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Computational Technologies
Vol 7, № 5, 2002
ON THE FOURIER TRANSFORM OF THE DISTRIBUTIONAL KERNEL KaßrfV RELATED TO THE OPERATOR ©k ' ' '
A. Kananthai, S. Suantai Department of Mathematics Chiangmai University, Thailand e-mail: malamnka@science.cmu.ac.th
Рассмотрено преобразование Фурье ядра , где а, в, 7, V — комплексные па-
раметры. Исследовано преобразование Фурье свертки * ', где
а, в, 7, V, а', в', 7', V' — комплексные параметры.
1. Introduction
The operator ©k can be factorized into the form
efc:
2 / p + q ^ \ 2
v—У-(v—
^8x2) \ ^ dx2
k
V — + г V —
dx2 dx2
r=1 ' j=p+1 j.
k
v — - i p+q —'
dx2 dx2
r=1 ' j=P+1 j.
k
(1.1)
where p + q = n is the dimension of the space cn, i = \J—\ and k is a nonnegative integer.
2 / \ 2
p ö2 \2 / p+q 02
The operator ( ) — S тт~2 I is first introduced by A. Kananthai [1] and named
r=1 dxr- / \j=P+1 dx2
the Dimond operator denoted by
2
P d2 \ 2 / P+q ö^2
♦ =( £ Ы -(^a^l- (12)
,r=1 '/ \J=P+1
Let us denote the operators L1 and L2 by
p d 2 p+q d2
Ll = £ + 8 J+dX2' (L3)
r=1 ' J=P+1 j P d2 p+q d2
L2 = - 8 2+dXf- (L4)
r=1 ' j=P+1 j
Thus (1.1) can be written by
0k = ♦ Lk Lk. (1.5)
© A. Kananthai, S. Suantai, 2002.
Now consider the convolutions RH (u) * RJ(v) * SY(w) * Tv (z) where R^ , Rj, SY and Tv are defined by (2.2), (2.4), (2.6) and (2.7) respectively. We defined the distributional kernel Ka,j,Y,v by
Ka,pa,v = RHH * Rj * SY * Tv. (1.6)
Since the function RH(u),Rfj3(v),SY(w) and Tv(z) are all tempered distribution see [1, p. 30, 31] and [6, p. 154, 155], then the convolutions on the right hand side of (1.6) exists and is a tempered distribution. Thus Ka,j,Y,v is well defined and also a tempered distribution.
In this paper, at first we study the Fourier transform or Ka,j,Y,v where KajYiV is
defined by (1.6).
After that we put a = 3 = y = v = 2k, then we obtain K2k,2k,2k,2k related to the elementary solution of the operator ©k.
We also study the Fourier transform of the convolution Ka,j,Y,v * Ka>,j>>Y>>v>.
2. Preliminaries
Definition 2.1. Let x = (xi,x2, ...,xn) be a point in the space cn of the n-dimensional complex space and write
u x 2 + x2 + ... + xp xp+i ... xp+q, (2.1)
where p + q = n is the dimension of cn.
Denote by = {x £ rn : xi > 0 and u > 0} the set of an interior of the forward cone and r+ denotes it closure and Rn is the n-dimensional Euclidean space. For any complex number a, define
( a — n
u 2
RHa (u)={ K(a) for x £ r+' (2.2)
0 for x £ r+, where the constant Kn(a) is given by the formula
n r f2^-^) V(l—a ) r(a)
2 + a - p\ r(p - a
Kn(a)
The function RH is called the ultra-hyperbolic Kernel of Marcel Riesz and was introduced by Y. Nozaki [5, p. 72].
It is well known that RH is an ordinary function if Re (a) > n and is a distribution of a if Re (a) < n. Let supp RH (u) denote the support of RH (u) and suppose supp RH (u) C r+, that is supp RH(u) is compact.
Definition 2.2. Let x = (xi,x2, ...,xn) £ rn and write
v = x\ + x2 + ... + x\. (2.3)
For any complex number ft, define
<(v) = 2-^n^r(-(l. (2.4)
rif
The function R^ (v) is called the elliptic Kernel of Marcel Riesz and is ordinary function for Re (ft) > n and is a distribution of ft for Re (ft) < n.
Definition 2.3. Let x = (x1; x2,..., xn) be a point of the space cn of the n-dimensional complex space and write
W — X2 + X2 + ... + Xp i (Xp+1 + Xp+2 + ••• + xp+j , (2.5)
where p + q — n is the dimension of cn and i — V —1. For any complex number 7, define the function
S (w) = 2-Yn-nr (^) ^. (2.6)
V 2 y
The function (w)is an ordinary function if Re (7) > n and is a distribution of 7 for Re(7) < n. Definition 2.4. For any complex number v, define the function
\ v — n
n — v \ Z 2
(;) — 2^ - U~JF(|)' (2-7)
where
z — X2 + X2 + ... + Xp + i (xp+i + Xp+2 + ... + Xp+J , (2.8)
X — (x1; x2,..., xn) g cn, p + q — n is the dimension of cn and i — v —1.
We have Tv(z)is an ordinary function if Re(v) > n and is a distribution of v for Re(v) < n. Definition 2.5. Let f (x) be continuous function on rn where x — (x1; x2,..., xn) g rn. The Fourier transform of f (x) denoted by of or f (C) and is defined by
of (x) — f (C) — J e-i(i'x)f (x)dX, (2.9)
Rn
where C — (Ci, C2, ...,Cn) g rn and (C,x) — C1X1 + C2X2 + ... + CnXn.
Definition 2.6. Let ^(x) be a tempered distribution with compact support. The Fourier transform of ^(x) is defined by
¿(C)—<Mx),e-i(«>x) >. (2.10)
Lemma 2.1. The functions R^, R^, and Tv defined by (2.2), (2.4), (2.6) and (2.7) respectively, are all tempered distributions. Proof see [1, p. 30, 31] and [6, p. 154, 155].
Lemma 2.2. The function (—1)kK2k;2k;2k;2k(x) is an elementary solution of the operator ®fc, that is ®fc(—1)kK2fc;2fc,2fc,2fc(x) — 6 where ®fc is defined by (1.1), K2fc;2fc,2fc,2fc(x) is defined by (1.6) with a — ft — 7 — v — 2k and 6 is the Dirac-delta distribution.
Proof see [4, p. 66].
Lemma 2.3. 1. The Fourier transform of the convolution RH(u) * R^(v) is given by the formula
a \ f ß
9 R(u) *Rß(v))
(i)q 2a+ß r( ay.2
n
Kn(a)Hn(ß)r| r(n '
ß
p p+q
Ee? -E ej
r=1 j=p+1
2 J \ 2
where RH (u) and R^(v) are defined by (2.2) and (2.4) respectively,
' 3 '
A
—ß
E«
r=1
(2.11)
Hn(ß)
r| 2 I 2ßn-
r
n - ß
and i = V -1.
In particular, if a = 3 = 2k then (2.11) becomes
9(RH(u) * R?k(v))
(-1)
p+q
EC2 - £ e?
, r=1
j=p+1
(2.12)
where k is nonnegative integer and (—1)kRHk(u) * Rf2k(v) is an elementary solution of the operator ♦ iterated k-times defined by (1.2).
Moreover |s(R2k(u) *Rf2k(v))| < M, where M is constant, that is s is bounded, that implies s is continuous on the space S' of the tempered distribution.
2. The Fourier transform of the convolution SY(w) * Tv(z) is given by the formula
2Y+VnnT{ ^ r 'V
9(Sy(w) * Tv(z))
hn(y)hn(v) r /n - tf n - v
x
x
A
p+q
—Y
E«?+
r=1
j=p+1
A
p+q
Ee? -
(2.13)
r=1
j=p+1
where SY(w) and Tv(z) are defined by (2.6) and (2.7) respectively,
Y
ri-j2^ n r( V-\2 n 2
Hn(Y) = ~^M-^ and Hn(v) = - V2
r
n - Y
r
n — v
In particular, if y = v = 2k then (2.13) becomes S(Sy(w) * Tv(z)) --
(e?+e? +...+e?)2 + (e?+1+e?+2 +...+eP+q)2
(2.14)
— a
k
2
2
p
1
—v
p
p
1
k
where k is a nonnegative integer and (—1)k(—i)2S2k(w) and (—1)k(—i)2T2k(z) are elementary solutions of the operators L and L2 defined by (1.3) and (1.4) respectively.
Proof: 1. To prove (2.11) and (2.12) see [2] and to show that ö is bounded, now
(u) * R2fc (v))| =
(—i)k
p+q
EC2) — ( EC2
r=i
j=p+i
<
EC
r=i
+
p+q
e e2 j=p+i
k —
< M,
where p + q = n for large g r (r = 1, 2,..., n).
That implies that s is continuous on the space S' of tempered distribution. For the case (—1)kR^c(u) * (v) is an elementary solution of the operator ♦, see [1]. 2. We have
(w)
w ^
Hn(Y) ■
where Hn(Y)
n 2 ) ^ n 2
r
n — y
and w — X-2 + X2 + ... + Xp i(Xp+i + Xp+2 + ••• + Xp+q).
Now, changing the variable xi — yi, x2 — y2, ..., xp — yp,
— yp+i — yp+2 xp+i ,-:, Xp+2
p+q
yp+q
Then we obtaill w — y2 + y2 + ... + yp + yp+i + yp+2 + ... + yp+q. Let p2 — y2 + y2 + ... + yp+q, p + q — n. Then
(w)
Hn(Y)
3 i(i'x)w^ dx
^Vt i e-i«'V-n d((^i,X2,...,Xra)) dyidy2...dy„ Hn (Y ^ d(yi,y2,...,yn)
Hn(Y )(—i)2
pY-rae-i(im +Î2y2 +...+ÇPyP+yp+l+...+^p+f yp+q )dy
Hn(Y )(—i)2
2Yn2v^2++...+ep + i(ep+i+ep+2 +...+ep+q)) 7 (2.15)
v — n
z 2
by [5, p. 194]
Similarly, for TV(z) — H / \ we have z — x1 + x2 + ... + xp + i(xp+1 + xp+2 + ... + xp+q).
Hn(v )
Putting X1 — y1, X2 — y2, ..., Xp — yp, Xp+1 — yp+1, ..., Xp+q — ^^. Thus z — y2 + y| +... + yp+„,
ii p + q — n. Let p2 — y2 + y| + ... + yp+q, p + q — n. Then
öTv (z)
Hn(v )
e i(i,x)zn dx
1
Hn(v )(i) 2
pV-rae-i(Çiyi+Ç2y2+...+ÇPyP+^P+i yp+l+...+^^ yp+q )dy
1
k
2
2
2
1
1
1
1
2v n 2
r
Hn(v)(i)2 Tf n - v
2 1 + ... + e? i(eP+1 + e?+2 + ...
p2+q
(2.16)
Since SY(w) and Tv(z) are tempered distributions, then SY(w) * Tv(z) exists and S(SY(w) *
Tv (z)) = 9(Sy (w))9(Tv (z)). Thus
9(Sy(w) * Tv(z)) = 9(Sy(w))9(Tv(z)) =
2Y+v nn
r
r
Hn(Y)Hn(v)r(n-Y^ ^n-^
p p+q
£c2 + ^e2
r=1 j=p+1
\
p p+q
£e,2 - ^ej
r=1 j=?+1
by (2.15) and (2.16). Now consider
2Y+v nn
r(2) r (2
Hn(Y)Hn(v) r f n - y\ rf n - v
r
Putting y = v = 2k, thus (2.18) becomes
(2.17)
(2.18)
24k nn
\ 2 I 2
/n - 2k\ /n - 2k
24knn M 2 JM 2
Hn(2k)Hn(2k)vin - 2k\ ^f n - 2k\ 24knn
T(k)T(k)
x
x
mm
n n - 2k\ v( n - 2k
1.
Thus, from (2.17)
9(S2k(w) * T2k(z))
Ee2) + ( E e2
i=1
j=p+1
(2.19)
— v
—v
2
2
1
k
2
2
3. Main results
Theorem 3.1. The Fourier transform of the distributional kernel (x) is given by the
formula
(
3K,
a,ß,Y,v
(x)
(n)2n(i)q
v r( aw 2 M YW2
p p+q
y ^ er e2
r=1 j=?+1
Kn(a)Hn(ß )Hn(Y )Hn(v)
x
x
C
r=1
-P
p+q
-Y
EC2 + i E C2
p+q
r=1
j=p+1
EC2 — i e C2
r=1
j=p+1
n — a \ r /n — ft\ r / n — y\ r / n — v
V
In particular, if a — ft — y — v — 2k then (3.1) becomes
(3.1)
3K,
a,P,Y,v
(x)
( —1)^
(C12 + C2 +... + Cp2)4 — (Cp2+1 + Cp+2 +... + Cp2+J4
(3.2)
Moreover (—1)kK2k)2k;2k;2k(x) is an elementary solution of the operator ©k defined by (1.1).
Proof. Now Ka,p;Y,v'(x) — RH(u) * Rp(v) * S7(w) * tv(z) by (1.6). Since RH,Rp,S7(w) and TV(z) are all tempered distributions by Lemma 2.1, thus 0Ka;p)7)V(x) — 0(RH(u) * Rp(v))0(S7(w) * TV(z)). By (2.11) and (2.17), we obtained (3.1) as required. For the case a — ft — y — v — 2k, by (2.12) and (2.19) we obtain (3.2) as required.
For (—1)kK2k)2k;2k;2k(x) is an elementary solution of the operator ©k see [4, p. 66].
Theorem 3.2. The Fourier transform of the convolution Ka;p)7)V(x) * Ka/;p/)7/)V/ (x) is given by the formula
(3.3)
0 (Ka,p)7,v (x) * Ka/,p/,7/,v/ (x)) — 3Ka,p)7)v (x)3Ka/)p/)7/ ^ (x),
where Ka;p)7)V(x) is defined by (1.6), a, ft, y, v, a', ft', y' and v' are complex numbers.
Proof. Now Ka;p;7;V(x) — RH(u) * Rp(v) * S7(w) * TV(z) by (1.6). Since Ka;p)7)V(x) is the convolutions of all tempered distributions, thus Ka;p)7)V(x) is also a tempered distribution and the convolution Ka;p)7)V(x) * Ka/,p/,7/,V/ (x) exists.
Since Ka;p;7;V (x) is a tempered distribution, then the Fourier transform
o (Ka,p)7)v(x) * Ka,p/,7/,v/(x)) — (oK„)ft7)v(x)) (3Ka/)p/i7/iv'(x)),
where o(Ka;p;7;V(x)) is given by (3.1).
Corollary 3.1. (The alternative proof of Theorem 3.1). The Fourier transform
OK
2k,2k,2k,2k
(x) —
(—1)k
¿=1
p+q
EC2 — EC2
kj=p+1
where k is a nonnegative integer and Ka,p)7)V(x) is defined by (1.6).
Proof. From Theorem 3.1 with the particular case a — ft — y — v — 2k, we can find OK2k)2k;2k;2k (x) directly from the elementary solution of the operator ©k defined by (1.1). Since (—1)kK2k,2k,2k,2k(x) is an elementary solution of the operator ©k.
Thus ©k ( —1)kK2k,2k,2k,2k (X) — 6 or (©k (—1)k6) * K2k,2k,2k,2k (x) — 6. By taking the Fourier transform both sides, we obtain
(
S( — 1)k6) * oK2k.2k.2k.2k(X) — 06 —1.
(3.4)
Now consider 0(©k(—1)k6). Since 6 is tempered distribution with compact support. Thus
(
:(—1)k6) —<
1)k6,e-i(i'x) >— < ♦LkLk(—1)k6,e-i(i'x) > by (2.10) where ©k
♦kLkLk by (1.5). Thus
< ♦ Lk Lk (—1)k 6,e-i(i'x) >—< ♦ L16, (—1)k Lk e-i(i'x) >=
— V
p
p
k
k
4
4
p
p
kk p+q k p p+q k
<
♦kL1Ö, (-1)k(-1)^c2 - i£ e2) e—'«>x) >=< s,[J2 er2 - i£ e2) «>x) >=
r=1 j=p+1
vr=1 j=?+1
=<
p p+q
♦ks, (E ei - i £ e2) (E ei+* E 2) (-d"^' >
k
p p+q
er + i / y e2
k
,r=1
j=p+1
,r=1
j=p+1
p 2 p+q 2
=< s, (-1)^ M + £ e2 ♦e—'«'x) >
r=1
p 2 p+q 2
-<s-(-«k Ee.2 + E e2
j=p+1
v
p 2 p+q 2
r2 -
r=1
j=p+1
r=1
p 4 p+ q 4
-<s.(-1)" EC2 - Ee
,r=1
j=p+1
e—i(£,x) >= /1
j=p+1
,r=1
>
p+q
(-1)k r2 -
j=p+1
Thus 9(®k(-1)"s) = (-1)" I ( ec2Y - ( E e2
r=1
Thus by (3.4) we obtain
9K2",2k,2k,2k (x)
j=p+1
j
(-1)k
p 4 p+q
Ee2 - E
i=1 j=p+1
k
References
[1] KANANTHAI A. On the solution of the n-dimensional Diamond operator // Appl. Math. and Comput. 1997. Vol. 88. P. 27-37.
[2] KANANTHAI A. On the spectrum of the Distributional Kernel related to the Residue // Intern. J. of Mathematics and Mathematical Sci.
[3] NozAKI Y. On Riemann — Liouville integral of Ultra-hyperbolic type // Kodai Mathematical Sem. Reports. 1964. Vol. 6, No. 2. P. 69-87.
[4] KANANTHAI A., SuANTAI S. On the weak solutions of the equation related to the Diamond operator // Comput. Technologies. 2000. Vol. 5, No. 5. P. 61-67.
[5] Gelfand I.M., Shilov G.E. Generalized Functions. Vol. 1. N.Y.: Acad. Press, 1964.
[6] DONOGHUE W. F. Distributions and Fourier transforms. N. Y.: Acad. Press, 1969.
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Received for publication January 15, 2002
