Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 908491, 16 pages
doi:10.1155/2011/908491
Research Article
On the Convolution Equation Related to the
Diamond Klein-Gordon Operator
Amphon Liangprom1 and Kamsing Nonlaopon1, 2
1
2
Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand
Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand
Correspondence should be addressed to Kamsing Nonlaopon, nkamsi@kku.ac.th
Received 11 July 2011; Revised 3 August 2011; Accepted 31 August 2011
Academic Editor: Elena Braverman
Copyright q 2011 A. Liangprom and K. Nonlaopon. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
k
k
We study the distribution eαx ♦ m2 δ for m ≥ 0, where ♦ m2 is the diamond Klein-Gordon
operator iterated k times, δ is the Dirac delta distribution, x x1 , x2 , . . . , xn is a variable in
k
Rn , and α α1 , α2 , . . . , αn is a constant. In particular, we study the application of eαx ♦ m2 δ
for solving the solution of some convolution equation. We find that the types of solution of such
convolution equation, such as the ordinary function and the singular distribution, depend on the
relationship between k and M.
1. Introduction
The n-dimensional ultrahyperbolic operator k iterated k times is defined by
⎞k
2
2
2
2
2
2
∂
∂
∂
∂
∂
∂
k ⎝ 2 2 · · · 2 − 2 − 2 − · · · − 2 ⎠ ,
∂x1 ∂x2
∂xp ∂xp1 ∂xp2
∂xpq
⎛
1.1
where p q n is the dimension of Rn , and k is a nonnegative integer. We consider the linear
differential equation of the form
k ux fx,
where ux and fx are generalized functions, and x x1 , x2 , . . . , xn ∈ Rn .
1.2
2
Abstract and Applied Analysis
Gelfand and Shilov 1 have first introduced the fundamental solution of 1.2, which
x
was initially complicated. Later, Trione 2 has shown that the generalized function RH
2k
defined by 2.2 with γ 2k is the unique fundamental solution of 1.2. Tellez 3 has also
x exists only when n p q with odd p.
proved that RH
2k
Kananthai 4 has first introduced the operator ♦k called the diamond operator iterated
k times, which is defined by
⎡
⎞2 ⎤k
2 ⎛ pq
p
2
2
∂
∂ ⎠⎥
⎢
−⎝
♦k ⎣
⎦ ,
2
2
i1 ∂xi
jp1 ∂xj
1.3
where n p q is the dimension of Rn , for all x x1 , x2 , . . . , xn ∈ Rn and nonnegative
integers k. The operator ♦k can be expressed in the form
♦k k k k k ,
1.4
where k is defined by 1.1, and k is the Laplace operator iterated k times defined by
k
∂2
∂2
∂2
2 ··· 2
2
∂x1 ∂x2
∂xn
k
.
1.5
Note that in case k 1, the generalized form of 1.5 is called the local fractional Laplace
operator; see 5 for more details. On finding the fundamental solution of this product, he
uses the convolution of functions which are fundamental solutions of the operators k and
k . He found that the convolution −1k Re2k x ∗ RH
x is the fundamental solution of the
2k
operator ♦k , that is,
δ,
♦k −1k Re2k x ∗ RH
x
2k
1.6
where RH
x and Re2k x are defined by 2.2 and 2.7, respectively with γ 2k, and δ
2k
x is called the
is the Dirac-delta distribution. The fundamental solution −1k Re2k x ∗ RH
2k
diamond kernel of Marcel Riesz. A number of effective results on the diamond kernel of
Marcel Riesz have been presented by Kananthai 6–12.
In 1997, Kananthai 13 has studied the properties of the distribution eαx k δ and
the application of the distribution eαx k δ for finding the fundamental solution of the
ultrahyperbolic equation by using the convolution method. Later in 1998, he has also studied
the properties of the distribution eαx ♦k δ and its application for solving the convolution
equation
eαx ♦k δ ∗ ux eαx
m
Cr ♦r δ.
1.7
r0
Recently, Nonlaopon gave some generalizations of his paper 6; see 14 for more details.
Abstract and Applied Analysis
3
In 2000, Kananthai 15 has studied the application of the distribution eαx k δ for
solving the convolution equation
eαx k δ ∗ ux eαx
m
Cr r δ,
1.8
r0
which is related to the ultrahyperbolic equation.
In 2009, Sasopa and Nonlaopon 16 have studied the properties of the distribution
eαx kc δ and its application to solve the convolution equation
eαx kc δ ∗ ux eαx
m
r0
Cr rc δ.
1.9
Here, kc is the operator related to the ultrahyperbolic type operator iterated k times, which
is defined by
⎞k
pq
p
2
2
∂
∂
1
⎠ ,
−
kc ⎝ 2
c i1 ∂xi2 jp1 ∂xj2
⎛
1.10
where p q n is the dimension of Rn .
In 1988, Trione 17 has studied the fundamental solution of the ultrahyperbolic KleinGordon operator iterated k times, which is defined by
m2
k
⎞k
p
pq
2
2
∂
∂
⎝
−
m2 ⎠ .
2
2
∂x
∂x
i1
jp1
i
j
⎛
1.11
The fundamental solution of the operator m2 k is given by
W2k x, m ∞
−1r Γk r r0
r!Γk
m2
r
−1r RH
2k2r x,
1.12
where RH
x is defined by 2.2 with γ 2k 2r. Next, Tellez 18 has studied the
2k2r
convolution product of Wα x, m ∗ Wβ x, m, where α and β are any complex parameter. In
addition, Trione 19 has studied the fundamental P ± i0λ -ultrahyperbolic solution of the
Klein-Gordon operator iterated k times and the convolution of such fundamental solution.
Liangprom and Nonlaopon 20 have studied the properties of the distribution eαx k
m2 δ and its application for solving the convolution equation
M
k
r
eαx m2 δ ∗ ux eαx Cr m2 δ,
r0
where m2 k is defined by 1.11.
1.13
4
Abstract and Applied Analysis
In 2007, Tariboon and Kananthai 21 have introduced the operator ♦ m2 k called
diamond Klein-Gordon operator iterated k times, which is defined by
♦ m2
k
⎤k
⎡
⎞2
2 ⎛ pq
p
2
2
∂
∂ ⎠
⎥
⎢
⎣
−⎝
m2 ⎦ ,
2
2
∂x
∂x
i1
jp1
i
j
1.14
where p q n is the dimension of Rn , for all x x1 , x2 , . . . , xn ∈ Rn , m ≥ 0 and nonnegative
integers k. Later, Lunnaree and Nonlaopon 22, 23 have studied the fundamental solution
of operator ♦ m2 k , and this fundamental solution is called the diamond Klein-Gordon
kernel. They have also studied the Fourier transform of the diamond Klein-Gordon kernel
and its convolution.
In this paper, we aim to study the properties of the distribution eαx ♦ m2 k δ and the
application of eαx ♦ m2 k δ for solving the convolution equation
M
k
r
eαx ♦ m2 δ ∗ ux eαx Cr ♦ m2 δ,
1.15
r0
where ♦ m2 k is defined by 1.14, ux is the generalized function, and Cr is a constant.
On finding the type of solution ux of 1.15, we use the method of convolution of the
generalized functions.
Before we proceed to that point, the following definitions and concepts require
clarifications.
2. Preliminaries
Definition 2.1. Let x x1 , x2 , . . . , xn be a point of the n-dimensional Euclidean space Rn . Let
2
2
2
u x12 x22 · · · xp2 − xp1
− xp2
− · · · − xpq
2.1
be the nondegenerated quadratic form, where p q n is the dimension of Rn . Let Γ {x ∈
Rn : x1 > 0 and u > 0} be the interior of a forward cone, and let Γ denote its closure. For any
complex number γ, we define the function
⎧ γ−n/2
⎪
⎨u ,
Kn γ
RH
x
γ
⎪
⎩0,
for x ∈ Γ ,
2.2
for x ∈
/ Γ ,
where the constant Kn γ is given by
π n−1/2 Γ 2 γ − n /2 Γ 1 − γ /2 Γ γ
Kn γ .
Γ 2 γ − p /2 Γ p − γ /2
2.3
Abstract and Applied Analysis
5
The function RH
γ x is called the ultrahyperbolic kernel of Marcel Riesz, which was
introduced by Nozaki 24. It is well known that RH
γ x is an ordinary function if Reγ ≥ n
H
and is a distribution of γ if Reγ < n. Let supp RH
γ x denote the support of Rγ x and
H
suppose that supp RH
γ x ⊂ Γ , that is, supp Rγ x is compact.
H
By putting p 1 in R2k x and taking into account Legendre’s duplication formula
1
Γ2z 22z−1 π −1/2 ΓzΓ z ,
2
2.4
we obtain
IγH x vγ−n/2
,
Hn γ
2.5
v x12 − x22 − x32 − · · · − xn2 , where
γ 2 − n γ .
Γ
Hn γ π n−2/2 2γ−1 Γ
2
2
2.6
The function IγH x is called the hyperbolic kernel of Marcel Riesz.
Definition 2.2. Let x x1 , x2 , . . . , xn be a point of Rn and ω x12 x22 · · · xn2 . The elliptic
kernel of Marcel Riesz is defined by
Reγ x ωγ−n/2
,
Wn γ
2.7
where n is the dimension of Rn , γ ∈ C, and
π n/2 2γ Γ γ/2
Wn γ .
Γ n − γ /2
2.8
Note that n p q. By putting q 0 i.e., n p in 2.2 and 2.3, we can reduce
γ−p/2
to ωp
, where ωp x12 x22 · · · xp2 , and reduce Kn γ to
γ−n/2
u
π p−1/2 Γ 1 − γ /2 Γ γ
.
Kp γ Γ p − γ /2
2.9
Using the Legendre’s duplication formula
Γ2z 2
2z−1
π
−1/2
1
,
ΓzΓ z 2
2.10
6
Abstract and Applied Analysis
1
1
Γ
z Γ
− z π secπz,
2
2
2.11
1 γπ Kp γ sec
Wp γ .
2
2
2.12
we obtain
Thus, in case q 0, we have
RH
γ x γπ uγ−p/2
γπ uγ−p/2
Reγ x.
2 cos
2 cos
2 Wp γ
2
Kp γ
2.13
In addition, if γ 2k for some nonnegative integer k, then
k e
RH
2k x 2−1 R2k x.
2.14
Lemma 2.3. The convolution −1k Re2k x ∗ RH
x is the fundamental solution of the diamond
2k
operator iterated k times, that is,
♦k −1k Re2k x ∗ RH
2k x δ.
2.15
For the proof of this Lemma, see 4, 12.
It can be shown that Re−2k x ∗ RH
x −1k ♦k δ, for all nonnegative integers k.
−2k
Definition 2.4. Let x x1 , x2 , . . . , xn be a point of Rn . The function Tγ x, m is defined by
⎛ γ⎞
r
⎝− 2 ⎠ m2 −1γ/2r Re x ∗ RH x,
Tγ x, m γ2r
γ2r
r
r0
∞
2.16
where γ is a complex parameter, and m is a nonnegative real number. Here, RH
γ2r x and
Reγ2r x are defined by 2.2 and 2.7, respectively.
From the definition of Tγ x, m, by putting γ −2k, we have
T−2k x, m ∞
k r0
r
m2
r
−1−kr Re2−kr x ∗ RH
2−kr x.
2.17
Abstract and Applied Analysis
7
Since the operator ♦ m2 k defined by 1.14 is linearly continuous and has 1-1 mapping,
this possesses its own inverses. From Lemma 2.3, we obtain
T−2k x, m ∞
k m2
r
r0
r
k
♦k−r δ ♦ m2 δ.
2.18
Substituting k 0 in 2.18 yields that we have T0 x, m δ. On the other hand,
putting γ 2k in 2.16 yields
−k T2k x, m 0
m2
0
∞
−k r1
r
−1k0 Re2k0 x ∗ RH
2k0 x
m
2
r
2.19
−1
kr
Re2k2r x
∗
RH
2k2r x.
The second summand of the right-hand side of 2.19 vanishes when m 0. Hence, we obtain
T2k x, m 0 −1k Re2k x ∗ RH
2k x,
2.20
which is the fundamental solution of the diamond operator.
For the proofs of Lemmas 2.5 and 2.6, see 23.
Lemma 2.5. Given the equation
♦ m2
k
ux δ,
2.21
where ♦ m2 k is the diamond Klein-Gordon operator iterated k times, defined by
♦ m2
k
⎤k
⎡
⎞2
2 ⎛ pq
p
2
2
∂
∂ ⎠
⎥
⎢
⎣
−⎝
m2 ⎦
2
2
∂x
∂x
i1
jp1
i
j
2.22
with a nonnegative integer k and the Dirac-delta distribution δ, then ux T2k x, m is the
fundamental solution of the diamond Klein-Gordon operator iterated k times ♦ m2 k , where
T2k x, m is defined by 2.16 with γ 2k.
Lemma 2.6. Let T2k x, m be the diamond Klein-Gordon kernel defined by 2.16, then T2k x, m is
a tempered distribution and can be expressed by
T2k x, m T2k−2v x, m ∗ T2v x, m,
2.23
8
Abstract and Applied Analysis
where v is a nonnegative integer and v < k. Moreover, if one puts l k − v and h v, then one
obtains
T2l x, m ∗ T2h x, m T2l2h x, m
2.24
for l h k.
3. Properties of the Distribution eαx ♦ m2 k δ
Lemma 3.1. The following equality holds:
k
eαx ♦ m2 δ Lk δ,
3.1
and eαx ♦ m2 k δ is the tempered distribution of order 4k with support {0}, where L is the partial
differential operator and is defined by
∂3
∂3
αi
L≡ ♦m
∂xi2 ∂xr
∂xi ∂xr2
r1
r1 i1
⎛
⎞
p
pq
pq
n n
2
2
∂3
∂3
∂
∂
⎠
αr 2
4 αr ⎝ αi
2
αj
−
αj
∂xi ∂xr jp1 ∂xj ∂xr
∂xj ∂xr
∂xj ∂xr2
r1 jp1
r1
i1
2
p
n
n 2
αr
αr − 2
⎛
⎞ ⎛
⎞
p
pq
p
pq
n
∂
∂ ⎠ ⎝
2 2 ⎠
2⎝
− 2 αr
αi
−
αj
αi −
αj ∂xi jp1 ∂xj
r1
i1
i1
jp1
3.2
⎛
⎞
⎞
⎛
p
pq
p
pq
n
n
∂
− 2⎝ α2i −
α2j ⎠ αr
⎝ α2i −
α2j ⎠ α2r .
∂xr
i1
jp1
r1
i1
jp1
r1
As before, is the ultrahyperbolic operator defined by 1.1 (with k 1), and is the Laplace operator
defined by
∂2
∂2
∂2
··· .
∂x1 ∂x2
∂xn
3.3
Proof. Let ϕ ∈ D be the space of testing functions which are infinitely differentiable with
compact supports, and let D be the space of distributions. Now,
eαx ♦ m2 δ, ϕx δ, ♦ m2 eαx ϕx ,
3.4
for eαx ♦ m2 δ ∈ D . A direct computation shows that
♦ m2 eαx ϕx eαx T ϕx,
3.5
Abstract and Applied Analysis
9
where T is the partial differential operator defined by
∂3
∂3
αi
T ≡ ♦m
∂xi2 ∂xr
∂xi ∂xr2
r1
r1 i1
⎛
⎞
p
pq
pq
n n
2
2
∂3
∂3
∂
∂
⎠
αr 2
4 αr ⎝ αi
−2
αj
−
αj
∂xi ∂xr jp1 ∂xj ∂xr
∂xj ∂xr
∂xj ∂xr2
r1 jp1
r1
i1
2
p
n
n 2
αr
αr 2
⎛
⎞ ⎛
⎞
p
pq
p
pq
n
∂
∂
⎠ ⎝ α2 −
2 α2r ⎝ αi
−
αj
α2j ⎠
i
∂x
∂x
i
j
r1
i1
jp1
i1
jp1
3.6
⎛
⎞
⎞
⎛
p
pq
p
pq
n
n
∂
2⎝ α2i −
α2j ⎠ αr
⎝ α2i −
α2j ⎠ α2r .
∂xr
i1
jp1
r1
i1
jp1
r1
Thus,
δ, ♦ m2 eαx ϕx δ, eαx T ϕx T ϕ0.
3.7
Since eαx ♦ m2 k δ, ϕx ♦ m2 k δ, eαx ϕx for every ϕx ∈ D and eαx ♦ m δ ∈ D , we have
2 k
♦m
2
k
αx
δ, e ϕx
k−1 2
αx
♦m
δ, ♦ m e ϕx
2
♦m
2
♦ m2
k−1
δ, e T ϕx
αx
k−2 δ, ♦ m2 eαx T ϕx
k−2
δ, eαx T T ϕx
k−2
δ, eαx T 2 ϕx .
♦ m2
♦ m2
3.8
Repeating this process ♦ m2 with k − 2 times, we finally obtain
♦ m2
k−2
δ, eαx T 2 ϕx
δ, eαx T k ϕx T k ϕ0,
3.9
where T k is the operator of 3.6 iterated k times. Now,
T k ϕ0 δ, T k ϕx Lδ, T k−1 ϕx ,
3.10
10
Abstract and Applied Analysis
by the operator L in 3.2 and the derivative of distribution. Continuing this process, we
obtain T k ϕ0 Lk δ, ϕx or eαx ♦ m2 k δ, ϕx Lk δ, ϕx. By equality of distributions, we obtain 3.1 as required. Since δ and its partial derivatives have support {0} which
is compact, hence, by Schwartz 25, Lk δ are tempered distributions and Lk δ has order 4k. It
follows that eαx ♦ m2 k δ is a tempered distribution of order 4k with point support {0} by
3.1. This completes the proof.
Lemma 3.2 boundedness property. Let D be the space of testing functions and D the space of
distributions. For every ϕ ∈ D and eαx ♦ m2 k δ ∈ D ,
k
αx
e ♦ m2 δ, ϕx ≤ M,
3.11
for some constant M.
Proof. Note that we have eαx ♦ m2 k δ, ϕx ♦ m2 k δ, eαx ϕx for every ϕx ∈ D
and eαx ♦ m2 k δ ∈ D . Hence,
♦m
2
k
αx
δ, e ϕx
k−1 k−1
2
αx
2
αx
♦m
δ, ♦ m e ϕx ♦m
δ, e T ϕx ,
2
3.12
where T is defined by 3.6. Continuing this process for k − 1 times, we will obtain
k
eαx ♦ m2 δ, ϕx δ, eαx T k ϕx T k ϕ0.
3.13
Since ϕ ∈ D, so ϕ0 is bounded, and also T k ϕ0 is bounded. It then follows that
k
αx
e ♦ m2 δ, ϕx T k ϕ0 ≤ M,
3.14
for some constant M.
4. The Application of Distribution eαx ♦ m2 k δ
Theorem 4.1. Let L be the partial differential operator defined by 3.2, and consider the equation
Lux δ,
4.1
where ux is any distribution in D , then ux eαx T2 x, m is the fundamental solution of the
operator L, where T2 x, m is defined by 2.16 with γ 2.
Abstract and Applied Analysis
11
Proof. From 3.1 and 4.1, we can write eαx ♦ m2 δ ∗ ux Lux δ. Convolving both
sides by eαx T2 x, m, we have
eαx T2 x, m ∗ eαx ♦ m2 δ ∗ ux eαx T2 x, m ∗ δ,
4.2
eαx T2 x, m ∗ ♦ m2 δ ∗ ux eαx T2 x, m,
4.3
then
or equivalently,
eαx
♦ m2 T2 x, m ∗ ux eαx T2 x, m.
4.4
Since ♦ m2 T2 x, m δ by Lemma 2.5 with k 1, we obtain
eαx δ ∗ ux eαx T2 x, m.
4.5
Moreover, since eαx δ δ, we have δ ∗ ux eαx T2 x, m. It then follows that ux eαx T2 x, m is the fundamental solution of the operator L.
Theorem 4.2 the generalization of Theorem 4.1. From Lemma 3.1, consider that
k
eαx ♦ m2 δ ∗ ux δ,
4.6
Lk ux δ,
4.7
or
then ux eαx T2k x, m is the fundamental solution of the operator Lk .
Proof. We can prove it by using either 4.6 or 4.7. If we start with 4.6, by convolving both
sides by eαx T2k x, m, we obtain
αx
e T2k x, m ∗ e
αx
♦m
2
k
δ ∗ ux
eαx T2k x, m ∗ δ,
4.8
or eαx ♦ m2 k T2k x, m ∗ ux eαx T2k x, m. Since ♦ m2 k T2k x, m δ by Lemma 2.5,
we have eαx δ ∗ ux eαx T2k x, m or ux eαx T2k x, m as required.
If we use 4.7, by convolving both sides by eαx T2 x, m, we obtain
eαx T2 x, m ∗ Lk ux eαx T2 x, m ∗ δ,
4.9
12
Abstract and Applied Analysis
or Leαx T2 x, m∗Lk−1 ux eαx T2 x, m. By Theorem 4.1, we obtain Lk−1 ux eαx T2 x, m.
Keeping on convolving eαx T2 x, m for k − 1 times, we finally obtain
ux eαx T2 x, m ∗ T2 x, m ∗ · · · ∗ T2 x, m eαx T2k x, m,
4.10
by Lemma 2.6 and 26, page 196.
In particular, if we put α α1 , α2 , . . . , αn 0 in 4.6, then 4.6 reduces to 2.21,
and we obtain ux T2k x, m as the fundamental solution of the diamond Klein-Gordon
operator iterated k times.
Theorem 4.3. Given the convolution equation
M
k
r
eαx ♦ m2 δ ∗ ux eαx Cr ♦ m2 δ,
4.11
r0
where ♦ m2 k is the diamond Klein-Gordon operator iterated k times defined by
♦ m2
k
⎛
⎞k
p
pq
2
2
∂
∂
⎝
−
m2 ⎠ ,
2
2
i1 ∂xi
jp1 ∂xj
4.12
the variable x x1 , x2 , . . . , xn ∈ Rn , the constant α α1 , α2 , . . . , αn ∈ Rn , m is a nonnegative
real number, δ is the Dirac-delta distribution with ♦ m2 0 δ δ, ♦ m2 1 δ ♦ m2 δ, and Cr
is a constant, then the type of solution ux of 4.11 depends on k, M, and α as follows:
1 if M < k and M 0, then the solution of 4.11 is
ux C0 eαx T2k x, m,
4.13
where T2k x, m is defined by 2.16 with γ 2k. If 2k ≥ n and for any α, then
eαx T2k x, m is the ordinary function,
2 if 0 < M < k, then the solution of 4.11 is
ux eαx
M
Cr T2k−2r x, m,
4.14
r1
which is an ordinary function for 2k − 2r ≥ n with any arbitrary constant α,
3 if M ≥ k and for any α one supposes that k ≤ M ≤ N, then 4.11 has
ux eαx
N
r−k
Cr ♦ m2
δ
rk
as a solution which is the singular distribution.
4.15
Abstract and Applied Analysis
Proof.
13
1 For M < k and M 0, then 4.11 becomes
k
eαx ♦ m2 δ ∗ ux C0 eαx δ C0 δ,
4.16
and by Theorem 4.2, we obtain
ux C0 eαx T2k x, m.
4.17
Now, by 2.2 and 2.7, Re2k x and RH
x are ordinary functions, respectively, for 2k ≥ n. It
2k
αx
then follows that C0 e T2k x, m is an ordinary function for 2k ≥ n with any α.
2 For 0 < M < k, then we can write 4.11 as
k
2
M !
eαx ♦ m2 δ ∗ ux eαx C1 ♦ m2 δ C2 ♦ m2 δ · · · CM ♦ m2 δ . 4.18
Convolving both sides by eαx T2k x, m and applying Lemma 2.5, we obtain
!
2
M
ux eαx C1 ♦ m2 T2k x, m C2 ♦ m2 T2k x, m · · · CM ♦ m2 T2k x, m .
4.19
It is known that ♦ m2 k T2k x, m δ, thus ♦ m2 k−r ♦ m2 r T2k x, m δ for r < k.
Convolving both sides by T2k−2r x, m, we obtain
k−r r
♦ m2 T2k x, m T2k−2r x, m,
T2k−2r x, m ∗ ♦ m2
4.20
or
♦ m2
k−r
r
T2k−2r x, m ∗ ♦ m2 T2k x, m T2k−2r x, m,
4.21
which leads to
♦ m2
r
T2k x, m T2k−2r x, m,
4.22
for r < k. It follows that
ux eαx C1 T2k−2 x, m C2 T2k−4 x, m · · · CM T2k−2M x, m,
4.23
or
ux eαx
M
r1
Cr T2k−2r x, m.
4.24
14
Abstract and Applied Analysis
Similarly, by case 1, eαx T2k−2r x, m is the ordinary function for 2k − 2r ≥ n with any α. It
follows that
ux eαx
M
Cr T2k−2r x, m
4.25
r1
is also the ordinary function with any α.
3 if M ≥ k and for any α, we suppose that k ≤ M ≤ N, then 4.11 becomes
k
k
k1
N !
δ · · · CN ♦ m2 δ .
eαx ♦ m2 δ ∗ ux eαx Ck ♦ m2 δ Ck1 ♦ m2
4.26
Convolving both sides by eαx T2k x, m and applying Lemma 2.5, we have
!
k
k1
N
ux eαx Ck ♦m2 T2k x, mCk1 ♦m2
T2k x, m· · ·CN ♦m2 T2k x, m .
4.27
Now,
♦ m2
M
M−k k
M−k
♦ m2 T2k x, m ♦ m2
T2k x, m ♦ m2
,
4.28
for k ≤ M ≤ N. Thus,
ux e
αx
e
αx
Ck δ Ck1 ♦ m δ Ck2 ♦ m
N
Cr ♦ m
2
2
r−k
2
2
δ · · · CN
N−k !
♦m
δ
2
4.29
δ.
rk
Now, by 3.1 and 3.2, we have
r−k
r−k
eαx ♦ m2
δ ♦ m2
δ terms of lower order of partial derivative of δ 4.30
for k ≤ r ≤ N. Since all terms on the right-hand side of this equation are singular distribution,
it follows that
ux eαx
N
r−k
Cr ♦ m2
δ
rk
is the singular distribution. This completes the proof.
4.31
Abstract and Applied Analysis
15
Acknowledgments
This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission, through
the Cluster of Research to Enhance the Quality of Basic Education, and the Centre of Excellence in Mathematics, Thailand.
References
1 I. M. Gelfand and G. E. Shilov, Generalized Functions, Academic Press, New York, NY, USA, 1964.
2 S. E. Trione, “On Marcel Riesz’s ultra hyperbolic kernel,” Trabajos de Mathematica, vol. 116, pp. 1–12,
1987.
3 M. A. Tellez, “The distributional Hankel transform of Marcel Riesz’s ultrahyperbolic kernel,” Studies
in Applied Mathematics, vol. 93, no. 2, pp. 133–162, 1994.
4 A. Kananthai, “On the solutions of the n-dimensional diamond operator,” Applied Mathematics and
Computation, vol. 88, no. 1, pp. 27–37, 1997.
5 X. J. Yang, “Local fractional integral transforms,” Progress in Nonlinear Science, vol. 4, pp. 1–225, 2011.
6 A. Kananthai, “On the convolution equation related to the diamond kernel of Marcel Riesz,” Journal
of Computational and Applied Mathematics, vol. 100, no. 1, pp. 33–39, 1998.
7 A. Kananthai, “On the convolutions of the diamond kernel of Marcel Riesz,” Applied Mathematics and
Computation, vol. 114, no. 1, pp. 95–101, 2000.
8 A. Kananthai, “On the diamond operator related to the wave equation,” Nonlinear Analysis: Theory,
Methods and Applications, vol. 47, no. 2, pp. 1373–1382, 2001.
9 A. Kananthai, “On the Fourier transform of the diamond kernel of Marcel Riesz,” Applied Mathematics
and Computation, vol. 101, no. 2-3, pp. 151–158, 1999.
10 A. Kananthai, “On the Green function of the diamond operator related to the Klein-Gordon operator,”
Bulletin of the Calcutta Mathematical Society, vol. 93, no. 5, pp. 353–360, 2001.
11 G. Sritanratana and A. Kananthai, “On the nonlinear diamond operator related to the wave equation,”
Nonlinear Analysis: Real World Applications, vol. 3, no. 4, pp. 465–470, 2002.
12 M. A. Tellez and A. Kananthai, “On the convolution product of the distributional families related to
the diamond operator,” Le Matematiche, vol. 57, no. 1, pp. 39–48, 2002.
13 A. Kananthai, “On the distribution related to the ultra-hyperbolic equations,” Journal of Computational
and Applied Mathematics, vol. 84, no. 1, pp. 101–106, 1997.
14 K. Nonlaopon, “The convolution equation related to the diamond kernel of Marcel Riesz,” Far East
Journal of Mathematical Sciences, vol. 45, no. 1, pp. 121–134, 2010.
15 A. Kananthai, “On the convolution equation related to the N-dimensional ultra-hyperbolic operator,”
Journal of Computational and Applied Mathematics, vol. 115, no. 1-2, pp. 301–308, 2000.
16 P. Sasopa and K. Nonlaopon, “The convolution equation related to the ultra-hyperbolic equations,”
Far East Journal of Mathematical Sciences, vol. 35, no. 3, pp. 295–308, 2009.
17 S. E. Trione, “On the elementary retarded ultra-hyperbolic solution of the Klein-Gordon operator,
iterated k-times,” Studies in Applied Mathematics, vol. 79, no. 2, pp. 127–141, 1988.
18 M. A. Tellez, “The convolution product of Wα u, m ∗ Wβ u, m,” Mathematicae Notae, vol. 38, pp. 105–
111, 1995-1996.
19 S. E. Trione, “On the elementary P ± i0λ ultrahyperbolic solution of the Klein-Gordon operator
iterated k-times,” Integral Transforms and Special Functions, vol. 9, no. 2, pp. 149–162, 2000.
20 A. Liangprom and K. Nonlaopon, “On the convolution equation related to the Klein-Gordon
operator,” International Journal of Pure and Applied Mathematics, vol. 71, no. 1, pp. 67–82, 2011.
k
21 J. Tariboon and A. Kananthai, “On the Green function of the ⊕ m2 operator,” Integral Transforms
and Special Functions, vol. 18, no. 4, pp. 297–304, 2007.
22 A. Lunnaree and K. Nonlaopon, “On the Fourier transform of the diamond Klein-Gordon kernel,”
International Journal of Pure and Applied Mathematics, vol. 68, no. 1, pp. 85–97, 2011.
23 K. Nonlaopon, A. Lunnaree, and A. Kananthai, “On the solutions of the n-dimensional diamond
Klien-Gordon operator and its convolution,” Submitted for publication.
24 Y. Nozaki, “On Riemann-Liouville integral of ultra-hyperbolic type,” Kodai Mathematical Seminar
Reports, vol. 16, no. 2, pp. 69–87, 1964.
16
Abstract and Applied Analysis
25 L. Schwartz, Theories des Distributions, vol. 1-2 of Actualites Scientifiques et Industriel, Hermann & Cie,
Paris, France, 1959.
26 R. P. Kanwal, Generalized Functions: Theory and Techniqu, Birkhäuser Boston: Hamilton Printing,
Boston, Mass, USA, 2nd edition, 1998.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014