Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol100.Issue1.1998:
Journal of Computational and Applied Mathematics 100 (1998) 33–39
On the convolution equation related to the diamond kernel
of Marcel Riesz
Amnuay Kananthai
Department of Mathematics, Chiang Mai University, Chiang Mai 50200, Thailand
Received 3 February 1998; revised 15 July 1998
Abstract
In this paper, we study the distribution e t ♦k where ♦k is introduced and named as the Diamond operator iterated
k-times (k = 0; 1; 2; : : :) and is de nded by
k
♦ =
@2
@2
@2
+
+
·
·
·
+
@t12
@t22
@tp2
2
−
@2
@2
@2
+
+
·
·
·
+
2
2
2
@tp+1
@tp+2
@tp+q
2 !k
;
where t = (t1 ; t2 ; : : : ; tn ) is a variable and = ( 1 ; 2 ; : : : ; n ) is a constant and both are the points in the n-dimensional
Euclidean space R n , is the Dirac-delta distribution with ♦0 = and p + q = n (the dimension of R n )
At rst, the properties of e t ♦k are studied and later we study the application of e t ♦k for solving the solutions of
the convolution equation
(e t ♦k ) ∗ u(t) = e t
m
X
cr ♦r :
r=0
We found that its solutions related to the Diamond Kernel of Marcel Riesz and moreover, the type of solutions such as,
c 1998 Elsevier
the classical solution (the ordinary function) or the tempered distributions depending on m; k and .
Science B.V. All rights reserved.
AMS classi cation: 46F10
Keywords: Diamond operator; Kernel of Marcel Riesz; Dirac delta distributions; Tempered distribution
1. Introduction
From [2, Theorem 3.1], the equation ♦k u(t) = has (−1)k S2k (t) ∗ R2k (t) as an elementary solution
and is called the Diamond Kernel of Marcel Riesz where S2k (t) and R2k (t) are de ned by (2.1) and
(2.2), respectively, with
= 2k where
is nonnegative.
c 1998 Elsevier Science B.V. All rights reserved.
0377-0427/98/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 2 8 - 9
34
A. Kananthai / Journal of Computational and Applied Mathematics 100 (1998) 33–39
Consider the convolution equation
(e t ♦k ) ∗ u(t) = e t
m
X
cr ♦r :
(1.1)
r=0
In nding the type of solutions u(t) of Eq. (1.1), we use the method of convolution of the
tempered distribution. Before going to that point, some de nitions and basic concepts are
needed.
2. Preliminaries
De nition 2.1. Let the function S
(t) be de ned by
S
= 2
−
−n=2
n−
2
|t|
−n
;
(
=2)
(2.1)
whereq
is a complex parameter, n is the dimension of R n , t = (t1 ; t2 ; : : : ; tn ) ∈ R n and
|t| = (t12 + · · · + tn2 ). Now S
is an ordinary function if Re(
)¿n and is a distribution of
if
Re(
)¡n.
2
2
De nition 2.2. Let t = (t1 ; t2 ; : : : ; tn ) be the point of R n and write v = t12 + t22 + · · · + tp2 − tp+1
− tp+2
−
2
n
· · · − tp+q , p + q = n. Denote by + = {t ∈ R : t1 ¿0 and v¿0} the set of an interior of the forward
cone and is the closure of .
For any complex number
, de ne
R
(t) =
(
−n)=2
V
Kn (
)
0
if t ∈
+;
if t ∈=
+;
where Kn (
) is given by the formula
Kn (
) =
(n−1)=2 ( 2+
−n
) ( 1−
) (
)
2
2
:
p−
2+
−p
( 2 ) ( 2 )
The function R
(t) was introduced by Nozaki [3, p.72]. It is well known that R
(t) is an ordinary
function if Re(
)¿n and is a distribution of if Re(
)¡n. Let supp R
(t) denote the support of
R
(t) and suppose that supp R
(t) ⊂ + .
Lemma 2.1. S
(t) and R
(t) are homogeneous distributions of order − n. Moreover they are
tempered distribution.
A. Kananthai / Journal of Computational and Applied Mathematics 100 (1998) 33–39
35
Proof. Since S
(t) and R
(t) satisfy the Euler equation
n
X
@R
(t)
ti
i=1
n
X
@S
(t)
ti
i=1
= ( − n)R
(t);
@ti
@ti
= ( − n)S
(t);
then they are homogeneous distribution of order − n by Donoghue [1, pp. 154,155] that proved
that every homogeneous distribution is a tempered distribution.
Lemma 2.2 (The convolution of tempered distribution). The convolution S
(t) ∗ R
(t) exists and is
a tempered distribution.
Proof. Choose supp R
(t) = K ⊂ + where K is a compact set. Then R
(t) is a tempered distribution
with compact support and by Donoghue [1, pp. 156–159], S
(t) ∗ R
(t) exists and is a tempered
distribution.
3. The properties of e t ♦k
Lemma 3.1.
e t ♦k = Lk ;
(3.1)
where L is the partial di erential operator of Diamond type and is de ned by
L≡♦+
n
X
r2
−2
r=1 i=1
r=1
+2
n X
r
X
p+q
n
X
X
r=1 j=p+1
@3
@3
r 2 + j
@tj @tr
@tj @tr2
p
n X
X
@2
+4
r j
−2
r=1
+
+
i=1
i2 −
i=1
p
X
i=1
!
p+q
n
X
X
p
p+q
X
X
@
@
j
−
r2 i
p
X
!
@2
−
r j
@ti @tr
@tj @tr
r=1 j=p+1
r=1 i=1
n
X
@3
@3
r 2 + i
@ti @tr
@ti @tr2
@ti
p+q
X
j=p+1
i2 −
p+q
X
j=p+1
@tj
j=p+1
j2 − 2
j2
n
X
r=1
r2 ;
p
X
i=1
i2 −
p+q
X
j=p+1
j2
n
X
r=1
r
@
@tr
(3.2)
36
A. Kananthai / Journal of Computational and Applied Mathematics 100 (1998) 33–39
P
2
P
2
@
where = pi=1 @t@ 2 − p+q
j=p @tj2 ; =
i
distribution of order 4k.
Pp
@2
i=1 @ti2 ;
p + q = n. Actually ♦ = and e t ♦k is a tempered
Proof. For k = 1, we have
he t ♦; ’(t)i = h; ♦e t ’(t)i;
where ’(t) ∈ S the Schwartz space. By computing directly we obtain
♦e t ’(t) = e t M’(t);
(3.3)
where M is the partial di erential operator of the form (3.2) whose the coecients of the third term,
the fourth term, the sixth term and the eighth term of the right-hand side of Eq. (3.2) have opposite
signs.
Thus h; ♦e t ’(t)i = h; e t M’(t)i = M’(0).
By the properties of and its partial derivatives with the linear di erential operator M , we obtain
M’(0) = hL; ’i where L de ned by Eq. (3.2). It follows that e t ♦ = L. Now
(e t ♦) ∗ (e t ♦) ∗ · · · ∗ (e t ♦) = (L) ∗ (L) ∗ · · · ∗ (L) ;
{z
}
{z
}
|
k -times
k -times
|
we have e t ( ∗ ♦k ) = ∗ (Lk ). Thus e t ♦k = Lk . It follows that, for any k, we obtain Eq. (3.1).
Since has a compact support, hence by Schwartz [4], and Lk are tempered distributions and
Lk has order 4k. It follows that e t ♦k is a tempered distribution of order 4k.
Lemma 3.2 (Boundedness property). |he t ♦k ; ’(t)i|6K where K is a constant and ’ ∈ S.
Proof.
he t ♦k ; ’(t)i=h♦k ; e t ’(t)i
=h♦k−1 ; ♦e t ’(t)i
=h♦k−1 ; e t M’(t)i;
where M is de ned by Eq. (3.3). By keeping on operate ♦k − 1 times we obtain
he t ♦k ; ’ti=h; e t M k ’ti
=M k ’(0):
Since ’(0) is bounded and also M k ’(0) is bounded. It follows that |he t ♦k ; ’(t)| = |M k
’(0)|6K.
4. The application of e t ♦k
Given u(t) is an distribution and by Lemma 3.1, we have
(e t ♦k ) ∗ u(t) = (Lk ) ∗ u(t) = Lk u(t);
where L is de ned by (3.2).
A. Kananthai / Journal of Computational and Applied Mathematics 100 (1998) 33–39
37
Theorem 4.1. Given the linear partial di erential equation of the form
(e t ♦k ) ∗ u(t) = Lk u(t) = :
(4.1)
Then u(t) = e t (−1)k S2k (t) ∗ R2k (t) is an elementary solution of (4:1) or The Diamond Kernel of
Marcel Riesz of (4:1) where S2k (t) and R2k (t) are de ned by (2:1) and (2:2) respectively with
= 2k.
Proof. By Kananthai [2, Lemma 2.4], (−1)k S2k (t) is an elementary solution of the Laplace operator
k iterated k-times and also by Trione [5], R2k (t) is an elementary solution of the ultra-hyperbolic
operator k iterated k-times (that is k (−1)k S2k (t) = and k R2k (t) = ).
Now ♦k = k k , consider e t ( k k ) ∗ R2k (t) = By Lemma 2.2 with
= 2k, (−1)k S2k (t) ∗ R2k (t)
exists and is a tempered distribution.
Convolving both sides of the above equation by e t [(−1)k S2k (t) ∗ R2k (t)] we obtain
e t [k (−1)k S2k (t) ∗
k
R2k (t) ∗ u(t)] = [e t (−1)k S2k (t) ∗ R2k (t)] ∗
(e t ) ∗ u(t) = e t (−1)k S2k (t) ∗ R2k (t):
It follows that u(t) = e t (−1)k S2k (t) ∗ R2k (t).
Theorem 4.2. Given the convolution equation
(e t ♦k ) ∗ u(t) = e t
m
X
cr ♦r ;
(4.2)
r=0
then the type of solutions of (4:2) depend on the relationship between k; m and are as the
following cases.
(1) If m¡k and m = 0 then (4:2) has the solution u(t) = e t [c0 (−1)k S2k (t) ∗ R2k (t)] and u(t) is an
ordinary function for 2k ¿n with any and is a tempered distribution for 2k¡n and for some
= ( 1 ; 2 ; : : : ; n ) with i ¡0 (i = 1; 2; : : : ; n).
(2) If 0¡m¡k; then the solution of (4:2) is
u(t) = e t
"
m
X
cr (−1)k−r S2k−2r (t) ∗ R2k−2r (t)
r=1
#
which is an ordinary function for 2k − 2r ¿n with any and is a tempered distribution if
2k − 2r¡n for some with i ¡0 (i = 1; 2; : : : ; n).
P
r−k
as
(3) If m¿k and for any and suppose that k 6m6M; then (4:2) has u(t) = e t M
r=k cr ♦
a solution which is the singular distribution.
Proof. (1) For m¡k and m = 0, then (4.2) becomes (e t ♦k ) ∗ u(t) = e t C0 = C0 e t = C0 . By
Theorem 4.1 we obtain
u(t) = C0 e t ((−1)k S2k (t) ∗ R2k (t)):
Now, by (2.1) and (2.2), S2k (t) and R2k (t) are ordinary functions respectively for 2k ¿n. It follows
that u(t) is an ordinary function for any constant . If 2k¡n then S2k (t) and R2k (t) are the analytic
38
A. Kananthai / Journal of Computational and Applied Mathematics 100 (1998) 33–39
functions except at the origin and by Lemma 2.1 S2k (t) and R2k (t) are tempered distributions and
by Lemma 2.2, (−1)k S2k (t) ∗ R2k (t) exists and is a tempered distribution.
Now, for some = ( 1 ; 2 ; : : : ; n ) with i ¡0 (i = 1; 2; : : : ; n) we have e t is a slow growth function
and also its partial derivative is a slow growth. It follows that e t [(−1)k S2k (t) ∗ R2k (t)] is also a
tempered distribution.
(2) For 0¡m¡k, we have
(e t ♦k ) ∗ u(t) = e t [c1 ♦ + c2 ♦2 + · · · + cm ♦m ]:
Convolving both sides by e t [(−1)k S2k (t) ∗ R2k (t)], we obtain
u(t) = e t [c1 ♦((−1)k S2k (t) ∗ R2k (t)) + c2 ♦2 ((−1)k S2k (t) ∗ R2k (t))
+ · · · + cm ♦k ((−1)k S2k (t) ∗ R2k (t))]
= e t [c1 (−1)k−1 S2k−2 (t) ∗ R2k−2 (t) + c2 (−1)k−2 S2k−4 (t) ∗ R2k−4 (t)
+ · · · + cm (−1)k−m S2k−2m (t) ∗ R2k−2m (t)]
=e
t
"
m
X
(−1)
r=1
k−r
#
S2k−2r (t) ∗ R2k−2r (t)
by
Theorem 4.1 and by Kananthai [2, Theorem 3.2] for r¡k. Similarly, as in the case (1) e t
Pm
[ r=1 (−1)k−r S2k−2r (t) ∗ R2k−2r (t)] is the ordinary function if 2k − 2r ¿n and for any , and is a
tempered distribution if 2k − 2r¡n and for some with i ¡0 (i = 1; 2; : : : ; n).
(3) For m¿k and for any , suppose that k 6m6M we have
(e t ♦k ) ∗ u(t) = ck e t ♦k + ck+1 e t ♦k+1 + · · · + cM e t ♦k :
Convoluing both sides by e t ((−1)k S2k (t) ∗ R2k (t)) and by Kananthai [2, Theorem 3.2] for k 6m6M ,
we obtain
u(t) = ck e t + ck+1 e t ♦ + ck+2 e t ♦2 + · · · + cM e t ♦M −k
P
r−k
or u(t) = e t M
. Since e t ♦r−k = Lr−k for k 6r 6M and L is de ned by (3.2). Thus Lr−k
r=k cr ♦
is a singular distribution. It follows that u(t) is a singular distribution. That completes
the proof.
Acknowledgements
The author would like to thank the Thailand Research Fund for nancial support and also
Dr. Suthep Suantai, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, Thailand, for many helpful discussions.
A. Kananthai / Journal of Computational and Applied Mathematics 100 (1998) 33–39
39
References
[1] W.F. Donoghue, Distribution and Fourier Transform, Academic Press, 1969.
[2] A. Kananthai, On the solution of the n-dimensional Diamond operator, Applied Mathematics and Computation,
Elsevier, 88 (1997), 27 – 37.
[3] Y. Nozaki, On the Riemann–Liouville integral of ultra-hyperbolic type Kodai Mathematical Seminar Reports 6(2)
(1964) 69 – 87.
[4] L. Schwartz, Theorie des Distribution, Vols. 1 and 2 Actualite’s, Scienti ques et Industrial, Hermann, Paris, 1957,
1959.
[5] S.E. Trione, On Marcel Riesz’s ultra-hyperbolic Kernel, Trabajos de Matematica 116 preprint, 1987.
On the convolution equation related to the diamond kernel
of Marcel Riesz
Amnuay Kananthai
Department of Mathematics, Chiang Mai University, Chiang Mai 50200, Thailand
Received 3 February 1998; revised 15 July 1998
Abstract
In this paper, we study the distribution e t ♦k where ♦k is introduced and named as the Diamond operator iterated
k-times (k = 0; 1; 2; : : :) and is de nded by
k
♦ =
@2
@2
@2
+
+
·
·
·
+
@t12
@t22
@tp2
2
−
@2
@2
@2
+
+
·
·
·
+
2
2
2
@tp+1
@tp+2
@tp+q
2 !k
;
where t = (t1 ; t2 ; : : : ; tn ) is a variable and = ( 1 ; 2 ; : : : ; n ) is a constant and both are the points in the n-dimensional
Euclidean space R n , is the Dirac-delta distribution with ♦0 = and p + q = n (the dimension of R n )
At rst, the properties of e t ♦k are studied and later we study the application of e t ♦k for solving the solutions of
the convolution equation
(e t ♦k ) ∗ u(t) = e t
m
X
cr ♦r :
r=0
We found that its solutions related to the Diamond Kernel of Marcel Riesz and moreover, the type of solutions such as,
c 1998 Elsevier
the classical solution (the ordinary function) or the tempered distributions depending on m; k and .
Science B.V. All rights reserved.
AMS classi cation: 46F10
Keywords: Diamond operator; Kernel of Marcel Riesz; Dirac delta distributions; Tempered distribution
1. Introduction
From [2, Theorem 3.1], the equation ♦k u(t) = has (−1)k S2k (t) ∗ R2k (t) as an elementary solution
and is called the Diamond Kernel of Marcel Riesz where S2k (t) and R2k (t) are de ned by (2.1) and
(2.2), respectively, with
= 2k where
is nonnegative.
c 1998 Elsevier Science B.V. All rights reserved.
0377-0427/98/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 2 8 - 9
34
A. Kananthai / Journal of Computational and Applied Mathematics 100 (1998) 33–39
Consider the convolution equation
(e t ♦k ) ∗ u(t) = e t
m
X
cr ♦r :
(1.1)
r=0
In nding the type of solutions u(t) of Eq. (1.1), we use the method of convolution of the
tempered distribution. Before going to that point, some de nitions and basic concepts are
needed.
2. Preliminaries
De nition 2.1. Let the function S
(t) be de ned by
S
= 2
−
−n=2
n−
2
|t|
−n
;
(
=2)
(2.1)
whereq
is a complex parameter, n is the dimension of R n , t = (t1 ; t2 ; : : : ; tn ) ∈ R n and
|t| = (t12 + · · · + tn2 ). Now S
is an ordinary function if Re(
)¿n and is a distribution of
if
Re(
)¡n.
2
2
De nition 2.2. Let t = (t1 ; t2 ; : : : ; tn ) be the point of R n and write v = t12 + t22 + · · · + tp2 − tp+1
− tp+2
−
2
n
· · · − tp+q , p + q = n. Denote by + = {t ∈ R : t1 ¿0 and v¿0} the set of an interior of the forward
cone and is the closure of .
For any complex number
, de ne
R
(t) =
(
−n)=2
V
Kn (
)
0
if t ∈
+;
if t ∈=
+;
where Kn (
) is given by the formula
Kn (
) =
(n−1)=2 ( 2+
−n
) ( 1−
) (
)
2
2
:
p−
2+
−p
( 2 ) ( 2 )
The function R
(t) was introduced by Nozaki [3, p.72]. It is well known that R
(t) is an ordinary
function if Re(
)¿n and is a distribution of if Re(
)¡n. Let supp R
(t) denote the support of
R
(t) and suppose that supp R
(t) ⊂ + .
Lemma 2.1. S
(t) and R
(t) are homogeneous distributions of order − n. Moreover they are
tempered distribution.
A. Kananthai / Journal of Computational and Applied Mathematics 100 (1998) 33–39
35
Proof. Since S
(t) and R
(t) satisfy the Euler equation
n
X
@R
(t)
ti
i=1
n
X
@S
(t)
ti
i=1
= ( − n)R
(t);
@ti
@ti
= ( − n)S
(t);
then they are homogeneous distribution of order − n by Donoghue [1, pp. 154,155] that proved
that every homogeneous distribution is a tempered distribution.
Lemma 2.2 (The convolution of tempered distribution). The convolution S
(t) ∗ R
(t) exists and is
a tempered distribution.
Proof. Choose supp R
(t) = K ⊂ + where K is a compact set. Then R
(t) is a tempered distribution
with compact support and by Donoghue [1, pp. 156–159], S
(t) ∗ R
(t) exists and is a tempered
distribution.
3. The properties of e t ♦k
Lemma 3.1.
e t ♦k = Lk ;
(3.1)
where L is the partial di erential operator of Diamond type and is de ned by
L≡♦+
n
X
r2
−2
r=1 i=1
r=1
+2
n X
r
X
p+q
n
X
X
r=1 j=p+1
@3
@3
r 2 + j
@tj @tr
@tj @tr2
p
n X
X
@2
+4
r j
−2
r=1
+
+
i=1
i2 −
i=1
p
X
i=1
!
p+q
n
X
X
p
p+q
X
X
@
@
j
−
r2 i
p
X
!
@2
−
r j
@ti @tr
@tj @tr
r=1 j=p+1
r=1 i=1
n
X
@3
@3
r 2 + i
@ti @tr
@ti @tr2
@ti
p+q
X
j=p+1
i2 −
p+q
X
j=p+1
@tj
j=p+1
j2 − 2
j2
n
X
r=1
r2 ;
p
X
i=1
i2 −
p+q
X
j=p+1
j2
n
X
r=1
r
@
@tr
(3.2)
36
A. Kananthai / Journal of Computational and Applied Mathematics 100 (1998) 33–39
P
2
P
2
@
where = pi=1 @t@ 2 − p+q
j=p @tj2 ; =
i
distribution of order 4k.
Pp
@2
i=1 @ti2 ;
p + q = n. Actually ♦ = and e t ♦k is a tempered
Proof. For k = 1, we have
he t ♦; ’(t)i = h; ♦e t ’(t)i;
where ’(t) ∈ S the Schwartz space. By computing directly we obtain
♦e t ’(t) = e t M’(t);
(3.3)
where M is the partial di erential operator of the form (3.2) whose the coecients of the third term,
the fourth term, the sixth term and the eighth term of the right-hand side of Eq. (3.2) have opposite
signs.
Thus h; ♦e t ’(t)i = h; e t M’(t)i = M’(0).
By the properties of and its partial derivatives with the linear di erential operator M , we obtain
M’(0) = hL; ’i where L de ned by Eq. (3.2). It follows that e t ♦ = L. Now
(e t ♦) ∗ (e t ♦) ∗ · · · ∗ (e t ♦) = (L) ∗ (L) ∗ · · · ∗ (L) ;
{z
}
{z
}
|
k -times
k -times
|
we have e t ( ∗ ♦k ) = ∗ (Lk ). Thus e t ♦k = Lk . It follows that, for any k, we obtain Eq. (3.1).
Since has a compact support, hence by Schwartz [4], and Lk are tempered distributions and
Lk has order 4k. It follows that e t ♦k is a tempered distribution of order 4k.
Lemma 3.2 (Boundedness property). |he t ♦k ; ’(t)i|6K where K is a constant and ’ ∈ S.
Proof.
he t ♦k ; ’(t)i=h♦k ; e t ’(t)i
=h♦k−1 ; ♦e t ’(t)i
=h♦k−1 ; e t M’(t)i;
where M is de ned by Eq. (3.3). By keeping on operate ♦k − 1 times we obtain
he t ♦k ; ’ti=h; e t M k ’ti
=M k ’(0):
Since ’(0) is bounded and also M k ’(0) is bounded. It follows that |he t ♦k ; ’(t)| = |M k
’(0)|6K.
4. The application of e t ♦k
Given u(t) is an distribution and by Lemma 3.1, we have
(e t ♦k ) ∗ u(t) = (Lk ) ∗ u(t) = Lk u(t);
where L is de ned by (3.2).
A. Kananthai / Journal of Computational and Applied Mathematics 100 (1998) 33–39
37
Theorem 4.1. Given the linear partial di erential equation of the form
(e t ♦k ) ∗ u(t) = Lk u(t) = :
(4.1)
Then u(t) = e t (−1)k S2k (t) ∗ R2k (t) is an elementary solution of (4:1) or The Diamond Kernel of
Marcel Riesz of (4:1) where S2k (t) and R2k (t) are de ned by (2:1) and (2:2) respectively with
= 2k.
Proof. By Kananthai [2, Lemma 2.4], (−1)k S2k (t) is an elementary solution of the Laplace operator
k iterated k-times and also by Trione [5], R2k (t) is an elementary solution of the ultra-hyperbolic
operator k iterated k-times (that is k (−1)k S2k (t) = and k R2k (t) = ).
Now ♦k = k k , consider e t ( k k ) ∗ R2k (t) = By Lemma 2.2 with
= 2k, (−1)k S2k (t) ∗ R2k (t)
exists and is a tempered distribution.
Convolving both sides of the above equation by e t [(−1)k S2k (t) ∗ R2k (t)] we obtain
e t [k (−1)k S2k (t) ∗
k
R2k (t) ∗ u(t)] = [e t (−1)k S2k (t) ∗ R2k (t)] ∗
(e t ) ∗ u(t) = e t (−1)k S2k (t) ∗ R2k (t):
It follows that u(t) = e t (−1)k S2k (t) ∗ R2k (t).
Theorem 4.2. Given the convolution equation
(e t ♦k ) ∗ u(t) = e t
m
X
cr ♦r ;
(4.2)
r=0
then the type of solutions of (4:2) depend on the relationship between k; m and are as the
following cases.
(1) If m¡k and m = 0 then (4:2) has the solution u(t) = e t [c0 (−1)k S2k (t) ∗ R2k (t)] and u(t) is an
ordinary function for 2k ¿n with any and is a tempered distribution for 2k¡n and for some
= ( 1 ; 2 ; : : : ; n ) with i ¡0 (i = 1; 2; : : : ; n).
(2) If 0¡m¡k; then the solution of (4:2) is
u(t) = e t
"
m
X
cr (−1)k−r S2k−2r (t) ∗ R2k−2r (t)
r=1
#
which is an ordinary function for 2k − 2r ¿n with any and is a tempered distribution if
2k − 2r¡n for some with i ¡0 (i = 1; 2; : : : ; n).
P
r−k
as
(3) If m¿k and for any and suppose that k 6m6M; then (4:2) has u(t) = e t M
r=k cr ♦
a solution which is the singular distribution.
Proof. (1) For m¡k and m = 0, then (4.2) becomes (e t ♦k ) ∗ u(t) = e t C0 = C0 e t = C0 . By
Theorem 4.1 we obtain
u(t) = C0 e t ((−1)k S2k (t) ∗ R2k (t)):
Now, by (2.1) and (2.2), S2k (t) and R2k (t) are ordinary functions respectively for 2k ¿n. It follows
that u(t) is an ordinary function for any constant . If 2k¡n then S2k (t) and R2k (t) are the analytic
38
A. Kananthai / Journal of Computational and Applied Mathematics 100 (1998) 33–39
functions except at the origin and by Lemma 2.1 S2k (t) and R2k (t) are tempered distributions and
by Lemma 2.2, (−1)k S2k (t) ∗ R2k (t) exists and is a tempered distribution.
Now, for some = ( 1 ; 2 ; : : : ; n ) with i ¡0 (i = 1; 2; : : : ; n) we have e t is a slow growth function
and also its partial derivative is a slow growth. It follows that e t [(−1)k S2k (t) ∗ R2k (t)] is also a
tempered distribution.
(2) For 0¡m¡k, we have
(e t ♦k ) ∗ u(t) = e t [c1 ♦ + c2 ♦2 + · · · + cm ♦m ]:
Convolving both sides by e t [(−1)k S2k (t) ∗ R2k (t)], we obtain
u(t) = e t [c1 ♦((−1)k S2k (t) ∗ R2k (t)) + c2 ♦2 ((−1)k S2k (t) ∗ R2k (t))
+ · · · + cm ♦k ((−1)k S2k (t) ∗ R2k (t))]
= e t [c1 (−1)k−1 S2k−2 (t) ∗ R2k−2 (t) + c2 (−1)k−2 S2k−4 (t) ∗ R2k−4 (t)
+ · · · + cm (−1)k−m S2k−2m (t) ∗ R2k−2m (t)]
=e
t
"
m
X
(−1)
r=1
k−r
#
S2k−2r (t) ∗ R2k−2r (t)
by
Theorem 4.1 and by Kananthai [2, Theorem 3.2] for r¡k. Similarly, as in the case (1) e t
Pm
[ r=1 (−1)k−r S2k−2r (t) ∗ R2k−2r (t)] is the ordinary function if 2k − 2r ¿n and for any , and is a
tempered distribution if 2k − 2r¡n and for some with i ¡0 (i = 1; 2; : : : ; n).
(3) For m¿k and for any , suppose that k 6m6M we have
(e t ♦k ) ∗ u(t) = ck e t ♦k + ck+1 e t ♦k+1 + · · · + cM e t ♦k :
Convoluing both sides by e t ((−1)k S2k (t) ∗ R2k (t)) and by Kananthai [2, Theorem 3.2] for k 6m6M ,
we obtain
u(t) = ck e t + ck+1 e t ♦ + ck+2 e t ♦2 + · · · + cM e t ♦M −k
P
r−k
or u(t) = e t M
. Since e t ♦r−k = Lr−k for k 6r 6M and L is de ned by (3.2). Thus Lr−k
r=k cr ♦
is a singular distribution. It follows that u(t) is a singular distribution. That completes
the proof.
Acknowledgements
The author would like to thank the Thailand Research Fund for nancial support and also
Dr. Suthep Suantai, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, Thailand, for many helpful discussions.
A. Kananthai / Journal of Computational and Applied Mathematics 100 (1998) 33–39
39
References
[1] W.F. Donoghue, Distribution and Fourier Transform, Academic Press, 1969.
[2] A. Kananthai, On the solution of the n-dimensional Diamond operator, Applied Mathematics and Computation,
Elsevier, 88 (1997), 27 – 37.
[3] Y. Nozaki, On the Riemann–Liouville integral of ultra-hyperbolic type Kodai Mathematical Seminar Reports 6(2)
(1964) 69 – 87.
[4] L. Schwartz, Theorie des Distribution, Vols. 1 and 2 Actualite’s, Scienti ques et Industrial, Hermann, Paris, 1957,
1959.
[5] S.E. Trione, On Marcel Riesz’s ultra-hyperbolic Kernel, Trabajos de Matematica 116 preprint, 1987.