Internat. J. Math. & Math. Sci.
VOL. 14 NO. 2 (1991) 269-274
269
CHARACTERIZATION OF HANKEL TRANSFORMABLE GENERALIZED FUNCTIONS
J.J. BETANCOR
Departamento de Anllsis Matemtico
Faciltad de Matemticas
Unlversidad de La Laguna
La Laguna, Tenerife
Canary Islands, Spain
(Received August i, 1988 and in revised form April 30, 1989)
In
ABSTRACT.
this paper we prove a characterization theorem for the elements of the
space H’ of generalized functions defined by A.H. Zemanlan.
KEY WORDS AND PHRASES.
Generalized functions, Hankel transforms.
1980 AMS SUBJECT CLASSIFICATION CODE. 46F12.
1.
INTRODUCTION.
The Hankel transformation defined by
h {f(x)}(y)=
where J
denotes
the
Bessel
f (xy)I/2j
function
(xy)f(x)dx
of
the
first
kind
and
order
,
has
been
extensively studied In recent years.
A
result
classical
point x
x
0.
Let f(x)
If
(LI(0,)
>-
h-l{F(y)}(x O)
the
Hankel
transformation
is
the
following
[I]).
inversion theorem (see
THEOREM I.
concerning
be of bounded variation in a neighborhood of the
and F(y)
f F(y)(XOY I/2j
0
h{f(x)}(y),
then
(x0Y)dy-1/2 {f(x 0 + 0)+f(xO-
0)}.
Another well known result is the Parseval’s equation (I) (see [I)).
THEOREM 2.
Let f(x) and G(y) be elements of
LI(0,).
If F(y) and g(x) are
respectlvely the direct and inverse -th order Hankel transforms of f(x) and G(y),
then
/
0
f(x)g(x)dx
/
0
F(y)G(y)dy,
for any
270
J.J. BETANCOR
Other conditions under which Parseval’s equation holds are given by P. Macaulay-
Owen [2].
The h -transform has
been extended to several spaces of generalized functions.
Apparently,
A.H.
Zemanian
introduced
the
space H
[I] was the first to extend the Hankel transform.
of
testing
differentiable complex-valued functions
,n
for every m,nN.
functions
consisting
defined on
I=(0,)
of
all
He
infinitely
and such that
xl
The Hankel transform is an automorphlsm onto H
For every
H’ (the dual space of H ), the generalized Hankel transformation H’f of f was
defined by the following generalization of Parseval’s equation
f
<h’f,>
>
<f,h
H
for every
h’ is an automorphism onto H’.
Later, E.L. Koh and A.H. Zemanian [3] defined the generalized complex Hankel
For a real number
transformation.
and a positive real number a the space
defined as the space of testing functions
- -
J
was
which are smooth on I and for which
xEl
where S
x
I/2 Dx2+IDx
C:llm yl<a,
R=
I/2.
For each complex number y in the strip
J
y e (- 0]}
contains the function
]
h -transform is now defined on the dual space
Let
DEFINITION.
be in the interval
< -.
(
(xy)
J(xy).
The
as follows:
Then, for every f
e
’, and
y e R,
(h’f) (y)= <f(x),
(xy)/2j
(xy)
E.L. Koh [4] showed that a distribution f
>
]’
can be written as a finite sum
of derivatives of continuous function of exponential descent.
More specifically, he
established:
THOEREM 3 ([4]).
k
d
i--0
where the
Ci(x)
Fi(x)
i
ype
functions
Mndez [7],...).
(e
-ax
ing’,.
Then f is equal to a finite sum
x-w-(ll2)-k+l Pi(x)Fi(x))
are continuous on
Other Hankel
generalized
Let f be
(0,-) and the P. (x) are polynomials of degree k.
transformations have been also extended to certain spaces of
(see G. Altenburg [5], L.S. Dube and J.N. Pandey [6], J.M.
271
CHARACTERIZATION OF HANKEL TRANSFORMABLE GENERALIZED FUNCTIONS
In this paper we pove a characterization theorem for the generalized functions
in H’
Our
proof
is
anaogous
to
method
the
employed in structure theorems
for
Schwartz distributions (see [8] and [4]).
In this paper a function (x) will be called of rapid descent if
xmDq(x)
tends
to zero, as x+, for every m,qeN.
2.
The space H’ of generalized functions.
A characterization theorem.
A useful result due to A.H. Zemanian (see [I]) is the following
PROPOSITION I.
Let f be in H’.There exist a positive constant C and nonnegatlve
integers r,k such that
<f’>l
C
maX{Ym,nlJ ();
Omr,Onk}, for every
We now present some new properties of the space H
PROPOSITION 2.
Let
be in H
The function x
(H.
of testing functions.
x--l/2 (x))
m.1 n
(D)
is
a) of rapid descent as x+=, and
b) in L I(0,-),
for every m,neN.
It is enough to take into account that
PROOF.
Ixm( D)n(x -u-I/2(x))
neN, C
C
m,n
I and m,
for every x
x
being a suitable positive constant.
m,n
The main result of this paper is the next.
THEOREM
4.
A
functional
measurable functions
gm
f
is
in
H’ if
and
only
if,
there
exist
bounded
(x) defined on I, for m--0,1,...,r and n=0,1,...,k, where r
and k are nonegative integers depending onf, such that
x-U- I/2
Z
(-lxl)n{xm(-D)gm,n(X)},(x)>
m,n
r,k
<f,>=<
for every
PROOF.
H
Let f be in H’.
In veiw of Proposition I, there exist a constant C
and nonnegative integers r and k depending on f such that
<f,>
(
C
maX{Ym,U n ();
=Cmax {sup
x61
for every
(2.1)
H
Ixm(1-D"
x
0
n
m
(
r,0(nk}
x-U- I/(x)
I;
O(m(r,0(n(k},
>
0
J.J. BETANCOR
272
Since
xmiD)’l n(x-U-I/2 @(x))
H
for every
is of rapid descent as x+ (Proposition
2),
we get
m,n,eN.
Hence
II ILl(O,.
where
denotes the norm on the space
LI(O,).
Then we can write
t<f,,>lCmax{tlD%{tm(tl--D)n(t-U-l(t))}l Ll(O,.);Omr,
for every
Onk}
eH
We now define the injective map
FH
F: H
(Dt{tm(It--D)n(t-- I/2 (x))})m=O,..., r
n=O,... ,k
If FH
A
r,k
is endowed with the topology induced in it by the product space
(0 )=(L I(0
))(r+l)(k+l)
G:FH
then
C
F
<f, @>
is continuous linear mapping.
By application of the Hahn-Banach Theorem, G can be extended to A r, k(0 ).
(r+l)(k+l)
(see F. Treves [I0])
Therefore, since A’ (0 ) is isomorhic to (L (0 ))
r,k
,r; n=0,.. ,k) such
(m--0,
there exist (r+l) (k+l) bounded measurable functions,
gm,n
that
r k
G(F)
<f
>
r
k
<
m=O, n=O
for every
e H
<gm,n (x) D{x m (D)n x- -I/2 #(x))}>=
x-U-I/m( -DIx )n [xm(-D)g m ,n (x)} (x)>
m=O, n=O
273
CHARACTERIZATION OF HANKEL TRANSFORMABLE GENERALIZED FUNCTIONS
is defined by
On the other land, if
To see
that
as v
v,
this,
it
,
0 as v
in
Ll(O,=),
(2) then
If_{@}veN, is a sequence in toH
/2,vkX)}vN converges
enough to prove that
is
the
then
ell’.
{xm( xI_D)n(
sequence
such
zero
This completes the proof of the theorem.
for every re,heN.
The Hankel-Schwartz transfqrm defined by the pair
f0 x2V+Ib V (xy)f(x)dx
F(y)=B f(x)}(y)
V
-
for
f(x)=B
,
,
formula.
setting V
z-J
(xy)F(y)dy
(z) and J
denotes
the
Bessel
[9],
was introduced by A.L. Schwamtz
of
function
who estbalished its
This integral transfommatlon has been extended by G. Altenburg
[7]
following
f0 y2+Ib
{F(y)}(x)
where b (z)
kind and order
Mendez
V
to
a
the
procedure
=-,
H’II’
analogous
space
of
functions
generalized
to
the
employed
one
(H=HI/2
by
A.H.
[n
the first
inversion
[5] and J.M.
their
Zemantan
notation)
[I].
By
we can deduce from Theorem 4 the next
COROLLARY.
The
functional
f
is
in
H’
if
and
only If,
there
exist
bounded
for m=O,...,r,n=O,...,k where r and k are
measurable functions gm,n(X) defined on I,
nonnegattve integers depending on f, such that
r
<f,>
k
<
m=O, n=O
(-)n[xm(-D)gm, n (x)},(x)>,
H.
REFERENCES
I.
ZEMANIAN, A.H. A distributional Hankel transformation, J. SIAM Appl. Math.
14(3) (1966), 561-576.
2.
Parseval’s theorem for Hankel transform, Proc. London. Math.
MACAULAY-OWEN, P.
Soc. 45 (1939), 458-474.
3.
KOH, E.L. and ZEMANIAN, A.J.
4.
KOH, E.L.
5.
ALTENBURG, G.
6.
DUBE, L.S. and PANDEY, J.N. On the transform of distributions, Tohoku Math.
Journal 27 (1975), 337-354.
7.
Mendez, J.M. On the Bessel transformation of the arbitrary order, Math. Nachr.
136 (1988), 233-239.
8.
TREVES, F. Topological Vector Spaces, Distributions and Kernels, Academic Press,
New York, 1967.
The complex Hankel and 1-transformatlons of
generalized functions, SIAM J. Appl. Math. 16 (1968), 945-957.
A representation of Hankel transformable generalized functions, SlAM
J. Math. Anal.
(1970), 33-36.
Bessel transformationen in raumen von grundfunktlonen uber dem
interval =(0,)und derem Dualraumen, Math. Nachr. 108 (1982), 197-218.
274
9.
J.J. BETANCOR
SCHWARTZ, A.L.
An inversion theorem for
Hanke[
transform, Proc. Amer. Math.
Soc. 22 (1969), 713-717.
I0. SCHWARTZ, L.
Theorie des Distributions, Vols. 1 and I[. Hermann, Paris,
and 1959.
1957
A treatise on Bessel functions and
their applications to Physics, McMillan and Co. Ltd., London, 1952.
GRAY, A., MATHEWS, G.B. and MACROBERT, T.M.
12. WATSON, G.N.
Theory of Bessel functions, Cambridge Universlty Press,
Cambridge, 1958.
13. ZEMANIAN, A.H. Generalized Integral Transformations, Interscience Publishers,
New York, 1968.