I ntrnat. J. Math. & Math.
(1983) 95-99
95
Vol. 6 No.
THE n-DIMENSIONAL DISTRIBUTIONAL
MELLIN TRANSFORMATION
SUNIL KUMAR SINHA
Department of Mathematics
Tata college Chaibasa, Singhbhum
Bihar-833202, INDIA
(Received revised copy July 20, 1981)
ABSTRACT.
The n-dimensional distributional Mellin transformation is developed
using the testing function space M
c,d
M’c,d"
and its dual
The standard theorems on
A necessary and sufficient
analyticity, uniqueness and continuity are proved.
condition for a function to be an n-dimensional Mellin transformation is proved by
the help of a boundedness property for distribution in
M’c,d"
Some operational
transform formulas are also introduced.
Distribonal MelVin Transformation, Distributions, and
KEY WORDS AND PHRASES.
Tst function spaces.
1980 AMS SUBJECT CLASSIFICATION CODES.
1.
46F12, 44.
N TRODUCT I ON.
The Mellin transformation was previously extended to certain generalized
functions by Zemanian [i] and Fung Kang [2].
In the present paper, we develop the
n-dimensional distributional Mellin transformation.
n
R
For the sake of brevity, we shall use the following notations.
respectively real and complex n-dimensional euclidean spaces.
n
stand for elements of C
respectively.
(Zl,Z 2
representing the n-triples
We take x
n,
R
t
n,
R
Rn,
Rn
and s
s
_{Sl,S 2,...,sn
and [e
-st
exp(-Sltl -’’’-sn nt ).
n
are
The symbols z and s
,Zn)
+
and
i
n
function on a subset of R shall be denoted by h(x)
2
s
Sn where
sl
,x
s]
L,...
Thus, [x
.,xn
n
mean the product Xl,X 2,
Xl x2
h(Xl,X
and C
,Xn).
By log x we mean
(Sl,S2,...,s n)
C
n
A
By [x] we
96
S.K. SINHA
{lOgXl,...
x
logx
{x s
s
n}
Xn
I
sn
and e
x
mean respectively
-<
y
{e -s t 1,...,e -Sntn }.
-st
x
and
<
y
n,
denote non-negative integers in R
Letting
k
k
I
{Xltl,X2t2,...,xn t n
and, by xt, we mean
i.e.,
+ kn’ Dkx
+ k2 +
(
The notation x _< y and x < y
1,2,...,n).
ku
m
and
Also
The letters k and m shall
are non-negative integers.
k
shall denote
Ox
x2k2
k
I
OXn kn
By a smooth function we mean a function that possesses partial derivatives of all
orders at all points of its domain.
2.
THE TESTING FUNCTION SPACE M
Let
cd"
R
denote the open domain 0 < x < oo.
H
Dc ,d (xv)
We define
n
function
=i
where
x-c
Ix-d
c,d(X)
x
x
if 0 <
if e <
Nc,d(X)
as the product
< i/e
<
R
is the linear space of all smooth functions f(x) defined on
c,d
I which satisfy the following set of inequalities
values in C
In fact
M
with
For each non-negative integer k,
lc,d(X)
Qk
[xk+l] DkQkx
f(x)
bers of M
v
c,d
are
Ix s-l]
for c -< Re s
d and [(log x)
k
x
s-l]
is in M
c,d"
Other mem-
for c < Re s < d.
represents a seminorm defi6ed by
max
sup
O_<[k[<
x
Of course, the collection
of seminorms.
f}
=i
Hence,
THEOREM 2.1.
c,d
Mc,d
[xk-l] Dkx
f(x)
I.
the topology generated by
is a Cauchy sequence in M
and, for each fixed k, the functions
oo.
lc d(X
(2.2)
is a multinorm, being a separating collection
Thus we can assign to M
A sequence {f
as v
R,
whose support is contained in
(f)
R$
(2.1)
denotes constants which depend upon the choices of k and f.
Any smooth function
n
0 < x < oo.
c,d
c,d(X) [xk+l]
{}
if and only if each f
Dkx fv(x..
M
c,d
converges uniformly on
is sequentially complete.
The mapping
f(x)
is an isomorphism from M
c,d
into L
[e -p] f(e -p)
c,d
wlere L
c,d
g(p)
(2.3)
denotes the testing function space
DISTRIBUTIONAL MELLIN TRANSFORMATION
97
defined by Sinha [3].
The inverse mapping is given by
Ix-i
g(p)
PROOF.
f(-log x)
(2.4)
The proof of this theorem is easy and is therefore omitted.
THE DUAL SPACE M’
c,d"
3.
M’
c,d
is the dual space of M
Multiplication by a complex number, equality,
c,d"
In fact, M’
is a linear space over
c,d
and addition are defined in the usual way.
C
f(x)
I.
M’
By <h, f> we mean a number that h
M
assigns to f
c,d
c,d"
If the support
R,
(,Miller [4], 1.6) of a distribution h is contained in a compact subset of
M’
h
c,d’ c,d
n
R
with c < d.
Also, every member of
Let us define a (weak) topology for
set of seminorms.
For every f
Mc,d,
for all f e M
c,d’
}o=l(h
M’
c, d)
is a distribution on
we define a seminorm
(h
f(h)
{<hv,f>
on
M’c,d
by
M’,d).c
is a Cauchy sequence in
the numerical sequence
n
R+.
by using the following separating
c,d
I<h,f>[,
f(h)
In fact, a sequence {h
M’
M’c,d
then
v=l
M’
c,d
if and only if,
converges
We can easily prove that M’
is sequentially complete
c,d
In view of Theorem i, we can relate to each h(x)
M’
a distribution
c,d
h(e -p)
L’
(see [3]) by
c,d
<h(e-P),
Conversly, if (p) e
L’c,d’
g(p)>
then (-log x)
<(-log x), f(x)>
<h(x), f(x)>.
M’
c,d
(3.1)
is given by
<(p), g(p)>.
(3.2)
Using (3.1) and (3.2), we can easily have the following theorem:
THEOREM 3.1.
_
M’
The inverse mapping is given by (3.2).
onto L’
c,d"
c,d
THE n-DIMENSIONAL DISTRIBUTIONAL MELLIN TRANSFORMATION M.
from
4.
h(e -p) defined by (3.1), is an isomorphism
The mapping h(x)
DEFINITION.
as the function
where
We define the n-dimensional distributional Mellin transformation
H(s) on
(Mh)(s)
2h
into C
H(s)
I
by
<h(x),
[xS-l] >
for
s e
$h’
(4.1)
is the tube of definition of the n-dimensional distributional Laplace
S.K. SINHA
98
transformation (see [3]).
M’
In fact, the R.H.S. of (4.1) has a meaning because the application of h
to
[x
s-l]
M
c,d
c,d"
[xs-l] and using Theorem
[x-1] g(-log x)
[e -sp] and f(x)
Setting g(p)
2.1, we can-have the following theorem:
The distribution h(x) is n-dimensional Mellin transformable if
THEOREM 4.1.
h(e -p) is n-dimensional Laplace transformable.
for ever s
Lh(e -p)
H(x)
In such a case, Mh(x)
5.
Using Theorems 2.1 and 3.1, we can have the following theorems:
then H(s)
h’
H(s) for s
If Mh
(The Analiticity Theorem).
THEOREM 4.2.
and
is analytic on
H
s
xu)
<(log
[xS]>,
h(p),
s e
(4.2)
h"
The proof is analogous to that given in [3].
for s
g,
g
n
if
in the same sense of equality in
M’c,d where
n
G(s) for s
H(s)
is non-void, and if
h
H(s) for s
If Mh
(The Uniqueness Theorem).
THEOREM 4.3.
c,d e
and c < d.
h
and Mg
g,
G(s)
then h=g
The proof is ana-
logous to that given in [3].
(The Continuity Theorem).
THEOREM 4 4
for some c,d e
R (c
c < Re s < d and
c < Re s < d to
PROOF.
< d) and if
Mhv H(s),
If {h
}=I
then Lh
converges in
M’
c,d
to h
H(s) exists for at least
{H (s)
converges pointwise in the tube of definition
=i
H(s).
Since
[xs] is in M
for each s satisfying c
Re s
the theorem
d,
c,d
follows from the definition of convergence in M’
and the fact that M’
is sec,d
c,d
quentially complete.
5.
A BOUNDEDNESS PROPERTY FOR DISTRIBUTIONS IN
For each h
constant c
R#
M’c,d,
M’
c,d"
there exists a non-negative integer r
such that, for all
in M
l<h,>l
]
R and a
positive
c,d’
<
c ().
(5.1)
DISTRIBUTIONAL MELLIN TRANSFORMATION
6.
99
A NECESSARY AND SUFFICIENT CONDITION FOR M(s) TO BE AN n-DIMENSIONAL MELLIN
TRANSFORM.
A necessary and sufficient condition for a function M(s) to be the n-dimen-
sional Mellin transform of a distribution h is that there be a tube c < Re s < d
(c < d) on which M(s) is analytic and bounded when
IM(s)
where
P(Isl)
is a polynomial in
<
(6.1)
sl.
It can be easily proved by using the boundedness property of Section 5 and
(Bochner [6], Theorem 60, p. 242 and 4, p. 244).
7.
SOME OPERATIONAL TRANSFORM FORMULAS FOR THE n-DImeNSIONAL DISTRIBUTIONAL
MELL IN TRANS FOPMAT ION.
Let us suppose that Mh(p)
H(s) for s e
h
R
and p
n.
C
We can easily
have the following operational transform formulas (Using Theorem 4):
kp h(p)
(ii)
M D
(12)
M{[x]}h
(13)
Mh (log x)
(14)
Mh{%(-log x)}
s
k
H(s), s e
,
,
h’
H(s + ), s + a e
H(-s), -s e
[T
-I]
H(s/T), s/
e
,
> 0.
Also, by using Theorem 5, we can have
(15)
M{(-log x) k h(-log x)}
(-)
Ikl
k
D
S
H(s)
s e
h
REFERENCES
N.Y., 1968.
i.
ZEMANIAN, A.H.
2.
FUNG KANG.
3.
4.
SINHA, S.K. The n-Dimensional Distributional Laplace Transformation, to appear.
MILLER, J.B. Generalized Function Calculi for the Laplace
Transformation,
Arch. Rational Mech. Analy. 12 (1963), 409-419.
5.
GELFAND, I.M.
6.
BOCHNER, S.
Generalized Integral Transformation, Interscience,
Generalized Mellin Transforms-l, Sci. Sinica 7
and SHILOV, G.E.
New York, 1964.
Lectures
Princeton, 1959.
on Fourier
(1958), 562-605.
Generalized Functions, Vol. i, Academic Press,
Integral.s, Princeton University Press,