545
Internat. J. Math. & Math. Sci.
VOL. 18 NO. 3 (1995) 545-550
CONVOLUTION OF HANKEL TRANSFORM AND ITS APPLICATION TO
AN INTEGRAL INVOLVING BESSEL FUNCTIONS OF FIRST KIND
vu KIM TUAN
Institute of Mathematics, P.O.Box 631, Bo
tto, Hanoi, Vietnam
MEGUMI SAIGO
Department of Applied Mathematics, Fukuoka University, Fukuoka 8l,l-0l, Japan
(Received August II, 1993)
Abstract
In the paper a convolution of the Hankel transform is constructed. The convolution is used
to the calculation of an integral containing Bessel functions of the first kind.
Key Words and Phrases: Hankel transform, Bessel function, Convolution.
1991 AMS Subject Classification Codes: 33B20.
1. Introduction
The convolution of a modified Hankel transform, introduced in [4], has been studied in [1], [4]
[7] in a space of generalized functions. For an another modified Hankel
transform the other convolution in some space of functions is obtained (see [5]).
The present paper is devoted to propose a definition of a convolution and to prove the convolution
property in the classical sense of the following standard Hankel transform (see [6], [8])
in classical sense and in
9[f](x)
yJ(yx)f(y)dy,
(1)
Re(,) > 2
As one of its applications, a formula of infinite interval of a product of Bessel functions of the first
kind is established.
2. Convolution of Hankel Transform
Set
h(x)
2-3’x-’
f/
V/.r( y + 1/2) ,,+v>,l,,-,,l<
x
(uv)’-"f(u)g(v)dudv,
x,
_
()
(0,
The function h(x) is called the Hankel convolution of the function f(x) with the function g(x). It
is easy to see that the convolution is a commutative operator of f and g.
Let L(R+;p.(x)) be a class of integrable functions f(x) with a weight p(x) > 0 in R + (0, oo).
The main aim of this section is to prove the following:
V. K. TUAN AND M. SAIGO
546
Theorem. Let Re(u) > and f(z),g(z) E L(R."
2
and there hoils the convolution property
9([h](z)
/’).
Then the function h,(z) in (’2) exists
x-9f[f](x)f.[g](x),
(3)
where :f,, is the Hankel transform (1).
Proof. It is well known [6, (2.12.42.15)] that
fo
-
J.( zt )J.( ut J( vt )dt
(4)
VTv( + /2)
where Re(u) >
2
and
+()
>/ v()’ v()
()
0,
< 0.
0,
Since
co,(
J,,(x)
2
4
) +o
(5)
o (),
+0),
(see [3]) it is easy to conclude that there exists such positive number C independent of z
that
and z-J.(x)
L(R+) when Re(u) >
(
.
(0, oo)
Therefore we have
t’-"J.(xt)J(ut)J.(vt)dt
(6)
<=
Cz()-
where C is independent of x, u, v and N. In particular,
the help of the estimate (6) we have
.
Ih()l N C (ince
I(),() e (.;
Cza)-
1
t-EJ(t), dt
mng use the formul (2) and (4) with
lf()(v)ldd <
Thus the function h() in
(2)
,
exists. urthermore, applying the
ubini theorem, e obtain
.. .
t-".(t)
Here we have used the estence N the
genkel
transform
defined by
; (R+; ) (see [2], [8]). Moreover, we notice the fct
x[I]()
from
[2,
t
(t) (t)()()
o (), (
+0) for
Ie
(1) Nr
()
Nnctions from
(,; )
p. 74]. Therefore, if we set
k(t)
t-"gf[f](t)rf.[g](t),
(S)
CONVOLUTION OF HANKEL TRANSFORM
W
547
htve
()
On the other hand
we
o("),
()
+0).
(
have
I:c.[/](t)l
V/’J
=<
u
v/’f
u du
,
C
<
e (0,),
(0)
and lherefore,
o (,--’), (
,()
Since Re(u) > 1/2, from (9),
can be rewritten in the form
(11)
we
+).
L
conclude that k(t)
(R+" ).
Therefore the form,,la (7)
C[]().
h(,)
ttence, by using the inversion formula of the Unkel transform in the class L(R+;
fir,.,
[3Q[k]] (z)
(see [2], [8]):
(12)
k(z),
we obtain
(13)
3f[h](z).
k(z)
As k(z) has the form (8), the formula (13) coincides with the formula (3). Thus the theorem is
proved.
3. Application
As
an
application of Theorem we consider the integral
j--1
with a, > 0 (j
0, 1,-.., n) and Re(y,)
=> 0 (j
n). We
1,.,9
f,*’’"’"" (a0, a,
a,,)
will prove that
0
when
ao>a,+.-’+a,, and
n-3
<Re(uo)< Re(u,)+.
We know that it is valid for n
(see [6, (2.12.31.1)] for the case Re(y,) 0, and [6, (2.12.35.12)]
for the case Re(ya) > 0). Suppose that it is valid for every k __< n. We have to prove it for the case
k n + 1. Put
fi (t2+y) -’12
gO,,,-.-,,
,,,.. ,,, (t, a,,... a,)
By using (5)
we have
(t
a,,)=O(t*)
gO,,,...,,(t,a
Yl ,’"Yn
+0)
(14)
Suppose that
2
Then from (14) we conclude that
formula (10) we obtain
0..--.
" (t,a ...,a,,)
9u,.Y!,,
.... (t,a,
o [a’’ ,,.
Lain,
Re(v,)+
n-3
2
")
L (R+; v/
(z
aa,...
tou,,.,u"
Since
< Re(vo) <
’’’,
a,)] ()
t ’’
.... (z,
+O,z
a,
Therefore, by using the
+).
a.)
(15)
548
V. K. TUAN AND M. SAIGO
the forr, mla
(15)
can ie real as
,,
But by
(z,a,...,,)
,o
+c).
+O,x
l}o assumplion we have
,n
,1,
,Yn
when z >
and by
(x
0
a
+’’" + an. Therefore
.... (x
(F, al,..-, an)
we have
t’’t’"+’
_
a,,) E L (R+; V’)
a,
(12)
Analogously,
0
(X an+l)
(R+. v/’)
i
and
under the conditions
2
....
Since
g,’’, ,.+, (t, al,
< Re(uo) < Re(u.+,)- 1.
a.+,)
/’
....
t-a"’’’o,, ,. (t,a,,
a.)a’’’’+’o.+, (t, a.+,),
then by using the theorem we obtain
21-3’x-
a.)] (t)] (x)
(y,a,
Yn+l
(16)
//
vr(,o + /2) u+v>z,lt,-vl<z
X
.... (u,a
Since
Yn
cr’0’r’l
.’".l/."r’r’ (it, al,
’/1
",
art)
fo,.+,
(v, a.+l du dv.
a,)=0 when u>a+..-+a,
and
f=’o,=’,,+, (V, an+l)
0
when v > a.+
provided that
n-3
< a(.o) <
3=1
we conclude from
Re(g0)- Re(tn+l) < -1,
(7)
+ a.+
(J8)
(16) that
f,.’o,,,,-,.+, (z
1,
under (17).
The formula
a(,) + --7--
(18)
,Yn+l
can
a,
ln+l
0
when z > a +--.
be analytically continued to the domain
"+z
-1 <
Thus we have proved
(.o) <
2 (",)+
3=1
n- 3
2
CONVOLUTION OF HANKEL TRANSFORM
549
Corollary. Let
-1
<Re(vo)<
Re(u.,)+
’)
% >O(j= 1,...,n)
with
ao >
a+..-+
anti
Th en
fot’+lJo(ao t) fi (t2 + y)-"’/2J,, (%t2 + y)dt
(19)
O.
)=0
The formula (19) is a generalization of the formulae (2.12.44.7) (the case
and (2.12.44.8) (the case Re(yi) > 0,--., Re(y,,) > 0)in [6].
y,
yl
0)
Acknowledgement. The work of the first author was supported, in part, by the National Basic
Research Program in Natural Sciences, Vietnam, and by the Alexander von Humboldt Foundation.
References
[1] F.M.
Cholewinski: Hankel Convolution Complex Inversion Theory, Mem. Amer. llalh. Soc.
Vol. 58, 1965.
Ditkin and A.P. Prudnikov: lntegral Transforms and Operational Calculus, Pergamon
Press, Oxford-London-Edinburgh-New York-Paris-Frankfurt, 1965.
[3] A. Erddlyi, W. Magnus, F. Oberhettinger and R.P. Soni: Higher Transcendental Functions,
[2] V.A.
Vol.
,
McGraw-Itill, New York-Toronto-London, 1953.
[4] I.I. Hirschman, Jr."
Variation diminishing Ha.nkel transforms, d. Analyse Math.
8(1960/61),
307-336.
[5] Nguyen Thanh Hal and S.B. Yakubovich:
The Double Mellin-Barnes Type Integrals and Their
Applications to Convolution Theory, World Scientific, Singapore-New Jcrsev-London-itong
Kong, 1992.
,
Special
de Sousa Pinto: A generalized Hankel convolution, SIAM J. Math. Anal. 16
(1985),
[6] A.P. Prudnikov, Yu.A. Brychkov
and O.I. Marichev: Integrals and Series, Vol.
Functions, Gordon & Breach, New York-London-Paris-Tokyo, 1986.
[7] J.
1335-1346.
[8] E.C. Titchmarsh:
1948.
Introduction to the Theory
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