III
Inter.at. J. Pla@h.
’+,L’. 6 N,.
P<ath. ScZ.
(1983) 111-118
IDENTITIES INVOLVING ITERATED INTEGRAL TRANSFORMS
CYRIL NASIM
Department of Mathematics and Statistics
The University of Calgary
Calgary, Alberta Canada T2N IN4
(Received August 18, 1982)
ABSTRACT.
A number of identities involving iterated integral transforms are establi-
making use of the fact that a function which is a linear combination of the
shed,
Macdonald’s function K (z), where z
KEY WORDS AND PHRASES.
is a complex
variable, is a Fourier kernel.
Macdonad’s run.ion, Fourier knP+,
<n
transform, Han-
kel trans f orm, Laplace trans form.
1982 <ATHEMATICS SUBJECT CLASSIFICATION CODE.
i.
4A20.
INTRODUCTION.
The object of this note is to establish various identities involving integral
operators.
function K
The integral operators are the integral transforms with respect to the
(z),
where 7/ (z) is the Macdonald’s function of order
complex variable.
and argument
z,
a
Some functional relations are deduced, as special cases, which
show the inter-relations among more familiar Fourier Sine, Fourier Cosine, and Laplace
transforms.
2.
THE KERNEL.
1/27/
Let
)2
Then
2
or
)2
we set @
ie
im /k
I/4
2y
m----x. 1/4] y
12_/41
0
-< m
a constant and
with
2
Whence
Now, if
(0),
k
82y,
)2 ky
2k-l,
k= 0 1 2
vl
<
I.
C. NASIM
112
y
then
satisfies a
x
2K(ie "m/kx),
k-fold Bessel equation:
D
92
1/4] k
x--’--- y
k
(-I) y
(2.1)
It is not difficult to see that if x is a complex variable, then every point of (2.1)
is regular except for a singularity at x
0.
Now consider a function of the form
2k-1
m
m=O
M
The functions are an extension of the functions which were first noted by Guinand.
As a special case when k
B0
2, chose the coefficients
i
B2
B
-
as
2
0 and B
Then we obtain
G2(x
1
1/2...
---x t (ix) + K (-ix) + 2cos(
cos
i
v
vr)Kv(x)}
K (x), say,
(2.2)
and we have
THEOREM 2.1.
kv(x)
y
is a solution of
.Z114
0 <
X <oo
the two-fold Bessel equation.
The function k (x) is of special interest to us here nd we shall develop its
properties further.
Using the representations [3]
Kv(x)
g
Y (x)
cosec v{cos w J (x) -J
cosec
v{I_v(x)
Iv(x)
and
where J
I
and Y
(x)},
are the usual Bessel functions, equation (2
kv(x)
i
x1/2{sin -vJv(x)
+
i
cos-v(Yv(x)
2),
can be written as
2
+-- Kw(x)) }.
These functions arise as kernels in divisor summation formulae of the Hardy-Landau
IDENTITIES INVOLVING ITERATED INTEGRAL TRANSFORMS
type, involving number theoretic function
divisor of
n, [4]
the number of kth powers of the
1
,we have
-2
If we put v
-(cos x
k+1/2(x)
ok(n),
113
-sin x
+
e
which obviously satisfies the differential equation
Dy
y.
I.
Next, the Mellin transform of
-
xK
h
m{x2K (ax)
where Re s >
is given by
IRe
i
w
h
-s-2
a
[5]
(x) is given by
I
I
i
i
i
I
2s_3/2 F(++)
(- +)
whence the Mellin transform of k (x) defined in (2 2)
28-3/2
I
I
+ i)
4
r( s +-f
i-s-1/2
ri
(-f
I
s
-
+1/4)
--f
-s -1/2
I
+ 2cos v).
+ (_i)
On simplifying, we have
2s+
k*(s)
for
Re s >
IRe I
By repeated use of the relation
r(z)r(1-z)
r(
i
+-f
-
I s
i
+- r(y
cos
i
(s
+
i
--f
I
+ y)
i
+)
I
cos (s_ + 1/2)
I
sin zz
it is not a difficult matter to see that
k*(s)k(l-s)
I.
Hence,
THEOREM 22.
The function k*(x) defined by equation (2.2) is a Fourier kernel, [6]
If we define the transformation
T[f] by
114
C. NASIM
I] k(xt)f(t)dt,
c[f]
C2[f]
then
f
and T is involutory since
F2
7, the identity transformation.
Mak-
ing use of the asymptotic expansions of the Macdonald’s function K (z), we have
k (x)
(x)
and
Now, if we take
(x)
x
O(x-’+1/2),
x
O.
for
lxl
L2(0,),
k (x)
Therefore
O(e -x),
L2 (0,),
the integral defining the transformation T exists
Thus T is a bounded transformation on
and is in fact absolute convergent.
for
Ivl
3.
THE OPERATOR.
I.
< i.
We shall now define the transformation T in operator notation.
rators
K
and
K
K
and
f
L 2 (0,) and
transformation
Denote the ope-
respectively by
K[f] =K{f(x);
where
L2-space
Kv{f(x);
.[f]
I,I
<
,
[ xaK(xt)f(t)dt
x}=
ix}
fr
K (ixt)f(t)dt
with K (z) being the Macdonald’ s function.
can be expressed, in operator form, as
and we can write, symbolically,
T-
1
--(K.,i + K,_z.
+ 2cos 1
vK
Then the
IDENTITIES INVOLVING ITERATED INTEGRAL TRANSFORMS
Since F 2
-
f, the identity transformation,
1
or
+ K
(K
+ 2cos
+
we have
i
vK)2
1
4cos 2
v,-.
v,.
"
.K + K
2cos-v(g
115
I
K2 + K
+ K
.K
.K
(3.1)
.K + K K
+ K K
The right-hand side is the linear combination of iterated transformations, which are
bounded on
L2-space
vj
for
I.
Now, using the standard result [5],
I
where
Ivl
< 1 and
tKv(at)Kv(Bt)dt
Re(u+B)
K
v,i
>
2(0,),
0, we have for example, if
Kv If]
K v (ixt)dt
0
v(ut)f(u).
(ut)
(xu)1/2f(u)du
=1
-(i)
tKv(ixt)Kv(ut)dt
(ix)2V u 2v du
() -vf( u)
-v
sn
_g g
D
2
--i
+2
If]
The change of order of integration can be justified by absolute convergence.
Thus,
we obtain our first identity
K
.K +KK
(3.2)
=0
The identity given by (3.2) can alternatively be established by making use of the
Mellin transform theory.
Kv,iKv[f],
That is, the Mellin transform of the iterated operator
is given formally by
m{K
v
iKv [/]
h
1/2
m{- 1 x K (ix); s }m{ i x-K (x); l-s}f*(s)
v
.-s-1/2
z
f*(s)
1
1
1
1
4sin
where
f*(s) denotes the
[f;}
-r(
Mellin transform of
v
1/2(s
)sn (
f(x).
1/2
m{- 1 x K (x); s}m{- 1
4sin
+
x1/2K
u
+
Also,
(-ix); l-s}f*(s)
(-i)
1
1
+ )sin u(8
+
v
+
1
+)
)
116
C. NASIM
.K + K K
m{ (K
Then
.) If]
O,
implying that
as shown above.
=o
.K +KK
K
Similarly, one can show that
KK
.K =o,
.+K
a sort of conjugate of the identity in
K(ix)
(3.3)
(3.2).
Consider the representation
(e-i1/2vJ -v (x)
2 sinwn
e
i1/2w
(x));
then
K
(xt)1/2Kvixt),(t)dt
[f]
I
2sin v
J;J
(xt)1/2j
-
2sin vr
where
H
e
i1/2vH
(xt)f(t)dt
(xt)
-e
If]
1
ei1/2H
1/2J
[]}
denotes the Hankel transform operator of order v.
K
,i
2sin
e
-iwH
and similaz ly
K ,-
2sin
Now, after substituting for K
e
-i1/2
(H_Kw
-
2sin
ei1/2WH -w
and
K
K)Hw)
2sin
in
e
i1/2v
e
i1/2
(xt) f( t)dt
Thus,
in operator
form,
H
(3.4)
e-i1/2H
(3.2) and rearranging, we have
(HK
KH_v).
By comparing the real and imaginary parts and solving, we obtain the identities
KH =H K
and
H K
KH
(3.5)
IDENTITIES INVOLVING ITERATED INTEGRAL TRANSFORMS
117
Now,
2j1/2 (xt) f(t) dt
H1/2 [f]
I
sin (xt) f(t) dt
S[f],
and similarly,
H_1/2[f]
C[f],
where S and C are the usual Fourier since and cosine transform repsectively.
Also,
I (xt)1/2K1/2(xt)f(t)dt
K1/2[f]
I11/2 Ie-Xtf(t)dt
c[yl,
where L[f] is the Laplace transform.
Hence, setting
and
+.I
-,
in
(3.4), we
f
CL
LC
SL.
obtain the relations,
(3.7)
Incidently, a general relation involving the operators K
HK
i
cosec { sin-(w-)
KvH +
sin
-g=(1
and
H
can be established:
T[K
H
On setting
_+v, (3.8) yields the identities. (3.5) and (3.6).
representation
(3.4),
(3.8)
Next, from the
we have symbolically
2
(3.9)
since
H2
H2
f, the identity operator, with Hw being the Hankel transform.
Similarly, one can show that
(3.10)
C. NAS IM
118
And, in the same vein, we have
K
.K
.K
+ K
v,-c v,c
v,c v,-c
Now, going back
(3.9)
in
2
2
Sin-
v(H H
cos
v-v
+H-v H v
(3.11)
(3.1) and using the results given by (3.2), (3.3), and
to equation
Ivl
(3.11) and simplifying, we have finally, for
[
H ) H --) +H--) H )
sin v
2K2
i,
An interesting relationship can be established by putting v
4
H1/2H_1/2 + H_ttH1/2
K1/22
(3.12)
2cos v F.
1
i
in
(3.12).
Then
0
or
_2
SC + CS
where
,C,
L2
(3.13)
and L denote the Fourier sine, Fourier consine and Laplace transforms
respect ively.
From (3.9) and (3.12), one can establish the identity
K2
which, on setting v
_K
-+
,i
,
i
_K 2
(3.14)
w,-i
yields (3.13).
REFERENCES
i.
GTTINAND, A.P., A class of
Fourier
Kernels, Ouart. J. Math. Oxford (2), 1 (1950),
191-3.
2.
EVERITT, W.N.
Fourier
3.
On a generalization of Bessel functions and resulting class of
kernels, 0uart. J. Math. Oxford (2), i0 (1959), 270-9.
WATSON, G.N., Theory of Bessel functions.
The University
Press, Cambridge,
England, Second Edition, 1944.
4.
NASIM, C. A
summation formula involving
951-64.
ok(n),
k
>
i, Can. J. Math.
2__1 (1960)
5.
ERDELYI et al., A., Tables of Integral Transforms, Vol. I, Bateman Manuscript
Project, McGraw-Hill Book Co., New York, 1954.
6.
TITCHMARSH, E.C., Introduction to the theory of "ourier Integrals, The Claredon
Press, Oxford, Second Edition, 1948.