Internat. J. Math. & Math. Sci.
Vol. 8 No. 4 (1985) 719-723
719
ON TWO NEW CHARACTERIZATIONS OF STIELTJES
TRANSFORMS FOR DISTRIBUTIONS
SUNIL KUMAR SlNHA
Department of Mathematics
Tata College Chaibasa, Singhbhum
Bihar-833202, INDIA
(Received September 27, 1981)
ABSTRACT.
Two new characterizations of the Stieltjes transform for distribution are
developed, using two transformations on the space of distributions viz., dilation
u
n
and exponential shifts
T
-p
The standard theorem on analyticity, uniqueness and
invertibility of the Stieltjes transform are proved, using the new characterization as
the definition of the Stieltjes transform.
KEY WORDS AND PHRASES. Stieltjes transform of distribution, iterated Laplace transform, dilations, exponential shifts.
1980 AS SUBJECT CLASSIFICATION CODE. 46F12.
INTRODUCTION.
Widder
([I], p. 325)
introduced the Stieltjes transform as an iteration of the
Laplace transform as follows:
-xt
f(x)
fo
e
where
(x)
fo
e
so that
f(x)
f=
o
(t)dt
$ )
-xt
(t)dt
(1.1)
(t)dt
(1.2)
(1.3)
More generally, (1.3) can be replaced by Stieltjes integral
f(x)
da(t)
(x + t)
(.4)
The integral (1.4) was originally considered by Stieltjes[2].
Various generalizations
of Stieltjes transform have been given by various authors viz., Widder[3], Pollard[4],
Sumner[5], Mishra[6], Pathak([7]-[8]), Rao[9], Varma[lO], Arya([11]-[15]), Ghosh([16][18]), Boas & Widder[19], and Dube[20].
The present paper is concerned with two new characterizations of Stieltjes trans-
formation for distributions by the help of dilation
T
-p
introduced earlier by Gesztelyi[21].
u
It is interesting to note here that
Gestzelyi considered two transformations viz., dilation
which are defined for ordinary fuctions
and exponential shifts
n
u
n
f, complex number
and exponential shifts
T
p, and positive integer n
-p
720
S.K. SINHA
by
r-Pf(t)
{u
the sequence function
n
T-Pf(t)}
(1.6)
f(t)
converges (in Mikusinski sense) as
f
classical Laplace transform of
to the
n
He then defines the Laplace transform of a
p
at
then
p
as the limit (whenever it exist in the sense of Mikusinski-
x
Mikusinski operator
-pt
e
is a function which has a Laplace transform at
f
Gesztelyi proves that if
(1.5)
n f(nt)
f(t)
u
{0
convergence) of the sequence
T
n
-p x}
and shows that his definition generalizes
the previous formulation of the Laplace transform of Mikusinski operators of G. Doetsch
Price[26] defined the Laplace transformation of a
[233 and V. A. Ditkin ([24]-[25]).
using sequences of the form
f
distribution
{u
n
T
-p f}
and shows that the new defi-
nition is equivalent to Schwartz’s extension of the transform to distributions.
B
also introduced spaces
f
distribution
B’o
in
{u. f}
sequepce
and
B
o
and their duals
B’
and
f
in
B’
f
is in
has a unique extension
converges to
< f,l>
whenever
B’
He
and shows that each
o
He also shows that the
B’o
Recently, working
characterizations of the
Weierstrass, Mellin, Hankel and K-transform for distributions(J28], [29], [30] & [31]).
2. TWO NEW CHARACTERIZATIONS OF THE STIELTJES TRANSFORMS.
on the same lines the present author has given two new
In the present section we give two new characterizations of Stieltjes transforms
for one-dimensional distributions.
We will say that a distribution
(,B)
interval
in
(a,B);
subspsace
such that whenever
T-Pf
of
(a,B)
If
f
is Stieltjes transformable if there is an open
p
o
is a distribution in
B’o
+
it
is a complex number with real part
where
B’
nition
,
f
open interval then the set
is the largest such
then for
p e
,
{NOTE:
f(t)
]im
S[](p)
(0) z 0
0
and
S[f](p)
we define the Stieltjes transform
T
-p
>
f
Jwith
f.
is a Stieltjes transformable distribution, where transform has domain defi-
S[f](p)
where
a
as defined in [27].
is called the domain of definition of the Stieltjes transform for
If
Bo
is the dual space of
o
R
t
of
f
at
p
by
(2 I)
R
for every positive
is a test function in
is another distribution.
For the existence of limit in (2.1) and its meaning one could focus
tion on Theorem 3.1, p. 24 [26] or
his atten-
[27].}
Thus we have another characterization also as
S[f](p)
where
tion.
f(t)
S[](p)
>
=<T-Pf
0
t
R
for every positive
(2.2)
R
and
is another distribu-
STIELTJES TRANSFORMS FOR DISTRIBUTION
.
S[f](p)
From (2.2) we see that
able
p
with domain
is a complex valued function of the complex vari-
+ bg)
=<T-P[af
a<T
a
S
l(p)
f
If
d
3.
p
and
g
and
I
,I+ b<T -pg
+ bg],
-pf
,i
S[f](p) + b S[g](p).
Q, then
and
in
S[f](p)
f
are complex numbers
b
and
a
is a distribution that is Stieltjes transformable in
is analytic function of
dp
P!OOF.
p
For, if
is linear.
and
p
is Stieltjes transformable at
S[af + bg](p)
TIEOREM 2.1.
S
In fact, the mapping
are distributions that are transformable at
then (af
721
S[-t f(t)](p)
The proof is analogous to that for Laplace transformation as given in [26]
TREATMENT OF THE CONVOLUTION OF TWO DISTRIBUTIONS.
Much of the usefulness of the Stieltjes transform is a result of the way it treats
We give here this important property of the
the convolution of two distributions.
transformation by the following theorem:
]’]EOREM
3.1.
If
f
and
are Stieltjes transformable distributions such that the
g
domain of their respective transforms have intersection
q
transformable in
PROOF.
For
p
and for every
in
T-Pf
Steltjes transformable at
and
p
p
in
T-Pg
f*g
then
S[f * g](p)
f.
B’o
are both in
f*g is
Therefore,
and from (2.2) and the definition of convolution we get
S[f * g](p)
=<T-P(f *
=<r-Pf
*
g) ,i
>
T-Pg ,I>
=<r-Pf(t)T-Pg()
,l(t +
=<T-Pf(t)(r-Pg()
l(t)
=<T-Pf ,I><T-Pg
,i
>
S[f](p) S[g](p)
4.
is Stieltjes
S[f](p) S[g](p).
Q.E.D.
INVERSION AND UNIQUENESS THEOREMS FOR THE STIELTJES TRANSFORM.
No theory of Stieltjes transform would be useful without the inverison and unique-
ness theorems.
We give Theorem 4.1 which includes both inversion and uniqueness
theorems as its corollary.
In what follows we will have as independent variables at various times the real
variable
t
and real and imaginary parts of complex variable
For reason we will
p
sometime indicate the particular independent variable for the space or an operation by
a subscript
e g.,
<f(), e-it>
where
f()
is in
B’o
and
T
m
is a parameter.
S.K. SINHA
722
THEOREM 4.1.
If
f
B’
is a distribution in
then
t
f(t)
=--2--
lim
r-
where the limit is taken in
PROOF.
t
-
{p:
is Stieltjes transformable in
lim
rwhere the limlt is taken in
COROLLARY 4.1(b):
If
(4.1)
d
(Inversion theorem).
f(t)
S[g](p)
e-i I
The proof is same as that of Laplace transform as given in [27].
COROLLARY 4.1 (a):
If
f-rr eitf__ (T)
and
t
yo+ir
c-Jr
ePt
Re p
}.
Then, as long as
S[f](p) dp,
(Uniqueness theorem).
g
are Stieltjes transformable distributions such that
on some vertical line in the common domain of transforms of
f
g
I.
WIDDER, D. V.
f
S[f](p)
and
g
then
as distributions.
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29.
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On two new characterizations of the K-transform for Distributions-
to appear.
On two new characterizations of the Hankel transform for Distributions-to appear.
30.
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