Abstract. Convolutions with a class of weight function for the Fourier
cosine integral transform in the spaces Lp α,β(R+) are studied. Existence
conditions for these convolutions, a Young’s type theorem and a Parseval
type equality are obtained. Applications to solve a class of the integral
equations and systems of integral equations with function coefficiences are
considered.
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JOURNAL OF SCIENCE OF HNUE
Natural Sci., 2010, Vol. 55, No. 6, pp. 90-100
THE CONVOLUTION WITH A WEIGHT FUNCTION
RELATED TO THE FOURIER COSINE INTEGRAL TRANSFORM
Nguyen Xuan Thao(∗)
Hanoi University of Science and Technology
Ta Duy Cong
The Broadcasting and Television College I
(∗)E-mail: thaonxfami@mail.hut.edu.vn
Abstract. Convolutions with a class of weight function for the Fourier
cosine integral transform in the spaces Lpα,β(R+) are studied. Existence
conditions for these convolutions, a Young’s type theorem and a Parseval
type equality are obtained. Applications to solve a class of the integral
equations and systems of integral equations with function coefficiences are
considered.
Keywords: Convolution, weight function, Fourier transform.
1. Introduction
The Fourier intergral transform of function f(x) is of the form (see [6, 9, 13,
14, 16])
(Ff)(y) =
1√
2pi
∞∫
−∞
e−xy f(x) dx, (1.1)
with Fourier convolution (see [13, 14])
(
f ∗
F
g
)
(x) =
∞∫
−∞
f(x− t)g(t)dt. (1.2)
The Fourier cosine intergral transform of function f(x) was studied in [3, 4, 5, 13, 16]
(Fcf)(y) =
√
2
pi
∞∫
0
cos yx f(x) dx. (1.3)
In 1951, Sneddon I. N. proposed the convolution of two functions f(x) and g(x) for
the Fourier cosine integral transform (see [7, 13, 14])
(
f ∗
Fc
g
)
(x) =
1√
2pi
∞∫
0
f(y)
[
g(x+ y) + g(|x− y|)] dy, (1.4)
90
The convolution with a weight function related to the Fourier cosine integral transform
which satisfies the following factorization property
Fc
(
f ∗
Fc
g
)
(y) = (Fcf)(y) . (Fcg)(y) ∀y ∈ R+, (1.5)
and the norm inequality
‖f ∗
Fc
g‖ ≤ ‖f‖ . ‖ g ‖. (1.6)
In 2004, Thao N. X. and Khoa N. M. studied the convolution with a weight
function γ = cos y of two functions f(x) and g(x) for the Fourier cosine integral
transform [15]
(
f
γ∗ g)(x) = 1
2
√
2pi
∞∫
0
f(t)[g(x+ 1 + t) + g(| x+ 1− t |) + g(| x− 1 + t |)+
+ g(| x− 1− t) |)]dt, (1.7)
and proved the factorization property
Fc
(
f
γ∗ g)(y) = cos y (Fcf)(y).(Fcg)(y), ∀y ∈ R+, f, g ∈ L(R+). (1.8)
Norm of the function f(x) in the space L(R+) is defined
‖f‖ =
√
2
pi
∞∫
0
|f(x)| dx. (1.9)
Then the convolution
(
f
γ∗ g)(x) belongs to the space L(R+) and satisfies the
convolution inequality
‖f γ∗ g‖ ≤ ‖f‖ . ‖ g ‖. (1.10)
Convolutions with a weight function in the space Lp(R) were studied [11, 12].
For instance, Young’s theorem and its corollary was proposed by himself.
Theorem 1.1 (A Young’s theorem). [2] Let p, q, r > 1,
1
p
+
1
q
+
1
r
= 2, and f ∈
Lp(R), g ∈ Lq(R), h ∈ Lr(R), then
∞∫
0
(
f ∗
F
g
)
(x) . h(x) dx
≤ ‖f‖ Lp(R) ‖g‖ Lq(R) ‖h‖ Lr(R). (1.11)
Corollary 1.1 (A Young’s inequality). [2] Let p, q, r > 1,
1
p
+
1
q
= 1 +
1
r
and f ∈
Lp(R), g ∈ Lq(R), then (f ∗
F
g
)
(x) ∈ Lr(R) and
‖(f ∗
F
g
)‖Lr(R) ≤ ‖f‖Lp(R) ‖g‖Lq(R). (1.12)
91
Nguyen Xuan Thao and Ta Duy Cong
In this paper, we study the convolution with a class of the weight function for
the Fourier cosine integral transform in the spaces Lpα,β(R+). We obtained conditions
for its existence, a Young’s type theorem and the Parseval type equality. Also,
in application, we solved a class of the integral equations and systems of integral
equation with function coefficients whose solutions can be presented in closed form.
2. The convolution and its properties in Lp(R+)
Definition 2.1. The convolution with a weight function γ = cosαy, α ∈ R+ of two
functions f(x) and g(x) for the Fourier cosine integral transform defined by
(
f
γ∗ g)(x) = 1
2
√
2pi
∞∫
0
f(t)[g(x+ α + t)
+ g(| x+ α− t |) + g(| x− α+ t |) + g(| x− α− t) |)]dt, x > 0. (2.1)
Remark 2.1. If α = 0 then the convolution (2.1) coincides with the convolution
studied in [4], if α = 1 then the convolution (2.1) coincides with the convolution
studied in [15].
Similar arguments as in [15], we obtain following results
Theorem 2.1. Let f(x), g(x) ∈ L(R+), then convolution (2.1) exists, belongs to
L(R+) and satisfies the factorization property
Fc
(
f
γ∗ g)(y) = cosαy (Fcf)(y).(Fcg)(y), ∀y ∈ R+. (2.2)
Further, convolution operator (2.1) is continuous, bounded and satisfies Par-
seval type equality
(
f
γ∗ g)(x) =
√
2
pi
∞∫
0
(
Fcf
)
(y).
(
Fcg
)
(y) cosαy cosxy dy. (2.3)
Definition 2.2. The functional spaces Lp(R+) and L
p
α,βR+ are respestively defined
by
Lp(R+) =
{
f :
( ∞∫
0
| f(x) |p dx
) 1
p
< +∞
}
; (2.4)
and
Lpα,βR+ =
{
f :
( ∞∫
0
tγe−βx | f(x) |p dx
) 1
p
< +∞
}
. (2.5)
92
The convolution with a weight function related to the Fourier cosine integral transform
Theorem 2.2. (A Young’s type theorem) Let p, q, r > 1, such that
1
p
+
1
q
+
1
r
= 2,
and let f ∈ Lp(R+), g ∈ Lq(R+), h ∈ Lr(R+), then
∞∫
0
(
f
γ∗ g)(x) . h(x) dx ≤
√
2
pi
‖f‖ Lp(R+) ‖g‖ Lq(R+) ‖h‖ Lr(R+). (2.6)
Proof. In view of the convolution of two functions f(x) and g(x) with a weight
function defined in (1.7), we have
∞∫
0
(
f
γ∗ g)(x) . h(x) dx
≤ 1
2
√
2pi
∫ ∞
0
∫ ∞
0
|f(t)| |g(x+ α+ t) + g(|x+ α− t|)| |h(x)| dt dx
+
1
2
√
2pi
∞∫
0
∞∫
0
|f(t)| |g(|x− α+ t|) + g(|x− α− t|)| |h(x)| dt dx. (2.7)
Let p1, q1, r1 be the conjugate exponentials of p, q, r respestively, it means
1
p
+
1
p1
= 1,
1
q
+
1
q1
= 1,
1
r
+
1
r1
= 1.
It is obvious that
1
p1
+
1
q1
+
1
r1
= 1.
Put
U(x, y) = |g(x+ α + t) + g(|x+ α− t|)| qp1 . |h(x)|
r
p1 ;
V (x, y) = |f(t)|
p
q1 . |h(x)|
r
q1 ;
W (x, y) = |f(t)| pr1 . |g(x+ α + t) + g(|x+ α− t|)| qr1 .
We have
U(x, y) . V (x, y) .W (x, y) = |f(t)| . |h(x)| . |g(x+ α+ t) + g(|x+ α− t|)|. (2.8)
Thus∫ ∞
0
∫ ∞
0
|f(t)|. |g(x+ α + t) + g(|x+ α− t|)| |h(x)| dt dx
=
∞∫
0
∞∫
0
U(x, y) . V (x, y) .W (x, y)dtdx.
93
Nguyen Xuan Thao and Ta Duy Cong
Applying the Ho¨lder’s inequality for three functions U(x, y) , V (x, y) , W (x, y),
we have
∞∫
0
∞∫
0
U(x, y) . V (x, y) .W (x, y)dydx ≤ ‖U‖Lp1(R2+) ‖V ‖Lq1(R2+) ‖W‖Lr1(R2+).
It follows that∫ ∞
0
∫ ∞
0
|f(t)| |g(x+ α+ t) + g(|x+ α− t|)| |h(x)| dt dx
≤ ‖U‖Lp1 (R2+) ‖V ‖Lq1(R2+) ‖W‖Lr1(R2+). (2.9)
In the space Lp1(R2+), we have
‖U‖p1
Lp1 (R2+)
=
∞∫
0
∞∫
0
|g(x+ α + t) + g(|x+ α− t|)|q . |h(x)|r dx dt
=
∞∫
0
( ∞∫
0
|g(x+ α + t) + g(|x+ α− t|)|q dt
)
|h(x)|r dx. (2.10)
Note that |t|q is a convex function, hence
∞∫
0
|g(x+ α + t) + g(| x+ α− t |)|q dt
≤ 2q−1
( ∞∫
0
|g(x+ α + t)|q dt+
∞∫
0
|g(| x+ α− t |)|q dt
)
= 2q
∞∫
0
| g(t) |q dt. (2.11)
From (2.11) and (2.10), we get
wwUwwp1
Lp1(R2+)
≤ 2q
∞∫
0
| g(t) |q dt
∞∫
0
|h(x)|r dx = 2q ‖g‖qLq(R+) ‖h‖rLr(R+). (2.12)
Similarly, in the space Lr1(R2+), we havewwWwwr1
Lr1(R2+)
≤ 2q ‖f‖pLp(R+) ‖g‖
q
Lq(R+)
. (2.13)
In the space Lq1(R2+), we havewwVww
L
q1(R2+)
= ‖f‖
p
q1
Lp(R+)
‖h‖
r
q1
Lr(R+)
. (2.14)
94
The convolution with a weight function related to the Fourier cosine integral transform
From formulas (2.12), (2.13) and (2.14), we havewwUww
Lp1 (R2+)
wwVww
Lq1(R2+)
wwWww
Lr1(R2+)
≤ 2. ‖f‖Lp(R+) ‖g‖Lq(R+)‖h‖Lr(R+). (2.15)
From (2.15) and (2.9), we have
∣∣∣∣
∞∫
0
∞∫
0
f(t)[g(x+ α+ t) + g(| x+ α− t |)] h(x) dt dx
∣∣∣∣
≤ 2. ‖f‖Lp(R+) ‖g‖Lq(R+)‖h‖Lr(R+). (2.16)
Similarly
∞∫
0
∞∫
0
|f(t)| |g(|x− α + t|) + g(|x− α− t|)| h(x) dt dx
≤ 2. ‖f‖Lp(R+) ‖g‖Lq(R+)‖h‖Lr(R+). (2.17)
From formulas (2.16), (2.17) and (2.7), we get
∞∫
0
(
f
γ∗ g)(x) . h(x) dx ≤
√
2
pi
‖f‖Lp(R+) ‖g‖Lq(R+)‖h‖Lr(R+).
The proof is complete.
Corollary 2.1. Let p, q, r > 1 such that
1
p
+
1
q
= 1 +
1
r
, and let f ∈ Lp(R+), g ∈
Lq(R+), then
(
f
γ∗ g)(x) ∈ Lr(R+) and
‖(f γ∗ g)‖Lr(R+) ≤
√
2
pi
‖f‖Lp(R+) ‖g‖Lq(R+). (2.18)
Theorem 2.3. Let f ∈ Lp(R+), g ∈ Lq(R+), 0 1 such that
1
p
+
1
q
= 1. Then the convolution ( 2.1) exists, is continuous and bounded. Further-
more,
(
f
γ∗ g) ∈ Lrα,γ(R+) and the following estimation holds
‖(f γ∗ g)‖Lrα,γ(R+) ≤
2q√
2pi
C ‖f‖Lp(R+) ‖g‖Lq(R+), (2.19)
where C = β−
γ+1
r .Γ
1
r (γ + 1), γ > −1.
Further, if f ∈ Lp(R+) ∩ L(R+), g ∈ Lq(R+) ∩ L(R+), the convolution (2.1)
satisfies the factorization property (2.2), and the Parseval’s type equality (2.3) holds.
This theorem can be proved easily basing on the Ho¨lder’s inequality for con-
volution (2.1), formula (3.225.3) in [10] and Riemann - Lebesgue lemma.
95
Nguyen Xuan Thao and Ta Duy Cong
3. Applications
3.1. Integral equations with function coefficient
Consider the integral equation
f(x) +
λ(x)
2
√
2pi
∞∫
0
f(t)ψ(g)(x, t) dt = h(x). (3.1)
Here, g, h are two given continuous functions in L(R+), λ is a continuous
function such that 0 < m ≤ |λ(x)|, f(x) is an unkown function and
ψ(g)(x, t) =
1
λ(t)
[
g(x+α+ t)+g(|x+α− t|)+g(|x+α− t|)+g(|x−α− t|)]. (3.2)
Theorem 3.1. Suppose that 1+cosαy
(
Fcg
)
(y) 6= 0, ∀ y ∈ R+, then there exists the
unique solution in L(R+) of equation (3.1), namely, a closed form of this solution
is of the form
f(x) = h(x)− (h
λ
γ∗ ϕ)(x).λ(x). (3.3)
where ϕ(x) is a continuous function, belongs to L(R+) defined by
(
Fcϕ
)
(y) =
(
Fcg
)
(y)
1 + cosαy
(
Fcg
)
(y)
.
Proof. We can rewrite equation (3.1) as follows
f(x) + λ(x)
(f
λ
γ∗ g)(x) = h(x). (3.4)
The given condition shows that λ(x) 6= 0 for all x ∈ R+, hence the equation
(3.4) is equivalent to
f(x)
λ(x)
+
(f
λ
γ∗ g)(x) = h(x)
λ(x)
. (3.5)
Applying the Fourier cosine integral transform to both sides of (3.5) and using
the factorization equality (2.2), we get
(
Fc
f
λ
)
(y)
[
1 + cosαy
(
Fcg
)
(y)
]
=
(
Fc
h
λ
)
(y).
By virtue of the hypothesis, we have
(
Fc
f
λ
)
(y) =
(
Fc
h
λ
)
(y)
1 + cosαy
(
Fcg
)
(y)
,
96
The convolution with a weight function related to the Fourier cosine integral transform
or equivalently,
(
Fc
f
λ
)
(y) =
(
Fc
h
λ
)
(y)
[
1− cosαy
(
Fcg
)
(y)
1 + cosαy
(
Fcg
)
(y)
]
. (3.6)
Due to Wiener-Levy’s theorem [1], there exists a unique function ϕ(x) ∈ L(R+)
such that (
Fcϕ
)
(y) =
(
Fcg
)
(y)
1 + cos y
(
Fcg
)
(y)
.
Therefore equation (3.6) becomes
(
Fc
f
λ
)
(y) =
(
Fc
h
λ
)
(y)
[
1− cosαy(Fcϕ)(y)]
=
(
Fc
h
λ
)
(y)− Fc
(h
λ
γ∗ ϕ)(y) (3.7)
Applying the inverse Fourier cosine integral transform to both sides of equation
(3.7), we get
f(x)
λ(x)
=
h(x)
λ(x)
− (h
λ
γ∗ ϕ)(x).
It follows the solution of equation (3.1)
f(x) = h(x)− (h
λ
γ∗ ϕ)(x).λ(x).
Note that conditions m < |λ(x)| and h ∈ L(R+) imply that h
λ
∈ L(R+). By
Theorem 2.1, we get f(x) ∈ L(R+).
The theorem is proved completely.
3.2. The integral equation system with function coefficient
Finally, we consider the following system of integral equations
f(x) +
λ1(x)
2
√
2pi
∞∫
0
g(t) h(ϕ)(x, t)dt = p (x)
λ2(x)
2
√
2pi
∞∫
0
f(t) k(ψ)(x, t)dt+ g(x) = q(x),
(3.8)
where, ϕ, ψ, p, q are continuous functions, belong to L(R+), λ1, λ2 are given such
that 0 < m ≤ |λj(x)|, j = 1, 2 for some positive m, and f , g are unkown functions
and
h(ϕ)(x, t) =
1
λ2(t)
[
ϕ(x+α+t)+ϕ(| x+α−t |)+ϕ(| x−α+t |)+ϕ(| x−α−t |)], (3.9)
97
Nguyen Xuan Thao and Ta Duy Cong
k(ψ)(x, t) =
1
λ1(t)
[
ψ(x+α+ t)+ψ(| x+α− t |)+ψ(| x−α+ t |)+ψ(| x−α− t |)].
(3.10)
Theorem 3.2. Suppose that the condition 1− cosαy Fc
(
ϕ
γ∗ ψ)(y) 6= 0 holds for all
y > 0. Then the system (3.8) has a unique solution (f, g) ∈ L(R+)× L(R+) defined
as follows
f(x) = p(x)− λ1(x).
( q
λ2
γ∗ ϕ)(x) + λ1(x).[( p
λ1
− ( q
λ2
γ∗ ϕ)) γ∗ ξ](x),
g(x) = q(x)− λ2(x).
( p
λ1
γ∗ ψ)(x) + λ2(x).[( q
λ2
− ( p
λ1
γ∗ ψ)) γ∗ ξ](x), (3.11)
Here, ξ(x) is a unique continuous function belonging to L(R+) and defined by
(
Fc ξ
)
(y) =
Fc
(
ϕ
γ∗ ψ)(y)
1− cosαy Fc
(
ϕ
γ∗ ψ)(y) . (3.12)
Proof. The equation system (3.8) can be rewritten as follows
f(x) + λ1(x).
( g
λ2
γ∗ ϕ)(x) = p (x)
λ2(x).
( f
λ1
γ∗ ψ)(x) + g(x) = q(x). (3.13)
Applying Fourier cosine transform Fc to both sides of each equations of (3.13),
we obtain
Fc
( f
λ1
)
(y) + cosαy Fc
( g
λ2
)
(y)
(
Fcϕ
)
(y) = Fc
( p
λ1
)
(y)
cosαy Fc
( f
λ1
)
(y)
(
Fcψ
)
(y) + Fc
( g
λ2
)
(y) = Fc
( q
λ2
)
(y).
(3.14)
Under the condition of the hypothesis, it follows Cramer linear equation system
(3.14) having a unique solution
Fc
( f
λ1
)
(y) =
Fc
( p
λ1
− ( q
λ2
γ∗ ϕ))(y)
1− cosαy Fc
(
ϕ
γ∗ ψ)(y)
Fc
( g
λ2
)
(y) =
Fc
( q
λ2
− ( p
λ1
γ∗ ψ))(y)
1− cosαy Fc
(
ϕ
γ∗ ψ)(y) ,
or equivalently,
Fc
( f
λ1
)
(y) = Fc
( p
λ1
− ( q
λ2
γ∗ ϕ))(y).[1 + cosαy Fc
(
ϕ
γ∗ ψ)(y)
1− cosαy Fc
(
ϕ
γ∗ ψ)(y)
]
Fc
( g
λ2
)
(y) = Fc
( q
λ2
− ( p
λ1
γ∗ ψ))(y).[1 + cosαy Fc
(
ϕ
γ∗ ψ)(y)
1− cosαy Fc
(
ϕ
γ∗ ψ)(y)
]
.
98
The convolution with a weight function related to the Fourier cosine integral transform
Due to Wiener-Levy’s Theorem [1], there exists a unique continuous function
ξ ∈ L(R+) such that
(
Fc ξ
)
(y) =
Fc
(
ϕ
γ∗ ψ)(y)
1− cosαy Fc
(
ϕ
γ∗ ψ)(y) .
Therefore
Fc
( f
λ1
)
(y) = Fc
[ p
λ1
− ( q
λ2
γ∗ ϕ)](y).[1 + cosαy (Fc ξ)(y)]
Fc
( g
λ2
)
(y) = Fc
[ q
λ2
− ( p
λ1
γ∗ ψ)](y).[1 + cosαy (Fc ξ)(y)].
We obtain
Fc
( f
λ1
)
(y) = Fc
[ p
λ1
− ( q
λ2
γ∗ ϕ)](y) + Fc[( p
λ1
− ( q
λ2
γ∗ ϕ)) γ∗ ξ](y)
Fc
( g
λ2
)
(y) = Fc
[ q
λ2
− ( p
λ1
γ∗ ψ)](y) + Fc[( q
λ2
− ( p
λ1
γ∗ ψ)) γ∗ ξ](y).
It shows that
f(x)
λ1(x)
=
[ p
λ1
− ( q
λ2
γ∗ ϕ)](x) + [( p
λ1
− ( q
λ2
γ∗ ϕ)) γ∗ ξ](x)
g(x)
λ2(x)
=
[ q
λ2
− ( p
λ1
γ∗ ψ)](x) + [( q
λ2
− ( p
λ1
γ∗ ψ)) γ∗ ξ](x).
Therefore, we obtain a unique solution of system (3.8)
f(x) = p(x)− λ1(x).
( q
λ2
γ∗ ϕ)(x) + λ1(x).[( p
λ1
− ( q
λ2
γ∗ ϕ)) γ∗ ξ](x)
g(x) = q(x)− λ2(x).
( p
λ1
γ∗ ψ)(x) + λ2(x).[( q
λ2
− ( p
λ1
γ∗ ψ)) γ∗ ξ](x)
Conditions 0 < m ≤ |λj(x)|, j = 1, 2 and p, q ∈ L(R+) imply that pλ1 ,
q
λ2
∈
L(R+). By Theorem 2.1, it is obvious that f(x) and g(x) belong to L(R+). The
proof is complete.
Remark 3.1. One can easily extend above results to weight functions cosαx, α ∈ R.
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House, Moscow.
[2] Adams R. A. and Fourier J. J. S., 2003. Sobolev Spaces, 2nd ed. Academic
Press/Elsevier Science, New York/Amsterdam.
99
Nguyen Xuan Thao and Ta Duy Cong
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