677
Internat. J. Math. & Math. Scl.
VOL. 13 NO. 4 (1990) 677-686
ABELIAN THEOREM FOR THE STIELTJES TRANSFORM OF DISTRIBUTIONS
BOGOLJUB STANKOVI(
Institute of Mathematics
University of Novi Sad
Yugoslavia
(Received February 13, 1989 and in revised form March 15, 1989)
ABSTRACT.
First, a definition is given of the Stieltjes transform of distributions
which contains some well-known. Then, a structural theorem for distributions having S-asymptotic is proved. This makes possible to prove two theorems of the Abelian type
valued for Stieltjes transforms of distributions given in different ways.
KEYS WORDS AND PHRASES. Stieltjes transform, S-asymptotic, Abelian type theorem, distribution, generalized function.
1980 AMS SUBJECT CLASSIFICATION CODE. 46F12.
I.
INTRODUCTION.
It is possible to define the Stieltjes transform of distributions in various ways.
One of them is the so-called direct approach in which we construct a basic space A D D
of smooth functions to which the set {(s + t)
Im s
0, r >- 0} belongs. Then, we
-r-l,
A’, by the expresion < T(t),
(s + t) -r-I >. In this paper we shall deal only with such definitions. If the constructed space A’ is such that T
A’ has the support belonging to [0,), we have the clasdefine a transform for T, belonging to the dual space
sical Stieltjes transform, extended to generalized functions in one dimension.
We shall give a definition of the Stieltjes transform in meny-dimensional case not
only to have a new generalisation of the Stieltjes transform, but to prove Abellan type
theorems valued for the Stieltjes transforms given in different ways.
NOTATIONS AND DEFINITIONS.
2.
N
...,x
+
iy
n
is the set of natural
+
iy
ai
z
i
n)
e
(I
and e
n
1-I z
i:l
i
a|
F will be a convex
2
acute cone with vertex at zero;
center at zero and with radius r
rior of F.
n
n,
C
z
z
N U {0}. If a,b
I), then: (a.b) i__’laibi;" z a II(x + iYi)
i=l
(a-a); a > 0 means a i 0 for all i
numbers, N O
>
(X + iy I
n
Izal
B(0,r) the closed,ball
0; S(0,1) the unit sphere, all in
Rn.
ir[__l Ix i +
n.
with the
is the inte-
678
BOGOLJUB
DEFINITION I.
Let h
h
F, lim G(h)
plex valued function G(h), h
there exists
If F
,
h(e)
> h
F. We say that h
e
2
STRIKOVI
IG(h)
F such that
A
A
e
in F if h
2
C, h
F, h
h2
+ r. For
if for every
a com> 0
Z h(e) in F.
< e when h
we can prove the following Lemma:
We denote by F
B(a,r) c
lim G(x+h)
A for every x R
A, then
heFl,h
Suppose F #
LEMMA i.
and
U
>0
n.
lim
G(h)
hF,h
AB(a,r). If
for every e > 0 there exists h(e) such that IG(h)-A[ <
By Definition
e
h(e) + F. Now x
h
h(e) +
B(B0a,80r if 80 .> l(x h(e)U/r, 80 depends on e.
e F c F. Since F is a convex cone, x
Hence x
h(e) +
h/(e) +
+ F c F + F
PROOF.
,
80a
Boa
c
F and x
+
80a
REMARK.
Boa
+ F
h(e) + F. It follows that
c
In Lemma
B(a,r)
we can choose the ball
between B(a,r) and any coordinate axis
>- c, > 0. Then
r .> e > 0, i
A
< e for h
n. We denote by
Rn
is positive,
u )B(a,r),
r2
+
0)=
h e
F2,
0’
for such
0
10)(lail
lUll
-,
lhil
0
[uil
> 0.
and for any M
then
d(B(a,r),X i)
z + ha + ly,
a +
for a h e F2, then u
0 + F2
0
r) -> ( +
It is easy to see that
>. ( +
M. Hence, if h
>.
0 we can find h with the property
0
constructed ball B(a,r). If u e h
where z,y e B(0,r);
Boa + FI.-
in such a way that the distance
n, in
i
Xi,
fail
IG(x+h)
,
n.
i
By P we denote the set of all the real and positive functions c. Notations for the
spaces of distributions are as in the book of L. Schwartz [1].
DEFINITION 2. A distribution T
D’ has the S-asymptotic in r related to the
function c e p and with the limit U
9’, if the following limit exists
lim
her,h
< T(x+h)/c(h),#(x) >
<
U,@ >,
e
(2.1)
D.
Then we write T(x+h)
c(h).U(x), h r and we say that T has the S-asymptotic in 9’
(Pilipovid and Stankovid [2] and Stankovid [3]).
We shall use the same definition for a T
B’ 9’(L’) and @ e 9(L ) stressing
that T has the S-asymptotic in B’.
If the interior of the cone
r
is not empty,
,
then we can give the analytical
expression of the limit distribution U:
a
0
9’ and has the S-asymptotic with the limit U # 0. Then there exists
Suppose that T
and t a
9 such that < U,@ 0 > # 0. For this
using Lenna I, we have
n,
0
c(h+t)
T(x+(h+t))
>
<
c(h)
c(h+t) ,@0(x)
hr2,h
lim
T((x+t)+h)
,#0(x) >.
c(h)
hr2,h
lim
<
(2.2)
From trhis relation it follows the existence of the following limit
c(h+t)/c(h)
lim
her 2, h
d(t), t
e
Rn
(2.3)
and that U satisfies the equation
d(t) < U,@ 0 >
<
U(x+t),o(X)
>, d(O)
I, t
e
R
n.
(2.4)
Since U, as a distribution, has all the derivatives, it follows from relation
(2.4) that:
[d(t+At 1)
d(t)] < U,$ 0 > < U(x+t+At I) U(x+t),$o(X) >
[d(At I) -d(0)]d(t) < U, 0 >, d(0)= I.
(2.s)
679
ABELIAN THEOREM FOR THE STIELTJES TRANSFORM OF DISTRIBUTIONS
It is now easy to prove that (Pilipovid and Stankovid
exp(a’t)
d(t)
3.
and
U(t)
[2]):
n.
C exp(a-t), t
(2.6)
STIELTJES TRANSFORM OF DISTRIBUTIONS
In the following we shall use the well-known function q e C’, w
E
0 (Vladimirov
[4]):
f
ng(x)
where
Dm
q(x)
-n
x
exp
w2_x
Rn
e
(3.1)
ql(t)dt
D
IxH
0
q
The function
qw(x)
qw(x-t)dt,
B(0,2)
0, Ilxll > 3;
w
has the properties: 0 <
IDkq(x)l Ck-(k’e)
Rn
q(x)
R
x
n
<
x e
1.
Rn
n(x)
The constants C
The Stieltjes transform of a distribution T
DEFINITION 3.
(3.2)
1,x
B(0,);
not depend on
k do
V’ (S -transform)
P
is
defined by the limit
lim
< T(x),n W (x)(s+x) -(p+e) >
n
if it exists for a p e
REMARK.
S (T)(s)
P
(I,
By e we denoted e
s e
(C\) n
(3.3)
,i)
s can belong to a larger set, as well. This depends on the support of T.
0.
So S (6)(s)
s -[p+e), s
P
We shall give a relation between Definition 3 of the Stieltjes transform and definitions of some other authors frequently used in many papers. All of them are in one
dimension. Let us start with the classical definition.
If T is defined by the function f, suppf c
[0,(R)), Definition
(-(R),0]), then
3 gives the clas-
sical Stieltjes transform, if it exists. Let s e (\
S (f)(s)
lim
f f(t)q(t)(s+t)-(0+l)dt f f(t)(s+t)-(P+l)dt +
+
f f(t)nw(t)(s+t)-(P+l)dt.
lim
0
(3.4)
0
3
lim
We have only to prove that:
lim
i f(t)q(t)(s+t) "(p+l)dt
0,
s e
(C \(-(R),0])
(3.5)
when the classical Stieltjes transform exists.
Since for w
.<
x < 3
2
q(x)
f q(x-t)dt f
-z
the function
Bw(x)
ql(Y)dy;
is positive and monotone decreasing in this interval.
value theorem, there exists a
,
(3.6)
(x-Z)l
0 <
q(t)Re[f(t)(s+t)-(P+l)]dt
The last integral tends to zero when
<
By the mean
2 such that
q()
,
f
Re[f(t)(s+t)-(P+l)]dt.
(3.7)
because we suposed that the Stieltjes trans-
form of function f exists. We have the same situation with the imaginary part of the
680
BOGOLJUB
STANKOVI
integral from relation (3.5).
[5-6] defined the Stieltjes transform of distributions belonging
J’(r),Rer>-1of’,which is used in many papers. A distribution T belongs to
J’(r) if and only if there exist m N such that T= DmG, where G is a locally inte0(xr+m-=), = 0. The Stieltjes
grable function having a support in [0,) and G(x)
’(r) is, by these two authors:
transform of T
Lavoine and Misra
to a subset
(r+l)...(r+m)JG(t)(s+t)-r-m-ldt
S (T)
r
(3.8)
C\(-,0].
s
0
’(r), r
For a T
p, relation
S (T)(s)
(-i) m
<
lim
Pm
(3.3) gives
T(x),q(x)(s+x)
-(p+I)
,m.
f (s+x)P+k+l
I ()(P+I)...(P+m)
>
G(x)
lira
k=l
D
(3.9)
m_k
q (x)dx
0
We have two types of integrals
G(x)
G(x)
(s+x)P+m+l q(x)dx
0
and
Dm_ k q(x)dx,
(s+x)Yk+1
(3.10)
m-k >-i.
For the first one we proceed as in the case of the classical Stieltjes integral.
We have only to start from
The second one tends to zero when
.
3
S(x)
(s+x )m-k Dm-k
(s+x)P+m+l
(3.11)
(x)dx
and to use the integration by parts.
In the next cases the authors followed the construction of the Stieltjes transform
as we mentioned in the Introduction. The basic spaces
complex valued smooth functions The space A contains
er then that induced on it by
For a p
A. The
A’
is defined by
@
retstriction T of an element
0 and s belonging to a subset of
form of a
A are topological vector spaces of
P and the topology of is strong-
C, (s+t)
-(p+e)
is in
A’
A. The Stieltjes trans-
{(s+t)-(P+e)q(t),
lim
<
O<t<(R)sup Xc ,d(t)l (tD)k/,(t)l
0
<
<
,
<
t
ken
and c
< I/2,
d
(s+x)-’p+e’- >
#(x),q(x)
By Zemanian [7] it is p
> -1/2.
No
The
No}
(s+t) -p+I)’"
in S for =
The topology in
S
for
=
< p+l when
=Bremermann [9]
The toplogy of 0
<_- p+l
S
S
{
we have:
t
c
]c,d
(R).
is that in.duced by
.
CO,’)’Pc,d,k(O)
d
{
< t < (R); X
c
C0 _)’Yk()
k’
{(s+t)-(P+l)(t)
sup
k
d(t)
t
Pc,d,k,
>
(l+xa)l(xD)k(x))l’"
N 0 The function
0} tends to (s+t) -+I’
(R),(k)(t)
{
c
He treated the case
introduced the basic space 0
(3.12)
is defined by the seminorms
is defined by the seminorms
but the family
0} converges in
>
< -T(x),(s+x) -(p+e) >
Xc,d(t)
topology of ]c,d
where
o
By Pandey [8] the basic space A is
k
’
0 and the basic space A is
k
<
’.
is in
< #(t),(s+t) -(p+e) >.
In all the cases, we shall list, the family
Then for a
A’ and T
A to (s+t) -(p+e) when
S (T)(s)
P
to
0, Im s
0(t )
N 0 }.
k
0 and
=
>. -I.
For the Stieltjes transform of distributions in many dimensional case see also [I0].
ABELIAN THEOREM FOR THE STIELTJES TRANSFORM OF DISTRIBUTIONS
4.
DISTRIBUTIONS HAVING THE S-ASYMPTOTIC
681
.
We shall prove a structural theorem for the distributions having the S-asymptotic
in a cone
r
with the nonempty interior,
l’U(x), h e F in ’, then
THEOREM i. Suppose T O e B’ and T0(x+h)
a) U C;
b) T O
z’ where F i are continuous functions belonging to L’;
i=O
c) For every 0
i & 2 functions F.(x+h) converge uniformly to a constant when x
1
=;
belongs to a compact set K and h e F, h
d) T has the S-asymptotic in B’
and with the limit U
as well, related to c
0
AikF.
=C in the coneF.
I.
PROOF. a) By relation (2.6) U has to be a constant because c
(Schwartz [I],
b) From the fact T O
it follows that
L" for every
II, p. 57) and the set of distributions Q
n} is weakly bounded and
{Th E T0(x+h), h
bounded in 9’.
In addition to Q, we shall construct an other bounded set of distributions. We dei’. Now,
note by S
9,
{@
,|L & i}. We have seen that for a fixed
for every
S:
(T0*)
’
(T0)
T0*, >I II(T0*)(t)(t)dtl
R
ITo*IL. IIIIL.
T0*, >I
<
(4.1)
<
Hence the set of regular distributions, defined by the set of continuous functions
5
H
S} is weakly bounded and bounded in 9’.
T0*,
the set of functions {T*,
A set W’ e 9’ is bounded if and only if for every = e
(Schwartz [I], II, p. 50).
T W’} is bounded on every compact set M belonging to
Hence {T*=, T e W’} defines a bounded set of regular distributions. In such a way
H} give two bounded sets of regular distributions. Now,
{Th*, T Q} and
{U
n
{U@*, U#
h
for these two sets we can repeat twice a part of the proof of Theorem XXII from Schvartz
[I], II, p. 51.
We denote by
Rn,
cl
K
an open neighbourhood of zero in
Rn
which is relatively compact in
a compact set. Then, by the mentioned part of the proof by Schwartz [i],
or (=,8)
0, such that the mappings (=,)
Z 0 and m2
or
into i; B is the ball B(0,r)where
equicontinuous and map
U#*(=*8)
there exist m
*(*8)are
r is a positive number. Hence,
(T0**8)(x+h)
K
U#*(=*8)
K 2,
e
e
0
L,
then there exists a neighbourhood
Z(0,) for e,8
2(m2,e2,K2) c 2,
K0
Th*
is continous.
Let Z(0,p) be a ball in
such that
..-olx 2x2
for every x
B and h
n the function (Th*=*8)(x)
such that
min(el,e 2)
Th*(=*8)
and m
l(ml,el,Kl),
U
VI(mI,EIKI)
in
9;
a neighbourhood
2(m2,e2,K2),
Th
Q. Let
We shall now use relation (VI, 6; 23)
2(0,p) for ,8
max(ml,m2).
H and
from Schwartz [I], II:
2k*(E,E,T) 2.ak*(E**T)
T0
+ (**T),
where E is a solution of the iterated Laplace equation;
supp belonging
to K
0
K
kE
6;
,
,
(4.2)
supp and
K 2. We have only to choose the number k large enough
so
682
STANKOVI
BOGOLJUB
D.m
NE
Now, we can take: F
2
these functions are of the from: F
that
We have to prove that
V(m, 0’,K 0) and
<
have
’
S
I[(T0*$)*(gi*i)](0)l (/0 )2
(T0*=i*Si), >1
0 be any element from i I, then
Now let
-< MNH/I
F.I
yE*,T and F
**T 0. All of
E*yE*T0; F
0
0
V(m,g
0
T0*i*i; =i,i
0’,K0), 0 >
the properties given in Theorem I. For =i,i c
which proves that
T0*(=i*Si)
/HHLI
S and
belong to i’. Since F
(4.3)
M
I< (T0*(=i*Si)), >l
-<
T0*(=i*i), =i,8i
i
(m,e 0’,K0), Fi are continuous and belong to i’.
c)We shall continue with investigations of the properties of F.. By the properties
1
.
Fi(x+h)
Fi *-h T0*(i *i )*-h Th*(i*i)
of the convolution we have that
m
where
8i
T-h is the translation operator.
D,
We have proved that the mappings (,)
map
m
m
xD
into
K, which
cl
converges to
i.
D
is a dense subset od
is dense in
Since
Th*=*8,
m
Th*(* )
(Schwartz
as well
C*=i* 8i,
m,
T
Q, are equicontinuous and
h
], II, p. 53).
*
D(L I) and T
T0(x+h),B(x)>,=<[! ,Fi(x+h)Aik(x),dx I Mif ,Aik(x),dx,
’,
(4.4)
<
n
0
i’
>. 0. We can construct a subset of
C** for
DxD, then
d) there remains to prove the last part of Theorem I. For
n6ting that F.
i’, we have:
,<
for
i=0
n.
Hence the set {T0(x+h), h n}, is weakly bounded in
suplFi(x)l’
B’. Since D is dense in D(i), by the Banach-Steinhaus theorem the limit:
where
M.1
x
<
lim
h F, h
exists, as well, and equals
5.
T0(x+h),(x)
>,
(4.5)
D(L),
< C,B >.
ABELIAN THEOREMS FOR THE STIELTJES TRANSFORM
D’, the cone F is with the nonempty interior and
THEOREM 2. Suppose that T
i) T(x+h)
c(h)-U(x), h r in D’;
(C\) n the distribution T(x)/(s0+x) r belongs to. ’;
ii) For a r > 0 and s o
F, h
iii) For the same r and So, c(h)/(s0+x+h) r converges to C # 0 when h
and x belongs to any compact set in
n.
Then T has the S -transform for all
(s0+x)r/(s+x) +e
>
PROOF.
n,
(T)(s)
<
T(x)/(s0+x)
r
and
lim
heF ,h
for a y
> r, S
Sp(T)(s-x(h)__
(5.1)
0, p > r.
C. Namely,
From supposition iii) it follows that the limit distribution U
n
(C\)
from the relation:
and s
x belonging to a compact set in
o
n
c(h+y)
c(h)
and from the Lemma
c(h+y)
(s0+x+h+y)r
(s0+x+h)r (s0+x+h+y)r
c(h)
(s0+x+h)r
with the Remark after Lemma I, we have
Now, relation (2.6) gives U
C.
lim c(h+y)/ c(h)
h,h
(5.2)
I.
ABELIAN THEOREM FOR THE STIELTJES TRANSFORM OF DISTRIBUTIONS
683
By Theorem X from [I], I, p. 72 (see also Pilipovid and Stankovid [2]), for a
it follows the existence of the limit:
T(x+h)/0+x+h)r
<
lim
hr,h
(x) >
In such a way we proved that
lim
<
T(x+h)
c(h)
c(h)
(x) >
’(s0+x+h)r
< cci,(x) >.
the distribution T(x)/(s0+x) r, s
o
sme properties as the distribution T
0
her,h
(C\)
(5 3)
n,
has the
and we can use the assertion of
from Theorem
this theorem.
We shall prove, now, the existence of the S -transfor for 9 > r.
9
the set of funck > 0, where C do not depend on
Since
k
r--e
>
r, in (ii), when
tions
converges to (s+x)
Moreover,
,
IDk(x)l Ck( -k’e),
D(x)(s+x)
(s0+x)r/(s+x) r,
S(T)(s)
lim
a
(C\R)
for s 0,s
n
belongs to D(L’); consequently
T(x),(x)(s+x)-9-e>
<
T(x)
(s0+x) r
>
(s0+x) is+x) r’n(x)(s+x)r-o-e
lim<r
a
T(x)](SO+x)r,(sO+x)r(s+x)-O-e >, 0 > r.
the distribution T(x)/(s0+x)r satisfies
<
We have seen that
.
r--e,
(5.4)
the conditions of Theo-
rem I, therefore
S (T)(s-h)
P
c(h)
I
c(h)
i(k’e)
(-I)
-i=o
The expression
c(h)
T(x+h)
r’
(s0+x+h) (s0+x+h)r(s+x)-P-e
<
n
Fi(x+h)A
Aik[(s0+x+h)r(s+x)-9-e
ik
(sO+x+h)r
>
(5 5)
dx.
(s+x)0+e
is given by the finite sum of elements which
have the following form:
H
p(SO+X+h)r-j+P(s+x)-O-e-P
/c(h) when x n and h
r.
C
j,P
We shall analyse
j,
j > p > 0
(5.6)
Hj ,P
First we prove two inequalities:
1 Is 0 i+xi+hil ri
I(So+X+h)r
Is 0
i=
1ri (ISo, i+hil + t)ri(Ixil+l)ri
i=1
i+hilri + IImso,il’ I (I x
S C I(so+h)r/c(h)lI (Ixil+l)ri;
r
i=i
i
+ 1) ri
(5.7)
<
i=l
I{ s0+x+h)p_ j
=iII__ lSo i+xi+hi lpi-ji <
IX
i:l
IIm So, i
pi-ji
C’
p,j"
Now
IHj ,p I/cCh)
< C
C’ C
j,p j,p r
This inequality shows that
i i,
when h e
lxl+)=
I(so+h)rl
C(hi I (si+xi)Oi+Pi+l’
Ij,pl/C(h)
i=1
P
z
(5.9)
O.
is bounded by a function which belongs to
F. Since F.(x+h) are bounded, as well, when x
I
n and h
e
r,
we can use
the Lebesgue theorem for the integral in (5.5)to obtain that S (T)(s-h)/c(h) tends to
zero when h e
r,
h
(R).
STANKOVI
BOGOLJUB
684
The next theorem presents more precisely how S (T)(s-h) tends to zero when h e F
p
2
F is defined in the Remark after Lemma I. We have seen that
2
n, if h/, h
F
lhil ,
-.
h
2.
Let T e
THEOREM 3.
i)
ii)
D’
(s0,j+xj+hj)T(x+h)
(s0,j+xj)T(x) e B’.
and s
%
< T(x+h) (s+x) p-e’ > +
e
n.
We suppose:
n
# 0
(5.10)
0
We can write:
< (s 0,j+xj+hj)T(x+h),(s+x) -p-e >
Sp((s 0,J+xJ)T(x))(s-h)
where
& j
h.S (T)(s-h)
3 P
lim
PROOF.
D’;
F in
Then, for p > 0 and F convex cone,
her2,h
(C\R)
(s0,1’’’S0,n)
E
o
I’C, h
I, i
Sp(T)(s-h)
j and e[
(s0, j -sj+hj
< T(x+h)(s+x)
p-e
>; e’
(5.11)
(el..
e’n
0. Hence
(s0,j-sj+hj)-l[Sp((s0,j+xj)T(x))(s-h)
First, we shall prove that S
P
< T(x+h),(s+x) -p-e’ >]
((So, j +xj)T(x))(s-h)
F
0 when h
2
(5.12)
By Theo-
h
it follows:
rem
<
S
lim
heF2,h P ((So, j +xj)T(x))(s-h)
C (s+x) -p-e
It remains to prove that < T(x+h),(s+x) -p-e’ >
0 when h
we know that our distribution T has the form
Theorem
T(x)
e
>
(5.13)
0
as well. By
F 2, h
(s0’j+xj)-li=0 ikFi(x)
Rn.
where F.(x) are bounded functions when x
1
(5.14)
Using this form of T, we have:
2
< T(x+h) (s+x) -p-e’ >
(-I) i(k-e) <
i=0
Fi(x+h), ik( s0,j+xj+hj)
-i
>
(s+x)
(5.15)
The second part of relation (5.15) consists in fact of a sum of integrals
F i (x+h) s
+hi
-m( s+x P-=dx
.
0 and
a.
fF
i
(x+h)
So, j +xj +hi )- sj+xj )- PJdx.
R, k # j and h
r 2. The function in
for every v > I. Let p be such that
ppj > and
for x
(5.16)
Z i, i # j. All of these integrals tend to zero when h e
We shall prove only the case: m
0. In other cases it is trivial
I, =.
Let us consider the integral
where m Z I;
h
=’3 z
0, j +xj
k
e
Xjl F i (x+h)(s 0Z
v
P
integral (5.17) is bounded by a function in h belonging to
r2,
5.17
j+xj+hjl-i
0. Then
(LU), !u
belongs to P’(Lv)
for x i
!+!
v
p
R, i # j,
(Schwa-
rtz [I], II, p. 60).
Now, we have to prove that we can find p, satisfying our conditions, and v > i,
such that the number u remains inthe interval
u <
(R).
Then, we have only to use the
ABELIAN THEOREM FOR THE STIELTJES TRANSFORM OF DISTRIBUTIONS
D(LU),
property of a
u < ; namely:
(x)
(Schwartz [1], II, p. 55)
0, Ilxll
r 2,
that integral (5.16) tend to zero when h
and it will follow
685
.
h
Therefore we
shall analyse two cases.
In ’case
Case 0 <
pj
> 1, we can take
< I. Since
+v
hence u
p
2v
2v
p
0, we have
>
1--p
v
For such an v the number 1/u is strictly positive, hence
.
pj,
u <
The next example shows that Theorem 3 cannot be proved for p
i
(x+a)(s+x)
(a-s)
lln
>
a s
(T)(s-h)
(hr)’3
P
This shows the following integral:
dt
e
where W
s
(s+t)
-[2
h e F 2, h
.exp(s12)W_12,(l_) ! 2( s)
,-1
pj
0
(5 18)
0
(s0,j+xj)rT(x)
There arises an another question: If we know that
is it true that S
<v< (1-
or
B’
for a r
>
1,
? The answer is in general negative.
s
-
s
> O,
s
,
(5.19)
is the Whittaker function.
ACKNOWLEDGEMENT. This material is based on work supported by the U.S.-Yugoslavia
Joint Fund For Scientific and Technological Cooperation, in cooperation with the NSF
under Grant (JF) 838.
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I.
SCHWARTZ, L.
2.
PILIPOVI,
3.
STANKOVI, B.
Thorie des distributions, T. I-II, Hermann, Paris, 1950-1951.
S. and STANKOVI, B. S-asymptotic of a distribution, Pliska Stud. Math.
Bulgar., Vol I0, (1989), 147-156.
Sci. Vol.
II
S-asymptotic expansion of distributions, Internat. J. Math. Math.
No. 3 (1988), 449-456.
Generalized Functions in Mathematical Physics, MIR Publisher,
4.
VLADIMIROV, V.S.
Moscow, 1979.
5.
6.
LAVOINE, J. and MISRA, O.P. Thormes Abliens pour la transformation de Stieltjes
des distributions, C. R. Acad. Sc. Paris S4r. I Math. T 279 (1974), 99-102.
LAVOINE, J. and MISRA, O.P. Sur la transformation de Stieltjes des distributions
7.
et son inversion au moyen de la transformation de Laplace, C. R. Acad. Sc. Paris
Sr. I Math. T. 290 (1980), 139-142.
ZEMANIAN, A.H. Generalized integral transformations, Intersciences, New York, 1986.
8.
9.
I0.
On the Stieltjes transform of generalized functions, Math. Prec.
Cambridge Philos. Soc....71 (1971), 85-96.
BREMERMANN, H. Distributions Complex Variables and Fourier Transforms AddisonWesley, Reading, Mass, 1965.
PANDEY, J.N.
PILIPOVI, S. and STANKOVI, B. Abelian theorem for the distributional Stieltjes
transform, Z. Anal. Anwendunen Bd. 6(4., (1978), 341-349.