I nternat. J. Math.
Math. Sei.
441
Vol. 2 # (1979) 441-458
THE STIELTJES TRANSFORM OF DISTRIBUTIONS
U. N. TIWARI
Department of Mathematics
John Abbott College
Montreal, Canada
J. N. PANDEY
Department of Mathematics
Carleton University
Ottawa, Canada
(Received September 28, 1978 and in revised form April 3, 1979)
ABSTRACT.
In the present work, two complex inversion formulas of Byrne and
Love for generalized Stieltjes transformation are shown to be valid for a
class of distributions.
This is accomplished by transfering the complex inversi(
formulas on the testing function space of a class of distributions and then
showing that the limiting process in the resulting formula converges in the
topology of the testing function space.
KEY WORDS AND PHRASES. Generalized Functions, Stieltj Transformation.
AMS (MOS) SUBJECT CLASSIFICATION (1970) CODES.
i.
P
46A40, Secondary 42A25.
INTRODUCTION.
Let p be any complex number except zero and the negative integers.
Then
442
U.N. TIWARI and J. N. PANDEY
for all s in the "cut plane", that is all complex numbers except those which are
negative real or zero, the
StleltJes
Transform in its general form is defined by:
(,+)P
The
follorl
inversion theorems for particular values of p and s are well known.
THEKEEM A (Widder).
If f(t) belongs to L(O,R) for every positive R and is
such that he integral
o
:+x
converges for x > O, then F(s) exists for complex s in the cut plane and
llm
f (-f -i) -F(-+i)
+)+f ( -)
2
2i
for any positive
at which
THBOREM B (Smer).
f(+)
If p
and
f(-)
> 0, f(t)
both exist.
is locally integrable in [0,=], the
improper Lebesgue integral.
o (+t)
converges (for a certain value of s in the cut plane and so for all), t
.
the limits
f(t_+0)
exist, then:
t
[(,:+o)+(,:-o)]
li,.
j’
0+
o
d
’
c
x
(z)P’lF’(z)
>
0
and
443
STIELTJES TRANSFORM OF DISTRIBUTIONS
where C
x
is a contour in the cut plane from
THEOREM 1.3 (Byrne and Love).
If Re p
-x-i
> I,
m
I(x+0)+ (x-0)}
-x+i.
f is locally integrable in [0,=],
> O; then,
improper Lebesgue integral (I.I)converges, and
x for which the Lebesgue limits
to
for each positive
f(x_+O) exists,
2i
O+
f (x+t)P’2{F(t’i)-F(t+i)}dt"
[2, p. 349]
-x
THEOREM I 4 (Byrne and Love).
If Re p
f(t)
> 1,
2- E L(0 ,=)
and the improper
l+t
Lebesgue integral
o
(+t)
p
converges, then for each positive x for which the Lebesgue limit
I
2
[f(x+O)+f(x-O)}
llm
-K)+
THEOREM i.i
and Pandey [14].
f-x (x+t)
p-2 |F
(t-i)-F (t+i)}dr
f(m0_ exists,
[2, p. 352]
has been extended to distributions by Pandey and Zemanlan
13]
Theorem 1.2 was extended to distribution by Pathak [15].
Our
object is to extend theorems 1.3 and 1.4 of Byrne and Love to generalized
functions
(dlstrlbutlons).
2. THE TESTING FUNCTION SPACE, S(1) AND ITS DUAL.
An infinitely dlfferentlable complex valued function (x) defined over
I
(0,) belongs to the testing function spaces S(1) if,
sup
0<x<
.l+x_
(x)
<
(R)
444
U.N. TIWARI and J. N. PANDEY
for all
k
0, I, 2,
Clearly, S(I) is
is a fixed real number.
where
The zero element
a vector space with respect to the field of complex numbers.
of the vector space
zero.
S(1)
is the function defined over I which is identically
The topology over S(1) is generated by the collection of semlnorms
}=
[24; p. 8]. We say that a sequence
|
converges in S(1) to (x) if for each fixed k,
tends to
space.
.
yk(
tends to zero as
The space B(1) is a vector subspace of S(1) and the topology of D(I)
of any member of
S(1) to D(1)
is in
D(1) by S(I) and as such the restriction
D’(1), where S(1) and D’(I) denote the dual
spaces of S(I) and D(I) respectively.
non-negative integer k
as
and
-
We say that a sequence
belongs to S(I) is a Cauchy sequence in S(I) if
3.
I)
The space S(I) is a locally convex Hausdorff tologlcal vector
is stronger than the topology induced on
other.
belongs to
where
yk(
1
where
) goes to zero for any
both tend to infinity independently of each
It can be readily seen that S(1) is sequentlally complete.
THE D.I.STRIB,UIONAL S,TIELT,J,E,S
.TNSFORMA.TION
For a complex s not negative or
zero.
1
(s+x)P
belongs to S’
where a
< Rep.
Therefore, the distributional Stieltes transformation F(s) of an arbitrary
element
F(s)
f
=n
S’,
a
< (x),
<
Re p, is defined by
I
(s+x)P
>
(3.)
where s belongs to the complex plane cut along the negative real axis including
the origin.
THEOREM 3.1.
(x)
If m and k both assume non-negatlve integral values and
is a
STIELTJES TRANSFORM OF DISTRIBUTIONS
445
compact set of the complex plane not meeting the negative real axis, then for
fixed non-negative integers m and k, there exists a constant B
satisfying
of the complex plane not meeting
uniformly for all s lying in the compact set
the negative real axis on the origin.
Using the compactness of the set
PROOF.
<
and the fact that
I
S’;
(s+x)e+k
Re p, the theorem is immediate.
THEOREM 3.2.
For an arbitrary
the equation (3.1).
d
(-s)
where
(-I)
m
(p)m
PROOF.
Then, for
(’)
m
m
< (x),
(+x)
p(p+l)(p+2)
f
S’
and
a
<
Re p let F(s) be defined by
I, 2,
(P)m >
p+m
(.)
(p+m-1).
p
If p is such that s does not have a branch cut in the complex plane,
the proof can be given in a way similar to that given in [3, Lemma 2a].
If p is
p
such that s has a branch cut (along the negative real axis for the sake of
deflnlteness), then choose the contour of integration as shown in the complexplane cut along the negative real axis below.
446
U. N. TIWARI and J. N. PANDEY
S/AS
Y
X
Here, C
I
and C
2
are arcs of two concentric circles with centre at origin and C is
The radii of C
the contour of integration as shown,
1
and C
2
and paths L
I and L2
are so chosen that the point s is contained in the region bounded by the contour.
Let
d
inf
l-sl
and choose
F(s+as)-F(s). < (t),
< f(t),
"P
2i
As
IAsl
d
<
NOW
>
z-s-s
(z+t)p
z-s
(z.s)2j
dz
>
where C is the contour shown in the diagram
< f(t),
8As >
where
eAs
As
2hi
f
C
1
(z+t )p
We now wish to show that
Using
1
(z-s )P(z-s-As)
8As
0
in
dz
S (l) as
c
AsO.
Theorem 3.1, we have
,yk(SAs ) <_ Bc IAsl
2
L
d
3
(3.4)
447
STIELTJES TRANSFORM OF DISTRIBUTIONS
where L is the length of the contour C and B
(p)k(l+t)a’tk
As-,
0
in
> 0.
(3.3) and using (3.4), we get
-p
< e(t),
’()
is the uniform botmd of
for all z lying on the closed contour Camd t
(z+t) +k
Letting
c
(+t)
>
+
Now; the theorem follows from induction on he order of the derivative of F(s).
THEOREM 3.3.
transform of
The function F
for real x where F(s) is the
0[x
as x
0[x
as x
0[ x
"k’Re
if cz
as x
0
Re p
Re p
if cz
o
P]
<
<_ Re
if
The proof is Insnedlate from the boundedness
4.
StleltJes
S’, satisfies the following relation:
f
E(m)(x)
(m)(x)
p
roperty of dlstributios
9, p. 18].
COI,LE INVERSION OREMS
We are now ready to prove our first inversion theorem.
THEOREM 4.1.
be the
im
StleltJes
<
!
2i
<1
For a fixed
transform of
and
Re p
> I,
let f
f(t) as defined by (3.1).
J (x+t)P -z[ F(t-i)-F(t+i)]dt,
Ten,
(x)>
-x
< f, > ur 11
PIOOF.
1
D(1)
First onslder
1
(x+t)P’2F(t-i)dt f
-x
-x
(d:)
p-2
1
< f(y),
(y+
-)
>
S’(1) and let F(s)
dt
U.N. TIWARI and J. N. PANDEY
448
For fixed x and t,
Since, in view of Theorem 3.2, F(s) is analytic in the cut plane, the left-hand
side integral in (4.1) is meaningful.
> 0,
can be shown that for
-x
By using the technique of Riemann sums it
(x+t)P "2F (t’i) dt
(x+t)
< f(y),
-x+
f(Y)’
p-2
(y+t-i)
dt
p
[ (y+A.i)P-l ]
cP’I
(x+x)P_ "I
I.
I
>
p---" (x-y+i)
(y-x+-i
[by Lenna 5,1, p. 333]
(say).
0+,
One can "easily check that as
and for fixed %,
and x.
I (x+t)P’2F(t’f) dt
cp -i
(-x+ -i)P -I
Therefore, letting
.
(x+t)P’2F(t-f)dt
>I >
0, we get:
(p-l) (y-x-i) (y+% -i)
In view of Lemma 3.5* [7, p. 12], it follows that:
Therefore, letting k
0 in S (I) for Re p
< f(y),
-x
for fixed x and
)p_i
as
p’I >
(4.2)
%
(4.3)
in
(4.2), we obtain
< f(Y)’
p--- y’-x-i >
-x
* The proof was provided by Professor E.R. Love.
(4.4)
449
STIELTJES TRANSFORM OF DISTRIBUTIONS
Using a similar argument, we can show that
f
(x+t)P -2F (t+i)dt
< f(y),
p.--
y-x+i
>
-x
Combining Equations (4.4) and (4.5), we get
J
2i
(x+t)P
-x
< f(y),
[
(y-x)2 2]
(4.6)
J>
Now using the technique of Riemann sums, we obtain
<
p-I
2
J (x+t )p-2 [(-+/-)-(+i)]a, (x) >
-X
b
< f(Y)’
(x) dx
__7
2a [y-x)2]
(x)
where the support of
>
D(1) Is contained in (a,b), b > a > 0.
same techniques as followed in proving Theorem 2 of
e
Usi
(3) one can show that
b
a
(y -x )
(y)
2.
in the topology of S
(4.8)
(I) as
0+.
Therefore, letting
0+
in (4.7), we
have
llm
<p-I
(x+t
2i
)p-2 [(:-i)-(+t)]dt,
(x) >
<
,>
This completes the proof of the theorem.
To prove our other inversion theorems, we require a couple of Lemmas.
EMM 4.2.
Let t, s,
> 0. Then,
b
lim
->+
(l+x)c
uniformly for all
J" (t_x)2+T2
a
x
> O.
for finite
b
>
a
>
0
and
< I,
U.N. TIWARI and J. N. PANDEY
450
PROOF.
Let
b
a
l(l+x)
Since sup
x>N>b
a<t<b
positive
>
N
and
b
(t-X)
<q<1
0
> 0,
is bounded, for
(t.x)24 2
there exists a
such hat
(4.9)
(0,q)
uniformly for all
and
x
> N.
Now assume that 6 is a positive number
_< rain (1,
a
)
and for 6
<_ x <_N
let us write.
I
a+x-6
(I+x)C
.
b
a+x+6
+
Denote
e ree eresslons
and
respectively.
+
I
on
e
t’xl
dt
rlghtd side of Eqn. (4.10) by
I1, 12
Now
a+x+5
a+x-s
2s
Now choose
(t’x)2 +
s() a
(
5
such that
5(I) <
Chs way.
and fix
Therefore
(4.)
uniformly for all
x
[6,N]
and
[0,q].
x
b
It-xl
dt
STIELTJES TRANSFORM OF DISTRIBUTIONS
There fore
(l+x)
(a+6)22
Therefore,
I
3
0
if
8
<x<
b
O+
as
(4. 2)
[5,N].
x
uniformly for all
,
451
Next,
a+x -5
(c-x)2 2
a
Now for
<_
5
x
<_
It-xlat
a
(a’S)2.l.q.l
(a-x
0
2
(4.3)
as
uniformly for all x lying in [5,a].
For
a
< x < N,
3’
I
-2
0
/,n
(a_:)2..
+ 3.2 n
k 2
O+ uniformly for all x
as
[a,N]
(l+x)
(4.14)
CombininE results (4. II) through (4.14) we have
lim
_<
<
x
0
Now, for
III <_
As
I11
x
uniformly for all
<_
8,
(l+) n
_>
0
(4.5)
we can see that
[(b’x)2+2]
0
(a-x)
O+ uniformly for all x
[0,].
(4.16)
U.N. TIWARI and J. N. PANDEY
452
Combining results (4.15) and (4.16), we get
lim
III <--
_+
x
uniformly for all
> 0.
Thus, the proof of the lemma is complete.
LEMMA 4.3.
numbers and (t)
D(1).
.
>I >
Let Re p
Assume that
t, x, I and
are all positive
Then,
I
S where the support
0+ in the topology of
as
contained in
PROOF.
0
as
(a,b);
>
b
a
of
(t) E D(1)
is
> 0.
We have to show that for each m
0, I, 2,
(l+x)LxmDl(x,)
0
uniformly.
It can be easily shewn that
uniformly for all
(l+x)xmDxm(x,)
uniformly for all
0
satisfying
0
as x
< I.
<
(R)
Therefore, for
(0,i).
So,
>
0
there exists
N > 0
such that
(4. I)
uniformly for all
x >N, and
Now consider the case
0
and complete the proof for
m
m= 0.
<
0
<
x
<_
.
< I.
I, 2, 3,
We will first give the proof for
m
by using the result for the case
0
453
STIELTJES TRANSFORM OF DISTRIBUTIONS
I.
O. Write
For m
(l+x)
(].+x)=: (x,])
2i
-1
l+
(x_t)2+T2 kx+X +i
First, consider
]: (x,,Q)
I l(x,).
(say)
]:2 (x,)
Since,
I
1
(x+x+) ’i
(x+
(p-l)
-)’
-
++ z "pdz
x+x
(xq.x)
<_ 21p.llT
Therefore
"l
(t)dt
err
X’m
Re P
Pl/ A Re p
we can find a constant B(p) independent of x and
such that
b
Z(x,)l
<_ B (+x)
j.
a
(x-t)2+
In view of Lemma 4.2, the rlght-hand side converges to 0 uniformly for
x
>
0
as
0+.
12(x,).
We now consider
(1)
For Re p
x++i
all
_>
2,
t
1
<
For
0
<
< b,
using Lemma 3.3 (7, p. 9) we get
x
<_ Ip-lle
t<x
2TT
(4. 8)
U.N. TIWARI and J. N. PANDEY
454
(ll)
For Re p
_>
2,
t>x
++)+ )
t+x+i
Re p -2
x+X + i
2
2+T
P’,[ e2 m P]’
Re p-1
X+l)Re p-2
(b+
(4.19)
(iii)
For Re p < 2,
t
<_
x and
ARe
For
<_
re p
2,
E [a,b],
b
>
a
> 0,
p-2
!P’ll’e2wllm Pl(a+A)Re
Re(p-2’)2
-I
(iv)
t
t
> x,
p-1
t
[l+x’i]
[a,b], b > a > 0,
0
< I.
<
(! t-x1+)
( i+
0 < T]
b)Re
p-2
(4.20)
X"
<
i
-1
(4.)
Therefore, from inequalities (4.19) through (4.21), it is evident that for
Re p
>I
and
0
<
< I,
there exists a positive constant K(X,p) independent of
t and x satisfying
-i
< K ,p)
(t-x) + ]
t { (a,b)
Similarly, under the same set of conditions
-1
<_ K(,p) (t-x) + ],
t
(a,b)
In view of these inequalities we, therefore, have established that
STIELTJES TRANSFORM OF DISTRIBUTIONS
455
a<t<b
That is
a
(t’x)2+2
a
(t’x)2+2
Since
(z+),
as
b
2
j,
dr.
(t-) +
’
o
0+ uniformly for all x
follows that
12(x,)
0
The case
m
> 0. Therefore,
O+ uniformly for all
as
in view of Lemma 5.2, it
x
> O.
1, 2, 3,
A careful computation along with integration by parts will show that
b
+
(Z+x)
x
(x+i)
Using the technique of induction, we obtain:
(xq -i) -p]
(t+A)p-2 (t)dt
U.N. TIWARI and J. N. PANDEY
456
"F"
+
[’(x++i)
"
(x+)p-2 (m-)
(x+-i) "p]
a
(t)p-2 (m-2) (t)dt
(4.22)
Denote the integrals on the right-hand side of (5.22) by
in that order.
0,
In view of case m
Jl0
as
To show that other integrals converge to 0 as
consider the most general integral
Jm+l"
(x+)’’m+l- (x--i)’P’ml
-)+
Jl’ 32’
Jm+l
uniformly for all x E (0,N).
-+ uniformly for
x E (0,N), we
As before,
(-p-re+l)
x+
z’’mdzl
f
x+ -i
--’eVl z’m Pl p+m
<_
(x+X)Re
Therefore, we can find a positive constant C independent of x and
(l+x) CZxm
0 as
such that
0+
(x
uniformly for all
x e (0,N) and each fixed
m
i, 2,
Thus, we have proved that
(l+x)xmDml(x,)
m
+
0 as
uniformly for all
0, l, 2, 3,
Combining this fact with inequality (4.17), we have
lira
l(l+x)zxmDmx(x,l)l
<
x
(0,N) and each fixed
STIELTJES TRANSFORM OF DISTRIBUTIONS
uniformly for all
x
>
0
and each fixed
m
457
O, I, 2,
since
is arbitrary
our claim is established.
THEOREM 4.4.
Let
Re p
>I >
and
S’.
f(t)
>
transform of f(t) defined by (3.1) then for
im
<P2i
f(y-i)-F(y+i)]
(y+t)P-2dy,
0
If F(s)
is the
StieltJes
and each
(t) >
0
X
PROOF.
By using the same technique as used in proving Theorem 4.1, it can be
shown that
where the support of (t) is contained in (a,b),
b
>
a
> 0,
< f(),
a
> (t)dt.
Letting
0+, the result follows in view of Lemma 4.3.
THEOREM 4.5.
the
For a fixed
StleltJes transform
llm
(By using Riemann’s sum technique)
<p-I
2i
< I < Re p,
let f(t)
of f(t) defined by (3.1).
J (x+t)P’2[F(t’i)’F(t+i)]dt’
S’(I) and let F(s) be
Then,
{(x) >= <f,{>
U.N. TIWARI and J. N. PANDEY
458
for all
6 D(1)
and A
> 0.
The result follows quite easily in view of Theorems 4.1 and 4.4.
PROOF.
The authors are grateful to Professor E.R. Love, for his
ACKNOWLEDGEMENTS.
valuable suggestions, in particular for providing the proof of Lemma 4.3.
The
work of the second author was supported by a National Research Council Grant
Number A52 98.
REFERENCES.
i.
Byrne, A. and Love, E.R. Complex inversion theorems for generalized
Stieltjes transforms, A.ustra.llan Jr. .o.f .Me..th.. (1975) 328-358.
2.
Pande)
h T., Zemanlan, A.H. Complex inversion for the generalized convolu"ansformation, Pacific Jr. of Math. 25 (1968) 147-157.
ti, n
3.
Pandey, J,N. On the Stieltjes transform of generalized functions,
Phil. Soc. 71 (1972)85-96.
4.
Pathak, R.S.
ceed.lns
5.
Proc.. C .ab...
A distributional generalized StleltJes transformation, Pr.._oof Edinburgh Mat.h..S0c. 20 (1976) 15-22.
Theorle des distributions., Vol. I and II (Hermann, Paris
Schwartz, L.
1957, 1959).
6.
An inversion formula for the generalized StleltJes transform,
55 (1949) 174-183.
Sumner, D.B.
Bulletin Americ.an Math. sot..
7.
Tix#ari, U.
Some distributional transformations and Abelian theorems,
Carleton University (1976).
.I, ..D.T.hesi.s,,
8.
Widder, D.V.
9.
Zemanlan, A.H.
(1968).
The Laplace transform_, Princeton (1976).
Generalized integral transformation, Intersclence
Pub.lls,h.e.r.s