1
0 (\BbbR +).
Theorem 2.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
ON THE FOURIER SINE AND KONTOROVICH – LEBEDEV GENERALIZED CONVOLUTION . . . 271
Theorem 2.2. Let 1 < p < \infty be a real number and q be its conjugate exponent, i.e., 1/p +
+ 1/q = 1. Then, for any h \in L - p,\beta
p (\BbbR +) and f \in Lq(\BbbR +), the generalized convolution h \ast
Fs,K
f
is a bounded function on \BbbR + . Moreover, h \ast
Fs,K
f belongs to L\alpha ,\gamma
r (\BbbR +), 1 \leq r < \infty , \alpha > - 1,
0 < \gamma \leq 1, and
\| (h \ast
Fs,K
f)\| L\alpha ,\gamma
r (\BbbR +) \leq C1/r
\alpha ,\gamma \| h\| L - p,\beta
p (\BbbR +)
\| f\| Lq(\BbbR +), (2.3)
where
C\alpha ,\gamma =
2r+\alpha - 1
\pi 2r\gamma \alpha +1
\Gamma 2
\biggl(
\alpha + 1
2
\biggr)
.
Proof. By using the integral representation (2.2) for the function K0(u), Hölder’s inequality,
and the fact that e - u cosh(x+v) + e - u cosh(x - v) \leq 2e - u for all positives u, v, and x, we get
| (h \ast
Fs,K
f)(x)| \leq 1
\pi 2
\int
\BbbR 2
+
\bigm| \bigm| \bigm| \bigm| h(u)u
\bigm| \bigm| \bigm| \bigm| | f(v)| [e - u cosh(x+v) + e - u cosh(x - v)] dudv \leq
\leq 1
\pi 2
\left( \int
\BbbR 2
+
\bigm| \bigm| \bigm| \bigm| h(u)u
\bigm| \bigm| \bigm| \bigm| p [e - u cosh(x+v) + e - u cosh(x - v)] dudv
\right)
1
p
\times
\times
\left( \int
\BbbR 2
+
| f(v)| q[e - u cosh(x+v) + e - u cosh(x - v)] dudv
\right)
1
q
\leq
\leq 2
\pi 2
\left( \infty \int
0
\bigm| \bigm| \bigm| \bigm| h(u)u
\bigm| \bigm| \bigm| \bigm| pK0(u) du
\right)
1
p
\| f\| Lq(\BbbR +).
Therefore, the generalized convolution is a bounded function. Moreover, in view of formula (2.16.2.2)
in [9] we get
\| (h \ast
Fs,K
f)\| L\alpha ,\gamma
r (\BbbR +) \leq
2
\pi 2
\| h\|
L - p,\beta
p (\BbbR +)
\| f\| Lq(\BbbR +)
\left( \infty \int
0
x\alpha K0(\gamma x) dx
\right)
1
r
=
=
2
\pi 2
(2\gamma ) - 1/r
\Bigl( \gamma
2
\Bigr) - \alpha /r
\Gamma 2/r
\biggl(
\alpha + 1
2
\biggr)
\| h\|
L - p,\beta
p (\BbbR +)
\| f\| Lq(\BbbR +), \alpha > - 1.
It yields (2.3).
Theorem 2.2 is proved.
By a similar argument as in the proof of Theorem 2.1, one can easily prove the following lemma.
Lemma 2.1. Let h \in L - 2,\beta
2 (\BbbR +), 0 < \beta < 1, and f \in L2(\BbbR +). Then the generalized
convolution (1.2) satisfies the factorization equality (1.3). Furthermore, the following generalized
Parseval identity holds:
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
272 T. TUAN
(h \ast
Fs,K
f)(x) =
\sqrt{}
2
\pi
\infty \int
0
(Kh)(y)(Fsf)(y) \mathrm{s}\mathrm{i}\mathrm{n}xy dy, (2.4)
where the integral is understood in L2(\BbbR +) norm, if necessary.
Next, we will prove a Young-type theorem for the generalized convolution (1.2).
Theorem 2.3 (Young-type theorem). Let p, q, r be real numbers in (1,\infty ) such that 1/p+1/q+
+ 1/r = 2 and let f \in L - p,\beta
p (\BbbR +), 0 < \beta < 1, g \in Lq(\BbbR +), h \in Lr(\BbbR +). Then\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\infty \int
0
(f \ast
Fs,K
g)(x)h(x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq 2
p - 1
p
\pi 2
\| f\|
L - p,\beta
p (\BbbR +)
\| g\| Lq(\BbbR +)\| h\| Lr(\BbbR +).
Proof. Let p1, q1, r1 be the conjugate exponents of p, q, r, respectively, it means
1
p
+
1
p1
=
1
q
+
1
q1
=
1
r
+
1
r1
= 1.
Then
1
p1
+
1
q1
+
1
r1
= 1. Put
F (x, u, v) = | g(v)|
q
p1 | h(x)|
r
p1 | e - u cosh(x - v) - e - u cosh(x+v)|
1
p1 ,
G(x, u, v) =
\bigm| \bigm| \bigm| \bigm| f(u)u
\bigm| \bigm| \bigm| \bigm|
p
q1
| h(x)|
r
q1 | e - u cosh(x - v) - e - u cosh(x+v)|
1
q1 ,
H(x, u, v) =
\bigm| \bigm| \bigm| \bigm| f(u)u
\bigm| \bigm| \bigm| \bigm|
p
r1
| g(v)|
q
r1 | e - u cosh(x - v) - e - u cosh(x+v)|
1
r1 .
We have
F (x, u, v)G(x, u, v)H(x, u, v) =
\bigm| \bigm| \bigm| \bigm| f(u)u
\bigm| \bigm| \bigm| \bigm| | g(v)| | h(x)| | e - u cosh(x - v) - e - u cosh(x+v)| . (2.5)
Furthermore, in the space Lp1(\BbbR 3
+) we obtain
\| F\| p1
Lp1 (\BbbR
3
+)
=
\int
\BbbR 3
+
| g(v)| q| h(x)| r| e - u cosh(x - v) - e - u cosh(x+v)| dudvdx \leq
\leq 2
\int
\BbbR 3
+
| g(v)| q| h(x)| re - u dudvdx =
= 2\| g\| qLq(\BbbR +)\| h\|
r
Lr(\BbbR +). (2.6)
On the other hand, by the fact that K0(u) \leq K0(\beta u), for 0 < \beta < 1,
\| G\| p1
Lq1 (\BbbR
3
+)
=
\int
\BbbR 3
+
\bigm| \bigm| \bigm| \bigm| f(u)u
\bigm| \bigm| \bigm| \bigm| p | h(x)| r| e - u cosh(x+v) - e - u cosh(x - v)| dudvdx \leq
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
ON THE FOURIER SINE AND KONTOROVICH – LEBEDEV GENERALIZED CONVOLUTION . . . 273
\leq
\int
\BbbR 2
+
\bigm| \bigm| \bigm| \bigm| f(u)u
\bigm| \bigm| \bigm| \bigm| pK0(\beta u)| h(x)| r dudx =
= \| f\| p
L - p,\beta
p (\BbbR +)
\| h\| rLr(\BbbR +), (2.7)
and, similarly,
\| H\| r1
Lr1 (\BbbR
3
+)
=
\int
\BbbR 3
+
\bigm| \bigm| \bigm| \bigm| f(u)u
\bigm| \bigm| \bigm| \bigm| p | g(v)| q| e - u cosh(x - v) - e - u cosh(x+v)| dudvdx \leq
\leq
\int
\BbbR 2
+
\bigm| \bigm| \bigm| \bigm| f(u)u
\bigm| \bigm| \bigm| \bigm| pK0(\beta u)| g(v)| r dudv =
= \| f\| p
L - p,\beta
p (\BbbR +)
\| g\| qLq(\BbbR +). (2.8)
Hence, from (2.6), (2.7) and (2.8), we have
\| F\| Lp1 (\BbbR
3
+)\| G\| Lq1 (\BbbR
3
+)\| H\| Lr1 (\BbbR
3
+) \leq 2
p - 1
p \| f\|
L - p,\beta
p (\BbbR +)
\| g\| Lq(\BbbR +)\| h\| Lr(\BbbR +). (2.9)
From (2.5) and (2.9), by the three-function form of the Hölder inequality [1], we have\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\infty \int
0
(f \ast
Fs,K
g)(x)h(x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq 1
\pi 2
\int
\BbbR 3
+
\bigm| \bigm| \bigm| \bigm| f(u)u
\bigm| \bigm| \bigm| \bigm| | g(v)| | h(x)| | e - u cosh(x - v) - e - u cosh(x+v)| dudvdx =
=
1
\pi 2
\int
\BbbR 3
+
F (x, u, v)G(x, u, v)H(x, u, v) dudvdx \leq
\leq 1
\pi 2
\| F\| Lp1 (\BbbR
3
+)\| G\| Lq1 (\BbbR
3
+)\| H\| Lr1 (\BbbR
3
+) \leq
\leq 2
p - 1
p
\pi 2
\| f\|
L - p,\beta
p (\BbbR +)
\| g\| Lq(\BbbR +)\| h\| Lr(\BbbR +).
Theorem 2.3 is proved.
The following Young-type inequality is a direct corollary of the above theorem.
Corollary 2.1 (Young-type inequality). Let 1 < p, q, r < \infty be such that 1/p + 1/q = 1 + 1/r
and let f \in L - p,\beta
p (\BbbR +), 0 < \beta < 1, g \in Lq(\BbbR +). Then the generalized convolution (1.2) is
well-defined in Lr(\BbbR +). Moreover, the following inequality holds:
\| (f \ast
Fs,K
g)\| Lr(\BbbR +) \leq
2
p - 1
p
\pi 2
\| f\|
L - p,\beta
p (\BbbR +)
\| g\| Lq(\BbbR +). (2.10)
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274 T. TUAN
3. A Watson-type theorem. An important class of integral transforms is unitary transforms. In
this section, for D = I - d2
dx2
, we give a necessary and sufficient condition for a kernel h such that
the generalized convolution transform
Dh : f \mapsto \rightarrow g = Dh[f ] =
\biggl(
1 - d2
dx2
\biggr)
(h \ast
Fs,K
f)(x)
is a unitary operator in L2(\BbbR +), and derive its inverse formula.
Theorem 3.1. Let h \in L - 2,\beta
2 (\BbbR +), 0 < \beta < 1. Then the condition
| (Kh)(y)| = 1
1 + y2
(3.1)
is necessary and sufficient to ensure that the transformation f \rightarrow g, given by formula
g(x) =
1
\pi 2
\biggl(
1 - d2
dx2
\biggr) \int
\BbbR 2
+
1
u
(e - u cosh(x - v) - e - u cosh(x+v))h(u)f(v) dudv (3.2)
is unitary in L2(\BbbR +). Moreover, the inverse transformation can be written in the conjugate symmetric
form
f(x) =
1
\pi 2
\biggl(
1 - d2
dx2
\biggr) \int
\BbbR 2
+
1
u
(e - u cosh(x - v) - e - u cosh(x+v))h(u)f(v) dudv. (3.3)
Proof. Sufficiency. Suppose that the function h satisfies condition (3.1). Applying Lemma 2.1,
it is easy to see that the generalized convolution transform (3.2) can be written in the form
g(x) =
\sqrt{}
2
\pi
\biggl(
1 - d2
dx2
\biggr) \infty \int
0
(Kh)(y)(Fsf)(y) \mathrm{s}\mathrm{i}\mathrm{n}xy dy,
or, equivalently,
g(x) =
\biggl(
1 - d2
dx2
\biggr)
Fs
\Bigl[
(Kh)(y)(Fsf)(y)
\Bigr]
(x).
It is well-known that h(y), yh(y), y2h(y) \in L2(\BbbR +) if and only if (Fh)(x),
d
dx
(Fh)(x),
d2
dx2
(Fh)(x) \in L2(\BbbR +) (Theorem 68 [11, p. 92]). Moreover,
\biggl(
1 - d2
dx2
\biggr)
(Fsh)(x) = Fs
\Bigl[
(1 + y2)h(y)
\Bigr]
(x). (3.4)
Condition (3.1) shows that (1 + y2)(Kh)(y) is bounded. Therefore (1 + y2)(Kh)(y)(Fsf)(y) \in
\in L2(\BbbR +), and formula (3.4) yields
g(x) = Fs
\Bigl[
(1 + y2)(Kh)(y)(Fsf)(y)
\Bigr]
(x) \in L2(\BbbR +).
Applying the Fourier sine transform to both sides of the above equation, we have
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ON THE FOURIER SINE AND KONTOROVICH – LEBEDEV GENERALIZED CONVOLUTION . . . 275
(Fsg)(y) = (1 + y2)(Kh)(y)(Fsf)(y).
Besides, from the Plancherel theorem for the Fourier sine transform \| Fsf\| L2(\BbbR +) = \| f\| L2(\BbbR +), and
condition (3.1), it is easy to see that \| f\| L2(\BbbR +) = \| g\| L2(\BbbR +), which implies that transform (3.2) is
unitary. Again from condition (3.1) we obtain
(Kh)(y)(Fsg)(y) = (Fsf)(y).
Thus, in the same manner as above it corresponds to (3.3) and the inversion formula of transform
(3.2) follows.
Necessity. Suppose that transform (3.2) is unitary in L2(\BbbR +) and the inversion formula is defined
by (3.3). Then, by using the Parseval-type identity (2.4), the Plancherel theorem for the Fourier sine
transform, and formula 4.5.68 in [2], we obtain
\| g\| L2(\BbbR +) = \| (Kh)(y)(Fsf)(y)\| L2(\BbbR +) = \| Fsf\| L2(\BbbR +) = \| f\| L2(\BbbR +).
The middle equality holds for all f \in L2(\BbbR +) if and only if h satisfies the condition (3.1).
Theorem 3.1 is proved.
4. A class of integrodifferential equations. Not many integrodifferential equations can be
solved in closed form despite their useful applications (see [4]). In particular, no applications of
convolution type transforms for solving integrodifferential equations were found in recent investiga-
tions [3, 7, 13, 15]. In this section, we apply the Fourier sine and Kontorovich – Lebedev generalized
convolution to investigate a class of integrodifferential equations, which seems to be difficult to be
solved in closed form by using other techniques.
To introduce a class of integrodifferential equation, we recall the generalized convolution for the
Fourier sine and Fourier cosine transforms, which is of the form (see [5])
(f \ast
1
g)(x) =
1\surd
2\pi
\infty \int
0
[f(x+ y) - f(| x - y| )]g(y) dy, x > 0. (4.1)
For f, g \in L1(\BbbR +), we have f \ast
1
g \in L1(\BbbR +), and the following factorization equality holds:
Fs(f \ast
1
g)(y) = (Fsf)(y)(Fcg)(y).
Here, the Fourier cosine transform is defined by [5, 11]
(Fcf)(y) =
\sqrt{}
2
\pi
\infty \int
0
f(x) \mathrm{c}\mathrm{o}\mathrm{s}xy dx.
We consider the integrodifferential equation
f(x) - f \prime \prime (x) + (Dhf)(x) = (h \ast
Fs,K
g)(x),
f(0) = 0,
\mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow \infty
f(x) = \mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow \infty
f \prime (x) = 0.
(4.2)
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276 T. TUAN
Here, h \in L - 1,\beta
1 (\BbbR +), 0 < \beta < 1, g \in L1(\BbbR +) are given functions, and f \in C2(\BbbR +) \cap L1(\BbbR +) is
the unknown function.
In order to get a solution of the above problem, note that, for f \in C2(\BbbR +) \cap L1(\BbbR +), such that
f(0) = 0, \mathrm{l}\mathrm{i}\mathrm{m}x\rightarrow \infty f(x) = \mathrm{l}\mathrm{i}\mathrm{m}x\rightarrow \infty f \prime (x) = 0, we have
(Fsf
\prime \prime
)(y) =
\sqrt{}
2
\pi
\infty \int
0
f
\prime \prime
(x) \mathrm{s}\mathrm{i}\mathrm{n}xy dx =
=
\sqrt{}
2
\pi
\left\{ f \prime (x) \mathrm{s}\mathrm{i}\mathrm{n}xy
\bigm| \bigm| \bigm| \infty
x=0
- y
\infty \int
0
f \prime (x) \mathrm{c}\mathrm{o}\mathrm{s}xy dx
\right\} =
= -
\sqrt{}
2
\pi
y
\left\{ f(x) \mathrm{c}\mathrm{o}\mathrm{s}xy
\bigm| \bigm| \bigm| \infty
x=0
+ y
\infty \int
0
f(x) \mathrm{s}\mathrm{i}\mathrm{n}xy dx
\right\} = - y2(Fsf)(y). (4.3)
Lemma 4.1. Let f \in C1
0 (\BbbR +) \cap L1(\BbbR +). Then g(x) = (f(y) \ast
1
e - y)(x) is twice differentiable,
g(0) = 0, and \mathrm{l}\mathrm{i}\mathrm{m}x\rightarrow \infty g(x) = \mathrm{l}\mathrm{i}\mathrm{m}x\rightarrow \infty g\prime (x) = 0.
Proof. We have
g(0) =
\infty \int
0
[f(0 + y) - f(| 0 - y| )] e - y dy = 0.
On the other hand,
g(x) =
\infty \int
0
[f(x+ y) - f(| x - y| )] e - y dy =
=
\infty \int
0
f(x+ y) e - y dy -
x\int
0
f(x - y) e - y dy -
\infty \int
x
f(y - x) e - y dy =
= ex
\infty \int
x
f(u) e - u du - e - x
x\int
0
f(u) eu du - e - x
\infty \int
0
f(u) e - u du =
= I1(x) - I2(x) - I3(x). (4.4)
Clearly, I3(x) \rightarrow 0 as x \rightarrow \infty . For I1(x) we obtain
| I1(x)| \leq
\infty \int
x
| f(u) e - (u - x)| du \leq
\infty \int
x
| f(u)| du \rightarrow 0 \mathrm{a}\mathrm{s} x \rightarrow \infty .
For any \epsilon > 0 choose N large enough such that
\int \infty
N | f(u)| du < \epsilon . Then, for x \rightarrow \infty ,
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ON THE FOURIER SINE AND KONTOROVICH – LEBEDEV GENERALIZED CONVOLUTION . . . 277
| I2(x)| \leq e - x
N\int
0
| f(u)| eu du+
x\int
N
| f(u)| du \leq e - x
N\int
0
| f(u)| eu du+ \epsilon \rightarrow \epsilon .
Thus, I2(x) \rightarrow 0, and, therefore, g(x) \rightarrow 0 as x \rightarrow \infty .
Next, from (4.4) we get
g\prime (x) = ex
\infty \int
x
f(u) e - u du+ e - x
x\int
0
f(u) eu du+ e - x
\infty \int
0
f(u) e - u du - 2f(x) =
= I1(x) + I2(x) + I3(x) - 2f(x). (4.5)
Since f \in C0(\BbbR +), then \mathrm{l}\mathrm{i}\mathrm{m}x\rightarrow \infty f(x) = 0, and, therefore, \mathrm{l}\mathrm{i}\mathrm{m}x\rightarrow \infty g\prime (x) = 0. From formula
(4.5) it is clear that g is twice differentiable.
Lemma 4.1 is proved.
Theorem 4.1. Suppose that the following condition holds:
1 + (Kh)(y) \not = 0 \forall y > 0. (4.6)
Then problem (4.2) has a unique solution f \in C2(\BbbR +) \cap L1(\BbbR +):
f(x) = ((\ell \ast
Fs,K
g) \ast
1
m)(x).
Here, m(x) =
\sqrt{}
\pi
2
e - x and \ell \in L - 1,\beta
1 (\BbbR +) is defined by
(K\ell )(y) =
(Kh)(y)
1 + (Kh)(y)
,
the generalized convolution (\cdot \ast
Fs,K
\cdot ) and the convolution (\cdot \ast
1
\cdot ) are defined by (1.2), (4.1), respectively.
Proof. Equation (4.2) can be rewritten in the form
f(x) - f \prime \prime (x) +
\Bigl(
1 - d2
dx2
\Bigr) \biggl\{
(h \ast
Fs,K
f)(x)
\biggr\}
= (h \ast
Fs,K
g)(x). (4.7)
Applying the Fourier sine transform to both sides of (4.7), and by virtue of the factorization equality
(1.3) and formula (4.3), we obtain
(1 + y2)(Fsf)(y) + (1 + y2)(Kh)(y)(Fsf)(y) = (Kh)(y)(Fsg)(y),
or, equivalently,
(1 + y2)(1 + (Kh)(y))(Fsf)(y) = (Kh)(y)(Fsg)(y).
From the condition (4.6) we get
(Fsf)(y) =
1
1 + y2
(Kh)(y)
1 + (Kh)(y)
(Fsg)(y).
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278 T. TUAN
By condition (4.6) the function \varphi (y) =
(Kh)(y)
1 + (Kh)(y)
satisfies conditions of the Wiener – Levy
theorem for the Kontorovich – Lebedev transform [14], and, therefore, there exists a unique function
\ell \in L - 1,\beta
1 (\BbbR +) such that
(K\ell )(y) =
(Kh)(y)
1 + (Kh)(y)
.
Moreover, note that
1
1 + y2
= (Fcm)(y) with m(x) =
\sqrt{}
\pi
2
e - x, we have
(Fsf)(y) =
\sqrt{}
\pi
2
(Fcm)(y)(K\ell )(y)(Fsg)(y) =
=
\sqrt{}
\pi
2
(Fcm)(y)Fs
\biggl[
(\ell \ast
Fs,K
g)
\biggr]
(y) =
=
\sqrt{}
\pi
2
Fs
\biggl[ \biggl( \biggl(
\ell \ast
Fs,K
g
\biggr)
\ast
1
m
\biggr) \biggr]
(y).
This implies f(x) = ((\ell \ast
Fs,K
g)\ast
1
m)(x). Since \ell \in L - 1,\beta
1 (\BbbR +) and g \in L1(\BbbR +), then by Theorem 2.1
we have \ell \ast
Fs,K
g \in C1
0 (\BbbR +) \cap L1(\BbbR +). Together with m \in L1(\BbbR +) it yields f = (\ell \ast
Fs,K
g) \ast
1
m \in
\in L1(\BbbR +). Lemma 4.1 implies f(0) = 0, \mathrm{l}\mathrm{i}\mathrm{m}x\rightarrow \infty f(x) = \mathrm{l}\mathrm{i}\mathrm{m}x\rightarrow \infty f \prime (x) = 0, and f \in C2(\BbbR +).
Theorem 4.1 is proved.
Remark. For p, q, r > 1 satisfying
1
p
+
1
q
= 1 +
1
r
, the following inequality holds [10]:
\| (f \ast
1
g)\| Lr(\BbbR +) \leq \| f\| Lp(\BbbR +)\| g\| Lq(\BbbR +), f \in Lp(\BbbR +), g \in Lq(\BbbR +).
Combining with inequality (2.10), if we assume that \ell \in L - p,\beta
p (\BbbR +), g \in Lq(\BbbR +), h \in Lr(\BbbR +),
and s > 1, such that
1
p
+
1
q
+
1
r
=
1
s
+2, we obtain an estimate for the solution of the problem (4.2)
in the space Ls(\BbbR +) as follows:
\| f\| Ls(\BbbR +) =
\bigm\| \bigm\| \bigm\| \bigm\| \biggl( (\ell \ast
Fs,K
g) \ast
1
m
\biggr) \bigm\| \bigm\| \bigm\| \bigm\|
Ls(\BbbR +)
\leq 2
p - 2
2p
\pi 3/2r1/r
\| \ell \|
L - p,\beta
p (\BbbR +)
\| g\| Lq(\BbbR +).
References
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2. M. Abramowitz, I. A. Stegun, Handbook of mathematical functions, with formulas, graphs and mathematical tables,
Nat. Bureau Standards Appl. Math. Ser., 55, Washington, D.C. (1964).
3. F. Al-Musallam, V. K. Tuan, Integral transforms related to a generalized convolution, Results Math., 38, 197 – 208
(2000).
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applications in mechanics and plasma physics, Lect. Notes Phys., 806, Springer, Dordrecht (2010).
5. I. N. Sneddon, Fourier transforms, McGray-Hill, New York (1951).
6. H. Bateman, A. Erdelyi, Table of integral transforms, vol. 1, McGraw-Hill Book Co., New York etc. (1954).
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7. L. E. Britvina, A class of integral transforms related to the Fourier cosine convolution, Integral Transforms Spec.
Funct., 16, № 5-6, 379 – 389 (2005).
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Received 22.12.16,
after revision — 01.08.17
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
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| id | umjimathkievua-article-2370 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:22:29Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/13/26172fb9830fea0f401dd08905cb2c13.pdf |
| spelling | umjimathkievua-article-23702020-04-07T12:16:11Z On the Fourier sine and Kontorovich–Lebedev generalized convolution transforms and applications Про синус-перетворення Фур’є i перетворення Конторовича–Лебедєва узагальнених згорток тa їх застосування Tuan , T. Tuan , T. UDC 517.5 We study a generalized convolutions for the Fourier sine and Kontorovich - Lebedev transforms $ (h\underset{F_s,K}\ast f)(x)$ in a two-parameter function space $L_p^{\alpha, \beta}(\Bbb R_+)$. We obtain several estimates for the norms and prove a Young-type inequality for this generalized convolution. We impose necessary and sufficient conditions on the kernel $h$ to ensure that the generalized convolution transform$$D_h\colon f\mapsto D_{h}[f] = \left(1-\dfrac{d^2}{dx^2}\right)(h\underset{F_s,K} \ast f)(x)$$is a unitary operator in $L_2(\Bbb R_+)$ (Watson-type theorem) and derive its inverse formula. Finally, we apply these results to an integrodifferential equation and obtain an estimate for the solution in the $L_p$-norm. УДК 517.5 Вивчається узагальнена згортка для синус-перетворення Фур'є i перетворення Конторовича - Лебедєва $ (h\underset{F_s,K}\ast f)(x)$ у двопараметричному просторі функцій $L_p^{\alpha, \beta}(\Bbb R_+)$. Отримано кілька оцінок для норм i встановлено нерівність типу Юнга для цієї узагальненoї&nbsp; згортки.Введено необхідні та достатні умови для ядра $h,$ за яких перетворення узагальненої згортки $$D_h\colon f\mapsto D_{h}[f] = \left(1-\dfrac{d^2}{dx^2}\right)(h\underset{F_s,K} \ast f)(x)$$– це унітарний оператор в $L_2(\Bbb R_+)$ (теорема типу Ватсона).Отримано формулу для оберненого перетворення.Крім того, ці результати застосовано до інтегро-диференціального рівняння та отримано оцінку для його розв'язку в $L_p$-нормі. Institute of Mathematics, NAS of Ukraine 2020-02-15 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2370 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 2 (2020); 267-279 Український математичний журнал; Том 72 № 2 (2020); 267-279 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2370/1564 Copyright (c) 2020 T. Tuan |
| spellingShingle | Tuan , T. Tuan , T. On the Fourier sine and Kontorovich–Lebedev generalized convolution transforms and applications |
| title | On the Fourier sine and Kontorovich–Lebedev generalized convolution transforms and applications |
| title_alt | Про синус-перетворення Фур’є i перетворення Конторовича–Лебедєва узагальнених згорток тa їх застосування |
| title_full | On the Fourier sine and Kontorovich–Lebedev generalized convolution transforms and applications |
| title_fullStr | On the Fourier sine and Kontorovich–Lebedev generalized convolution transforms and applications |
| title_full_unstemmed | On the Fourier sine and Kontorovich–Lebedev generalized convolution transforms and applications |
| title_short | On the Fourier sine and Kontorovich–Lebedev generalized convolution transforms and applications |
| title_sort | on the fourier sine and kontorovich–lebedev generalized convolution transforms and applications |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2370 |
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