Internat. J. Math. & Math. Sci.
Vol. 9 No. 3 (1986) 521-524
521
TAUBERIAN THEOREM FOR THE OISTRIBUTIONAL
STIELTJES TRANSFORMATION
D.
NIKOLI-DESPOTOVI and S. PILIPOVI(
Institut of Mathematics
Novi Sad, Yugoslavia
(Received March 31, 1985 and Revised July 5, 1985)
ABSTRACT.
In this paper we use the notion of L-quasiasymptotic at infinity of distri-
butions to obtain a final value Tauberian theorem for the distributional Stieltjes
transformation.
KEY WORDS AND PHRASES. Slowly varying function, the quasiasymptotic behaviour at infinity, distributional Stieltjes transform.
1980 AMS SUBJECT CLASSIFICATION CODE. 44A05, 46F12.
i. NOTIONS AND NOTATION.
L
Throughout this paper,
tion on [A, )
will denote real valued, positive and measurable func-
such that
lim
L(t)
e(t)
for each
%
tion
0, is called slowly varying and the class of such functions is introduced
a
0.
Function
L
which is regularly varying with index of regular varia-
and investigated by J. Karamata.
The quasiasymptotic behaviour at infinity of tempered distributions with supports
in
[0, ) (denoted by
Definition i.
the power
a e R
S$)
was defined by Zavijalov
A distribution
f
and with the limit
(see for instance [I]).
S+ has the L-quasiasymptotic
S+ g # 0, if for every
at infinity of
g e
S
(S
is the
the space of rapidly decreasing functions)
lim
k/
f(kt)
kaL (k)
<g(t). (t) >.
(t)
From the properties of homogeneous distributions it follows ([i]) that if this limit
exists then
for some
g
is a homogeneous distribution of degree
C # O, where
fa+l (t)
,
H(t)ta
F(a+1)
a
Namely,
g(t)
a>-i
Dnf a+n+l (t
a<-l, a+n
-I.
Cfa+l(t)
D. NIKOLIC-DESPOTOVIC and S. PILIPOVIC
522
As usual,
H
(0, =)
is the characteristic function of the interval
D stands
and
for the distributional derivative.
We use the definition of the distributional Stieltjes transform given in [2],
[3] in a little different notation ([4]).
N u {0})
Space J’(p), p e R\(-N o) (No
supports in [0, )
(a)
DkF;
f
k e N
iff there exist
with the support in [0, =)
F
grable function
f e J’(0)
such that
is the space of distributions with
such that
f= IF(t) l(t+B)-O-kdt
(b)
and a locally inte-
o
for
B
0
(i.I)
0
(D is the distributional derivative).
(b) we suppose that there exist
If instead of
IF(x)
(c)
C(l+x) 0+k-l-e
<
I’(0)
J’(0),
0 e
C(F) and
x
0,
e
c(F)
0
such that
I’().
the corresponding space is denoted by
Obviously
if
C
R\(-No).
It was proved in [4] that:
J’(0),
f
If
e
0 (k is from (a)), then
p+k
I’()
f
for
0
and
R-N o ).
If
J’(0),
f
R-N o
(I 2)
f e I’()
0, then
0+k
(k is from (a)) and
-k
for
).
(1.2)
S
The Stieltjes transform
of index
0
of a distribution f J’ (0)
R\(-N o
0
0,
with the properties given in (I.i) is a complex-valued function given by
(Sp{f})(s)
(0)
where
0(0+I)
(0)
s
J’(0)
f
0
e
s
(0)
and
(S {f})(s)
C-=,)]
in the domain
Observe that
f0 (t+s)0+k,
(0+k-l), k
It is proved in [3] that
variable
F(t)dt
k
C(-,0]
{i.3)
o
is a holomorphic function of the complex
f
J’(0).
J’(0+n), n e N.
provided that
f
implies that
The following equality holds:
(S0{Dnf})(s),
(0)n(S0+n{f})(s)
J’(0)
f
and
n e N.
We shall need the following theorem ([5], p. 339, Macaev and Palant)
THEOREM A.
Let us suppose that for some
de(k)
m+l
(+x)
f
0
f
0
0
m
and
x
d()
(+x)
m+l
and the following conditions are satisfied:
I)
Functions
2)
lim
and
are defined for
x
0
and are non-decreasing-
(x)
x+
3)
For any
C
that for any
there are constants
x
y
(x)
N
is
c ()
Y
and
N
0
y
m,
N
0
such
-
TAUBERIAN THEOREM FOR THE STIELTJES TRANSFORMATION
+,
Then, for
() ().
(This means
2. TAUBERIAN THEOREM.
THEOREM.
s > i,
Let us suppose that
(S {f}) (x)
L
o(e)’)
F (from (l.l)(a))
x/
P
where
<
if
l’(p) and
p+k-s-i > 0, f e
Moreover, let
is non-decreasing function.
e
523
xSL(x)
[A,=),
is slowly varying function in some interval
such that
x
0+k-s-I
L(x)
is non-decreasing function.
Then
has the
f
L-quasiasymptotic of the power
(-s)
p-l-s
and with the limit
xP-l-s
k
(P)k
Let us put
PROOF.
xP+k-S-iL(x)
x > A
0
x<A
(x)
has the
Then
the limit
f=
0
x
p+k-s-I ([i] Theorem I) and with
L-quasiasymptotic of the power
p+k-s-i
By [6] it is
de(t)
(x+t)
(2.1)
(p+k-l)
p+k-I
(t)dt
p+k
(x+t)
f
0
(p+k-l)
Now we show that the conditions of Theorem
0 < y < p+k-2
only to show that for some
A hold for
and every
(2.2)
x
xSL(x)
and
C
F
In fact we have
N
there exists
0
such that
(ky)
(y)
Let us put
C kY
k
for
where we choose
p+k-s-l+e
(2 3)
N
y
and
e
0
such that
y
N
where
0
e
and
s-l.
After substituting (2.1) in (2.3) we obtain
L(ky) < C
and this inequality is true if
From the
assumption that
(P)k f
0
(Sp{f})(x)
dF(t)
(x+t)
p+k
kL(y)
k
and
l’(p)
f
0
depends on
dF(t)
f
(P)k-i
N
C
(see[6]).
and (2.2) we have
(x+t)
p+k-I
s
x L (x)
X
It implies
(P)k f
0
dF (t)
(x+t)
p+k-1
d
f
(x+t)
+k-1
and by Theorem A it implies
(P)k F
Thus. we obtain that
Since p+k-s-i
F(x)
x
x
p+k-s-1L (x)
O, it follows ([I]) that
x+
F
p+k-s-I
has the L-quasiasymptotic of the
power p+k-s-I
and with the limit
([i]) that
has the L-quasiasymptotic of the power
f
(p+k-s-l)...(p-s) x p-s-I
(P)k
x
Since
A
O-s-1
DkF
it easily follows
and with the limit
D.
524
COROLLARY.
Let us suppose that
(S{f}(x)
where
L
NIKOLI-DESPOTOVI6
x
and S.
PILIPOVI
faJ’(p) and for some
,
$
> p
sR\(-N O
I,
s
xSL(x)
x-S+kL(x)
[A,). Further, suppose -s+k>O and
k+l
D
(f F(t)dt)).
(Consequently, fel" () and f(x)
is slowly varying function on
is non-decreasing in
[A,).
O
Then
f
has the L-quasiasymptotic of the power -1-s and with the limit
(-s)
k+l
()
k+1
REFERENCES
1.
DROZINOV, JU.N. and ZAVIJALOV, B. I.
Quasiasymptotic Behaviour of Generalized
Functions and Tauberian Theorems in Complex Domain (In Russian). Mat. Sbornik,
Tom. 102(144), No. 3, (1977), 372-390.
2.
LAVOINE, J. and MISRA, O. P. Theoremes Abelians pour la Transformation de Stieltjes
des Distributions, C.R. Acad. Sci. Paris 279 (1974), 99-102.
3.
LAVOINE, J. and MISRA, O. P. Abelian Theorems for the Distributional Stieltjes
Transformation, Math. Proc. Camb. Phil. Soc. 86 (1979), 287-293.
4.
PILIPOVI,
5.
KOSTJACENKO, A. G. and SARAGOSJAN, I. S. Distribution of Eigenvalues.
Ordinary Differential Operators (In Russian) Moscow, Nauka, 1979.
6.
NIKOLI-DESPOTOVI,
S. An Inversion Theorem for the Stieltjes Transform of Distributions,
Proc. Edinburgh Math. Soc. (Work accepted for publication).
Self-Adoint
D. and TAKACI, A. On the Distributional Stieltjes Transformation,
International Journal of Math. and Math. Sci. (Work accepted for publication),
East Carolina University.