Internat. J. Math. Math. Sci.
Vol. 2 No. 4 (1979) 693-701
693
A DISTRIBUTIONAL HARDY TRANSFORMATION
R.S. PATHAK
Department of Mathematics
Banaras Hindu University
Varanasl, India
J.N. PANDEY
Department of Mathematics
Carleton Unlverslty
Ottawa, Canada
(Received January 23, 1979)
_ABSTRACT:
The Hardy s F-transform
0
is extended to distributions.
f(x)
The corresponding inversion formula
C (ix) t F(t)dt
0
is shown to be valid in the weak distributional sense.
This is accomplished by
transferring the inversion formula onto the testing function space for the
generalized functions under consideration and then showing that the limiting
R. S. PATHAK AND J. N. PANDEY
694
process in the resulting formula converges with respect to the topology of
the testing function space.
KEY WORDS AND PHRASES.
Integral Transform, Hardy Transform, Hankel Transform,
Funcio ns
%strib’zio ns’ Genzed
AMS (MOS) SUBJECT CLASSIFICATION (1970) CODES.
Primary 44A20, Secondary 46F05.
I. INTRODUCTION.
The Hardy transforms with their inversion formulae are represented by the
following two integral equations:
f(x)
F (tx)tdt
0
0
and
()
C (ty)yf(y)dy
(2)
0
where
C (z)
0
cosp J (z) + slnp Y (z)
(3)
and
m=0
F(p
22 -v-2p
+ m+ I) F(P + m+
+ I)
s
{r(p) r(v +p)} [2, p. 40].
+
I,
2p
v(z)/
(4)
The theory of the inversion formulae (i) and (2) has been given by Cooke [I].
The Hankel transform with its inversion formula can be deduced as a special case
of both (I) and (2) by taking
p
0.
The Y-transform [3, p. 93] is a special
case of (i) whereas H-transform [3, p. 155] is a special case of (2) for
p
%.
Recently the inversion formula (I) was proved to be valid for the generalized
function space
H’ (I) by
Pathak and Pandey [7] in the weak distributional sense.
It turns out that the kernal y F (ty) of F -transform does not belong to the space
DISTRIBUTIONAL HARDY TRANSFORMATION
695
H (I) and therefore the inversion formula (2) cannot be proved to be valid for
the space of distributions directly as a corollary to theorems proved in [7].
We
will therefore extend briefly the inversion formula (2) to a generalized function
space essentially by following the techniques and results proved in [7].
2. TESTING FUNCTION SPACE
H
’p
(I).
the function defined in (4) and let
For -1/2
_<
0, I, 2,
k
and real p let F (z) be
+
be a fixed number satisfying
Assume that 8 is also a fixed number satisfying
For each
_< 1/2
_>
max (v
+
2p
define a positive and continuous function
+ 2p >_ 0.
2, -1/2).
k(X)
on
satisfying
x 2k+
k(x)
x-8
O<x<l
-2
x>l.
An infinitely differentiable complex-valued function
H
said to belong to the space
k ()
(I) if
Akx ( )
k(X)
sup
,P
,8
(x) defined over I is
x
<
O<x<
for each
where A
0, i, 2, 3,
k
x
stands for the differentiation operator
2
D
x
+-x
space.
D
D
x
x
x
The topology over
Iklk--O [9; p.
A sequence
1
dx
H
’p
It can be readily seen that H
V’p
,8
(I) is a vector
(I) is generated by the sequence of semlnorms
8].
in this space is said to converge to the element
if
R. S. PATHAK AND J. N. PANDEY
696
@)
yk(
0 as
for each
is said to be a Cauchy sequence if
of each other.
0, i, 2, 3,
k
Yk (
m
@n )
A sequence
0 as m, n
Frchet space.
the topology of D(1) is stronger than that induced on
Since
in H
’p,8
(I)
independently
It is a simple exercise to verify that the space
sequentially complete and so it is a
@9
H
v’p
(I) is
D(1) c H ’p (I) and
D(1) by H’p (I),
it
,8
follows that the restriction of any
H,P
f
(I) to D(1) is in D’ (I).
In view of the fact that
A
(xt)
x
(-I
where
k
P(x,t)
v+2p
t
a
i=l
a
k
F (xt) -P(x,t)
t
x+2P -2i
i
t
2k-21
being certain constants depending on
(5)
and p, and the asymptotic orders
[9, p. B45]
o
Izl" +
Izl
o
Izl =
zl
(z)
o
(6)
where
max (
+ 2p
2,
-)
it follows that for fixed
t
treated as a function of x.
function
f e
H,P
,8
[8, pp. 347, 351]
> O,
x
F (ix)
belongs to the space
’p,8
H
(I) when
Therefore, Hardy’s F -transform F(y) of a generalized
(I) can be defined by
F) =<f(x), x F
(xy)>
y
> 0.
(7)
By following the technique as used in [7] it can be shown that F(y) is differentlable for each
F’(y)
y
<f(x),
>
0 and that
{x F (xy)}>
(8)
697
DISTRIBUTIONAL HARDY TRANSFORMATION
Note that
Y
’p
(I).
(xy)] also belongs to H
[x F
We now state some results which will be used in the sequel.
Define
N
H
N
c
(t,)--
(tx) C
(ylyay
o
N
X
[x Cv+(xN)
C (iN)
t C
2 h
t
2
v+l
(iN) C (xN)
-Q (x,t)
where
2 sin p
Q (x, t)
sin
ff
n
sin (p
+)
n
t (x
x
sin (p+)
2 sin p
n sin
L.
2
2
t
when
t
(Io)
x# t
2)
x
x
[8, p. 466].
Using the technique employed in proving Lemma 2 in [7] it can be proved
t, x Q (x t)
that for fixed
H
,P
(I).
It is now a simple exercise to prove for
c
_>
II,
8 _<
4
and
@
D(1) that
b
x
’p
Q(x,y) @ (y) y dy also belongs to H
LEMMA 2.
<
t
k(t)
sup
k ()
0<t<
gk (ty)
PROOF.
0
Let
(I).
be defined as in section 2.
=y
mln (8
+
2,
o
2k
Then
k
for
0
<y < I
O, I, 2,
The result follows by dividing the t- llne into three parts
< I, I <
sup
0<t<:
t
< l/y,
i/y < t <
and considering the corresponding
698
R. S. PATHAK AND J. N. PANDEY
Le__t C (z) be the
LEMMA 3.
< %,
-%_<
F
<_ 3/2, 8 >
as defined
2, -1/2 ).
+ 2p
max
i_.p.n (3) and let
> O,
Then for flxed x
t
<_
2p
-v
function
(xy) ydy
(ty) C
0 in
H
’p
(I) as
0
+
0
The lemma can be proved by using lemma 2 and a variation of the
PROOF.
technique used in proving lemma 4 of
LEMMA 4.
H ,P
,8
Let
,
and p
8,
be restricted as in Lemma 3 and let
then
N
N
< f(t),
(ty) >
t F
C
(xy) ydy-
< f(t),
F (ty) C (xy) ydy >.
t
0
0
The result follows in view of Lemma 3.
PROOF.
The details of the
technique to be used can be found in [7, Lemma 5].
Le_.t
LEMMA 5.
defined by (9)
llm
and
b
>
(i0).
a
>
0
(t,x),
and
Then
b
..H
Q (t,x) b_e th__e functions a_.s
(t,x)+ Q (t,x)
xdx
I
0
t e
t 4
a,b]
[a,b].
a
See Leuna 6 in
PROOF.
LEMMA 6.
b
>
a
> 0.
Assume that
Then
b
-
7].
Let the support of
D(1) be contained in (a,b) where
(t,x), Q (t,x) be the functions as defined in (9) and (i0).
Let
<_
v < 1/2, max (-
2p, v)
_<
< 3/2
and
_> max (v + 2p
2, -1/2).
DISTRIBUTIONAL HARDY TRANSFORMATION
PROOF.
699
The proof can be given only by using Lemma 3 and a simple variations
of the techniques used in proving [7, Lemma 7] and so the details are
omitted.
3. INVERSION OF THE DISTRIBUTIONAL
F
transform:
We now state and prove our main result.
Let
THEOREM.
( + 2p
Assume
-% <_
<_
3/2
8 _> max
and
H%)’P
f
(I) as defined
Then
F(y) C
Let the support of
-
Since F(y) C
N /
>
f,
for each
@
D(1).
\
(xy)y dy,
where
b
>
a
>
0.
F
t
(ty) C
N
b
(x)
/
(t),
F(y)C%)(xy)
(x)dx
0
a
(x) dx
(xy) y d
0
Lena 4]
(t,x) + QN(t’x)
(t), t
@ (x)
dx
[7, Lemma 8]
2
>
be [a,b]
(xy) y generates a regular distribution we have
a
b
@
%)
<F(y)
C%)
\
>
<
(x,y) y dy, # (x)
0
PROOF.
N
Iv l)
2p,
that F(y) is the. distributional F -transform of
N
for
max (-9
2, -1/2).
by (7).
0
1/2,
(t),
0
t
(t,x) + q(t,x)
9 (x)
[I, Lenna
dx
p. 394]
y dy
R. S. PATHAK AND J. N. PANDEY
700
(t),
( x) +Q( x
x dx
9, p. 148]
by Riemann Sums technique
Lemma 6]
This completes the proof of the theorem.
Taking
0
p
COROLLARY I.
8 _> "%-
Define
t
f
Hv’0 (I1
,8
where "%<-
<- %’ )l <- <- 3/a
and
th.._e distributional Hankel transform of f by
<f(t),
F(y)
p
1/2 in the above theorem we derive
and
(ty)>
t J
then
F(y) y
N
fo__y_r al__l
F(y)
D(I).
Define
Let
f
H’% (I)
where
"% <
9
< %
II
<
< 3/2
th__e distributional .Struve tr,ansform (H -transform) of f by
<f(t),
then
lira
fo__r al.._l
(x
0
COROLLARY 2.
8 _> "1/2.
(xy) dy,
j
D(I).
t
Hv (t-y)>
and
DISTRIBUTIONAL HARDY TRANSFORMATION
A.KNOW.LEDGMEN.T.
No. A5298.
701
This work was supported by National Research Council Grant
The first author is thankful to the Banaras Hindu University for
granting him study leave.
REFERENCES
I. Cooke, R.G. The nversion formulae of Hardy and Titchmarsh, Proc. London Math.
Soc. 24 (1925) 381-420,
2. Erdelyi, A. (Editor) Higher Transcendental Functi0.ns, Vol. II, McGraw-Hill
Book Co., Inc., New York, 1953.
3. Erdelyi, A. (Editor) Tables of Integral Transforms, Vol. II, McGraw-Hill Book
Co., Inc., New York, 1954.
4. Hardy, G.H. Some ormulae in the theory of Bessel functions, Proc. London
Math. Soc. (2), 23,(1925) ixi-lxlii.
5. Pandey, J.N. and Zemanian, A.H. Complex inversion for the generalized
convolution transformation, Pacific J. Math. 25, (1968) 147-157.
6. Pandey, J.N. An extension of Haimo’s form of Hankel convolutions, Pacific
J. Math. 28 (1969) 641-651.
7. Pathak, R.S. and Pandey, J.N. A distributional Hardy transformation,
Camb.. Phil...S0c. 76 (1974) 247-262.
Proc.
8. Watson, G.N. .A. treatise on the theory of Bessel funct.lons, Cambridge University
Press, 2nd edition, 1962.
9. Zemanian, A.H.
1968.
Generalized integral transformations, Intersclence Publishers,