Internat. J. Math. & Math. Sci.
VOL. 12 NO. 4 (1989) 673-684
673
QUASIASYMPTOTIC EXPANSION OF DISTRIBUTIONS
FROM
S’+ AND THE ASYMPTOTIC EXPANSION OF THE
DISTRIBUTIONAL STIELTJES TRANSFORM
S. PILIPOVIC
Institute of Mathematics
University of Novi Sad
21000 Novi Sad
Yugoslavia
(Received December 13, 1988)
ABSTRACT.
By following the approach of Droinov, Vladimirov and Zavialov we investigate
the quasiasymptotic expansion of distributions and give Abelian type results for the ordinar asymptotic behaviour of the distributional Stieltjes transform of a
distribution
with appropriate quasiasymptotic expansion.
KEY WORDS AND PHRASES. Stieltjes transform of distributions, quasiasymptotic behaviour of
distributions.
1980 AMS SUBJECT CLASSIFICATION CODES. 44A15, 46F12.
i.
INTRODUCTION.
In the papers [9,10] authors followed the definition of the distributional Stieltjes
transform given in [7] which enabled them to use the strong theory of the space of tempered distributions S’. In fact, they generalized slightly the definition of Lavoine and
Misra. Using the notion of the quasiasymptotic behaviour of distributions from ’= {fe
[0,)}, introduced by Zavialov in [15], they obtained more general results than
supp f
in [6,7,2] for the asymptotic behaviour of the distributional Stieltjes transform at
Let us notice that the notion of the quasiasymptotic behaviour of distributions
and 0
was studied by Droinov, Vladimiraov and Zavialov in several papers (see [13] and references there) in which they obtained remarkable results in the quantum flela theory.
S+
+.
McClure and Wong [3,14] studied the asymptotic expansion of the generalized Stieltjes
transform of some classes of locally integrable functions characterized by their asymptotic expansions at
and 0
+
Our approach to the asymptotic expansion of the distributional Stieltjes transform
which we study in this paper is quite different from the approach given in [3,14].
In the first part of the paper we slightly extend and investigate the definition of
the quasiasymptotic expansion at
of a distribution from
S++ given
we give the definition of the quasiasymptotic expansion at 0
in
[4, p. 385]. Also,
of an element from
PILIPOVI
S.
674
This enables us to obtain, in the second part of the paper, the asymptotic expansions at
and at 0
+
S+.
of the Stieltjes transforms of appropriate distributions from
in this paper on which the Stieltjes transform is defined do not
[14] and
Domains in
The examples given at the end of this paper show some advantages of
contain each other.
our approach in the case when a distribution belongs to the intersection of the mention-
ed domains.
2.
,
NOTATION
As usually,
C, N,
are the spaces of real, complex and natural numbers;
The space of rapidly decreasing functions is denoted by S and by S m, m
pletion of S under the norm
IIII m =sup{ (I +
x2)m/21(i)(x)l;
R,
x
i
NO=NU{0}.
N O, the com-
m}.
A positive continuous function L defined on (0,) is called slowly varying at
>
if for every a
0
(lim L(at)/L(t)
lim L(at)/L(t)
t
t0 +
i).
the set of all slowly varying (in short s.v.) functions at
Z (Z0+)
We denote by
(0 +)
(0+).
For the properties of s.v. functions we refer the reader to [11].
-
(0+),
If L is an s.v. function at
X
< L(x)<
x
g
(.x
-g
then ([ii]) for every e > 0 there is A g > 0 so that
g
> L(x) >
if
X
x
(0 < x < A ).
> A
This property of L and the corresponding properties of
Sm ([12, p. 93]) imply the
following assertion which we shall use in parts 6 and 7:
Let G
[0,),
supp G
Llo c,
=
Then G(kx)/(k L(k))
(If t
Recall
=
for
>
=+I and G
> -i
x
x+,
S[
H(x)x
k
then
=
> -I
,
in
and G(x)
S t for t
%
>
x=L(x)
0+).
(x
as x
(2
#
+I
G(x/k)/((I/k)=L(I/k))
x
k
,
H is Heviside’s function. (The symbol
in
%
i)
St
is related to
the ordinary asymptotic behaviour.
The following scale of distributions from S’ has been used in investigations of the
quasiasymptotic behaviour of distributions:
Hta/r(+l),
f+l
Dn f=+n+l,
> -1
([12, p. 88])
--< -i’ =+n
>
-I
where D is the distributional derivative.
3.
THE q.a.b. OF DISTRIBUTIONS
We shall repeat in this section some well-known facts about the quasiasymptotic be-
[13,10].
It is said that f has the quasiasymptotic behaviour (in short q.a.b.) at
if
with the limit g # 0 with respect kL(k), L
Z ((i/k)L(i/k), L Z0),
haviour fom
Let f
(0 +)
S+.’
,
QUASIASYMPTOTIC EXPANSION OF DISTRIBUTIONS AND THE STIELTJES TRANSFORM
< f(kt)/(kL(k)),#(t) >
lim
< g(t),(t) >,
675
S
(3.1)
f(t/k)/((i/k))h(i/k)),(t) >
(lim <
< g(t),(t) >,
k
Let us notice that in [i0] we reformulate the definition of the q.a.b, at 0
We need in the paper the following structural theorem (for the q.a.b, at
and for the q a b. at 0
+
0+
with respect to kL(k) ((i/k)L(I/k)).
[0,), C # 0 and m N0 m+ > -I,
L foe’ supp F
%
Ckm+=L(k),
(F(I/k) % C(i/k)m+L(I/k),
F(k)
k
g at
Then there exist F
DmF,
such that f
EXAMPLES
For
-i,
H(x_l)x
LI()
qa. 6
Moreover, if f
2. H(x-l)/x
qa.
-e+l
supp f
at
c
6(x) at
[0,),
ff
#
with respect to k
0
then f
%
4. H(x-l)e
x
m
1/x
q4a.
C6 at
with respect to k
-I
-I ink;
with respect to k
-m
for
7. For -n
8. For n
qa.
H(x) at with respect to k 0
I;
m-I
(_i)
(m-l)
6
at with respect to k -mlnk
(m-l)!
+
at 0 with respect to (I/k)inm(i/k);
-i, m
No, (xinmx)+ qa.
> -n-l, (x%inmx)+ qa. r(l+l)Dnfl+n+l at 0 + with respect (I/k)llnm(I/k);
(-I) n-I
(n-l) at 0 + with
6
N, m No, (x -n inmx)+ qa
respect to
(m+l)(n-l)!
(i/k)-nlnm+l(i/k).
For the
(m
x
6. For
No,
(3.2)
N;
some m
n
rJ
i;
with respect to k
C
3. Any distribution with compact support has the q.a.b, at
5
[13]
see
[i0]):
see
q.a.
Let f
+ from [13].
definition of distributions
see [5, pp. 338, 339].
xm,
m
N,
(xilnmx)+,
-n >
> -n-l,
We remark that the q.a.b, at 0 + is a local property while the q.a.b, at
-nInmx)+,
is a global
S+.
property of an f
Namely, it is proved in [13] that if f
and L
lim
k+
(x
f(x/k)((i/k)L(i/k))
Clearly this does not hold for
0 in
0 in a neighbourhood of zero then for any
S’.
(see example 3.).
As it was shown in [9,10] the notion of q.a.b, is much more appropriate for the in-
I,
vestigations of the Stieltjes transform. For example if f
L
supp f
[0,) and
I/x 5 as x
f
this ordinar asymptotic does not imply the behaviour of its Stieltjes
,
transform.
The behaviour of the transform is determined by the quasiasymptotic behaviour
of f (see example I.).
4.
THE q.a.e. OF DISTRIBUTIONS
We extend slightly the definitions of the closed and open quasiasymptotic expansion,
in short the q.a.e, at
,
given in [4] and using the same idea we give the definition of
676
PILIPOVI
S.
+.
the q.a.e, at 0
=
Let
l (L 10). We put
H(t)L(t)t/r(=+l), =
and L
e
{ Dn(fL)+n+l
(fL)+l
-l,
>
(4.1)
+n
<- -I,
>
-I,
where n is the smallest natural number such that =+n > -1.
q.a.
(0
at
with respect to
Obviously,
(fL)+l
+)
f+1
S+ has
DEFINITION I. We say that an f
E
Z(R)((=,L)
L0(i/k))
exist
E
0)
and of lenght E, 0
if f has the q.a.b, at
.
L
1
_-< a
Z
N)
r.0+)
(L i
Z
kL(k) ((I/k)L(I/k)).
the closed q.a.e, at
=<
<
,
with respect to
(0 +) with respect to k=L(k)
C, i
,N, N N
c
i
(0 +) of order (,L)
((I/k) +E
k=-L0(k)
((i/k)=L(i/k))and
i
>
>
>
2
if there
N (I 2
and that f is of the form
N
_>ci(f Li)i+l(t) + h(t)
f(t)
(4.2)
i=l
such that
lim
h(kt)
<
k- L 0
x
(lira
,(t) >
h(t/k)
<
1/k)
k
(4.3)
S
0,
k
+L0
,(t) >
S).
0,
(i/k)
Obviously, we shall assume that c. # 0 and that
I
N>
-
(N <
=+)"
Since the sum of two slowly varying functions is the slowly varying one we can and we
shall always assume that in the representation (4.2)
N ).
Namely,
+
(fLj)8+
(fLj)81+l
(fLj+gk)8+ I.
(FLk)8+
(fLk)81+l
and
=l
>
2
>
(0 +) iff
have the same q.a.b, at
>
81
N (I
82
and
<
Lj
=2
%
<
Lk.
So,
we have
PROPOSITION I.
Let f
e
S+
satisfy conditions of Definition
and assume that there
are two representations of f
N
I ci(fLi)i
I i(fni)i
f(t)
+l
+ h(t),
i=l
M
f(t)
(t)
+I +
j=l
for which all the assumptions given above hold. Then M
LN {’N’ I ’ LI
the following notation
{’i
N,
L.
We shall use
f
for the f
S+
from Definition i:
N
qe.
ci(fLi)i +I
at
(0 +) of order (,L)
(4.4)
i=l
with respect to
EXAMPLES
H(x-1
9. We haw that
r=l
H(x-l)e I/x
r!x r
uniformly converges to H(x-l)e
q4 e" H(x) + ((inx)+)’
at
I/x but
of order (0,L--I)
(4.5)
677
QUASIASYMPTOTIC EXPANSION OF DISTRIBUTIONS AND THE STIELTJES TRANSFORM
oi
with respect to k
H(x-l)e 1/x
Ink and
qe(x)
+
((Inx)+)’+
(4.6)
6(x)
r! r
r=2
of the order (0,L =-i) with respect to k
q.e.
10. H(t-1)/t
q4 e"
H(t l)/t n
+1
(n-l)
6
(n-2)1!
6
(-1)J-16(j-l)
+
(4.7)
(n-1)(j-l)!
(n-l)
(-I) n-2 (n-2)
6
+
(n-2)!
+
+ 6’ +
(n-2)l!
of order (-1,L i) with respect to
at
6’ +
i) with respect to k -j"
of order (-1.L
at
n-i
q4e"
H(t-l)/t n
l) with respect to k 2 ink. Moreover,
of order (-I,L
6 + 6’ at
2; then for j
let n
-i
(-l)n-16 (n-l)
(n-l)!
k-nlnk.
11.
H(l-x)x
at 0
+ of
q4 e (_l)m_
m
(m-l)!
(Inx)m)
+
m-i
(_l)m_
order (-m, ln(I/k)) with respect to (i/k)
(4.9)
"(x)
?-!
i=l
-m.
Following [4] we define the open q.a.e.
An f has the open q.a.e, at
DEFINITION 2.
Z
0)
and of length s, 0 < s
order (=,L) and of length
,
,
iff for every
with respect to
(0 +) of order (,L)
,
R
Z) ((,L)
< s, f has the closed
0
q.a.e,
e
Rx
of
k-L(k) ((I/k)=+L(I/k)).
By the same arguments as for Proposition
one can prove the following proposi-
tion:
PROPOSITION 2.
let 0 &
I<2
N
f
q4e"
f
q4 e"
I
Suppose that
s.
a
of order (,L) and of length s and
Let f have the open q.a.e, at
i
(fL)
i i +1
y bi(f
L i i +I
at
(0 +) with respect to
at
(0 + with respect to
k=-1L1(k) ((I/k)+1L1(1/k))’
k-2L2(k) ((I/ k)+2L2 (l/k)).
i=l
Then, M
N and a
bi, =i
i
$i’ Li
%
i’
,
f has the open q.a.e, at
lar conclusion holds for the point
0t as
N.
of order (=,L) with respect to
Let us note if f has the closed q.a.e, at
then for any s &
i,
i
of order
[=,L) and of leQgth
k=-L(k)
s. The simi-
well.
Proposition 2 implies:
CORROLARY 3.
(0 +) of order (,L) and of lenght s.
Let f have the open q.a.e, at
Then f may be asymptotically expanded into a series
f(x)
q4 e"
Y ci(fLi)i+l
at
(0 +)
i=l
where
I
>
>
=n
>
(
i
<
<
n
<
...), so tha for any 0
< s and
L
678
S. PILIPOVIC
(f
I ci(fLi)=i+l)(kx)/(k-L(k))
I ci(fL )ai )(x/k)/((l/k)a+L(l/k))
S’),
0 in
((f-
i
k
0 in S’
+
,)
k
i;1
(Note that here c. can be equal to zero for i
i
THE DISTRIBUTIONAL STIELTJES TRANSFORM
5.
There are several definitions of the Stieltjes transform of generalized functions. We
follow the definition given by Lavoine and Misra [7]. Some advantages of this definition
[8].
J’(r), r
were mentioned in
The space
Lo c,
N and F
m
f
(-N),
supp F
c
is a subspace of
[0,),
S+
J’(r) if there
such that f
exist
such that
DmF,
(5.1)
i IF(t)l(t+)-r-m-ldt
(5.2)
> 0.
for
0
The Stieltjes transform S
in
(5.1) and (5.2)
,
J’(r) with the properties given
an f
r
(Srf)(z)
(If p
(-N), of
r
r
is a complex valued function S f defined by
(r+l)
fF(t)(t+z)-r-m-ldt,
m
z
(-,0].
C
t e
(P)t
N,
(p+t-1),
p(p+l)
(P)O
1.)
It is proved in [7] that S f is a holomorphic function in C
r
then
Dmf
(5.3)
0
(-,0]. If f
J’(r+m),
J’(r) and
Sr(Dmf) (r+l)m(Sr+mf).
(5.4)
One can show easily that
Dm(Sr f) (-l)m(r+l)m(Sr+mf),
6.
f e J’(r), m
(5.5)
N.
ON THE BEHAVIOUR OF S f
r
(0 +) with respect to k=L(k) ((I/k)L(i/k)). Then for some
Let f have the q.a.b, at
N O, m+= -i, and F L loc’ supp F
m
[0,), (3.2) holds. In the case of the q.a.b, at
this implies that f
(-N). In the case of the q.a.b, at
J’(r) for r > =, r E
0+,f
ought not belong
J’(r) for r
to
=, r
follows from [12, p. 93]. So, for p
F[kt)/k=+mLk))"
(Let f
J’(r); then
C
For a given z
A(z) if there is an
0 _-< D(t)
Cf=+m+
-<
e
in
S’
k
we denote by
0 < 2e <
i, q(t)
Clearly, for a given z
r+m+l
(-N). If f J’(r) then F S’r+m+l; this
=+m+l (2.1) and (2.2) imply:
.........P
Ft/k)/i/k)=+mLi/k))
(-,0]
e
E
in
Rez I, such that
(-(R),0]
S’
p
A(z) the set of all q(t)
for t > -e, n(t)
C
Cf+m+
and every
0 for t < -2e.
n
A(z)
}
k
C
such that
(6.1)
QUASIASYMPTOTIC EXPANSION OF DISTRIBUTIONS AND THE STIELTJES
TRANSFORM
E
D(t)(t+z) -r-m-I
t
for p
<
679
(6.2)
r+m+l.
For the main results of this section we need the following assertion from [I0] ([9]):
J’(r). We have (x > 0, t > 0),
Let f
x-(r-)-iL(x) as x
((r+l)l(r-))x-(r-)L(x) as x
(S r f)(x)
%
x(r+l)J(Sr+if)(xu)du,
+) with
0+).
(x
(Sr+if)(x)
if
and
(Srf(tx)
0
(x
r
>
,
then
(6.3)
Now, we are ready to prove:
Let f have the closed q.a.e, at
THEOREM 4.
of order (,L) and of length
ke-L0(k)
,r
(see the notation in Definition i).
(-N). Then
J’(r),
N;
(fLi)i+l J’(r), i
(ii) If we put
(fLi)i +l(x) si ,Li(x), i I,...,N, then for
F(ri
Sci,Li(X) N si,Li(X) % F(r+l x i-rL i (x) x
F (r-e i
ei
xa_ 0 (x)) x
(iii) (Srf)(x)ci F(r+l) x rL I.(x) 0(
respect to
Let r
(i) f
>
with
Sr
[i Li
.
L
(6.4)
i=l
We shall prove the theorem by using the similar idea as in the proof of the
PROOF.
main theorem in
[9].
Obviously, (i) follows from (3.2).
< r-l, x
(ii) Let
L
Z. Let m be the smallest element from N O such that
,
8+m > -i. Then
Sr(fL)B+l(X)
(r+l)m0
f8 +m+l
(t)L(t)
(x+t)r+m+l
dt
(r+l) m <
N(t)
fS+m+l(t)L(t),(x+t)r+m+l >,
A(x),
e
where
<
f+m+l(t)L(t),
D(t)
r+m+ >
x+t
is observed as a pair from
Since
r+m
(S’r+m,Sr+m).
Obviously this pair does not depend on D
A(x).
> 8+m+l, we have
Sr(fL)8+l(kX)/k-rL(k)
(r+l)
<
m
(r+l)m < fS+m+l
f+m+l (kt)L(kt)
k+mL(k)
D(kt)
(x+t)r+m+l
t)L(t)
D(t)
,kB+m+l L(k)(x+t/k) r+m+l
>
(kt)L(kt), D(t)r+m+l
(r+l)m < fs+m+l
kS+mL(k)
(x+t)
If k
, from (6.1) follows
Sr(fL)8+l(kX)/kS-rL(k) < fS+m+l(t), (x+t).(t)r+m+l
it
t+mdt
(r+l) m
r(8+m+l)J(x+t)
r+m+l
F(r-8)
r(r+l)
x- r.
o
On putting x
we obtain that
(ii) holds for all =.
1
r-1. Let us suppose that
r-l8<r.
PILIPOVI6
S.
680
Then, by the same arguments given above, we have
F(r+l-8)
r(r+2)
(Sr+l(fU)8+l)(X)
xS-r-iL(x),
.
x
Now by (6.3) we complete the proof of (ii).
Sr+l( fL)8+l
serve firstly
I ci(fLi)ai+l
f
(6.2) implies that
=
r-I -<_
and after that to use (6.3).
(iii) We can assume that
< r-i because if
(ii), to oh-
< r we have, as in
Since
S’r+m’
e
(S’r+m,Sr+m)
in the sense of the dual pair
we have
N
[
{(Srf)(kx)
Sr(fhi )ai + 1) (kx)}/(ka--rLo(x))
c
-
i=l
N
ci(fui)(kt)}/(k L0 (k))
< {f(kt)
i=l
q(t)(x+t) -r-m-I >+0ask+.
the assertion (iii) follows.
On putting x
+
The similar assertion holds for the closed q.a.e, at 0 but with more restrictive assumpt ions.
Let f have the closed q.a.e, at 0 + of order (=,L) and of length
J’(r) then
If =+ < r and f
THEOREM 5.
respect to
with
(i/k)=+L0(i/k).
(Srf)(x)
c
r(r-=i)
i
x ai-rL i (x)
r(r+l)
0(xa+-rL0(x))
(6 5)
O.
x
The proof of this theorem is very similar to the proof of Theorem 4. We only notice
that we must observe firstly
have that F e
(D(t)(x+t)
7.
S’r+m+l
-r-m-2
Sr+if
and after that to use
Sr+m+
as a function of
If z
a
Rei#;
{a +
ape
+ Re iv
’z + t’r+l
A
>
a,E
R >- 0, -+e .<
and t e
r+l
a
>
=
> -1 and
F(x)
%
x
=
we have
+ t) r+l.
(7.1)
This follows from the elementary inequalities:
(a+t) 2 + 2(a+t)Rcosb + R 2 >- (a+t) 2
(a+t) 2 + R 2 + ((a+t) 2 +
>=
((a+t)2
+ R 2)(1 + cose)
Assumptions on F imply
F(x)
<
C(I +
xa),
xO.
as x +.
-< -e}.
[0,)
(l-s’)--(R +
(Sm+l, Sr+m+l).
t.)
THE UNIFORM BEHAVIOUR OF S r f
Let F be a continuous function with supp F c [0,), r
a > O, e > 0, a subset of C defined by
Denote by h
A
J’(r) we
(6.3). Namely, from f
and this implies that we have to observe the dual pair
2(a+t)Rcos + R 2
R2)cose (a+t+R)2cose >=
2((a+t)2 + R 2)cos ((a+t)2
+ R 2)(I
cose).
QUASIASYMPTOTIC EXPANSION OF DISTRIBUTIONS AND THE STIELTJES TRANSFORM
For z
A
and suitable C
a,g
)
if (z+t)r+l dtl ( licole (r+l)/2 (R+a+t)r+
F(t)
(1,t
2
S C
j
c
at
( + P(r-)P(&+])(J[)rr(r+l)
0
(7.2)
So, we have proved the following lemma:
LEMMA 6.
Let F satisfy the conditions given above. The function
bounded in A
681
0, e
a
zr-=(S r F)(z)
is
0.
We use this lemma for the proof of the following Theorem:
Let f satisfy the conditions of Theorem 4 and let all the slowly varying
functions in Theorem 4 are equal to i. Then
THEOREM 7.
N
Af,r(Z) ((Srf)(z)
(i)
F(r-i)
--r
ci r(r+l) z i-r) Iz
i=l
is a bounded analytic function in any A
Af,r (z)
(ii)
[l,p.
a > 0, e > 0;
converges uniformly to zero in
Aa,e
when
zl
.
It follows from the structural theorem (3.2) and Lemma 6.
PROOF (i).
(ii)
a,E
It follows from (i) and Theorem 4 which enable us to use the Montel Theorem
53.
Let f satisfy the conditions of Theorem 5 and let all the slowly varying
THEOREM 8.
functions in Theorem 5 be equal to i. Let
N
Af, r (z)
((S r f)(z)
c
F(r_i
i
F(r+l
z i-r )/z
=+-r.
i=l
Then
Af,r(Z)
(i)
{z;Izl
(ii)
is a bounded function in
A0,eNB(0,R),
e > 0, R > 0, where
B(0,R)
R};
<
Af,r (z)
converges uniformly to zero in
A0,e
when
Izl
0.
For the proof of this theorem we need:
LEMMA 9.
Let F
flF(t)(z+t)-r-lldt0
R>O.
PROOF.
Take M
Lo
e
<
supp F
c
z e
’
A0,e
[0,(R)), r > =
N B(0,R). Then
M
if.(z+t)r+l dt]
&
0
!
t
Iz+tl r+l
dt
+
r-=f IF(t)Idt
(t+) r+l’
9 < R.
So, it follows that
r-=
M
zr-=(SrF)(z)
Iz+t[ r+1
(7.1) implies that
M
-I, F(x)
O. For suitable C we have
F(t)
9
>
Iz+tl r+
is bounded, and we have to prove the same for
dt.
x
=
x
0
+ and
is bounded in
A0,e
s( o ,R),
682
PILIPOVI
S.
M
z
r-=
!i z+tt,17+I
dt.
p.e i
On putting z
0
=<
R, -+e _-<
p
-e we have
NIp
N
t
[.
d o r+llei + lr+
P
p
r-a
(u
O
2
+
+ i) (r+l)12
2ucose
udu
(
r-
udu
__i
dt-
J
u
2
2ucose
+
(r+l)/2
1)
This implies the assertion.
PROOF OF THEOREM 8. (i) From the structural theorem (3.2) and Lemma 9 it follows that
Af,r(Z)
f,r(Z)
The function f,r(),
(ii) Let
Af,r(i/z),
f,r(X)
(-,0].
C
z
n {m;ll
A0,
As well, we have
n B(0,R).
Ao,e
is bounded in
I/R},
A0,
8.
as
> 0
Af,
zl
as
Izl
is analytic
in the domain A
r
0. So by the Montel Theorem it follows that
a > 0, R
a,
R
and bounded
.
0 as x
This implies that the same assertions hold for
in h
0
e
Further on, this implies that
a,g
N {m;
Iml
I/R}
>
f,r(Z)
f,r (z) converges uniformly to 0
converges uniformly to 0,
in
and so, that the assertion (ii) holds.
EXAMPLES
By examples 9., i0., Ii. and Theorems 4., 5. we have:
12. For r > 0 (r
(S r
e
R
(-N))
(H(t-l)el/t))(x) (_F(r)
Lr(r+l
=0(x
-l-r
),
x
(
+ x -l-r lnx-1 +
-r
I
r! (r-l)
)
l-r
)
x+.
13. For r > -I
(S r (H(t-1)/tn))(x)
+
F(r+n)(-l)n-I
r(r+l)(n-1)!
14. For r
x
(n-l)
-r-n
o(
x
-r-1
x-n
1
S+
x
-m-r
-r-n+l
q4 e"
inx
+
(I ijx
I-m-r)
0(
x-m-r ),
x
0
+
i=l
and have the compact support. It is proved in [4, p. 386] that f has
-I and of length -,
of order (,i) with
C)
f(t)
x
-m
the open q.a.e, at
c.
r(r+l)(n-2)!
.
inx), x
(Sr(H(l-t)t)Xx)-(-l)m-l(m-l)! F(r+l)F(r+m) (
15. Let f
r(r+n-1)(-1)n-2
(r+l) -r-2
x
(n-2)
+
[ cifai+l(t)
at
,
i
e
N
i.e. (with suitable
QUASIASYMPTOTIC EXPANSION OF DISTRIBUTIONS AND THE STIELTJES TRANSFORM
Corollary 3 and Theorem 4 imply that for r > =, r
_(Srf)(x)
c_
F(r+i
r(r+l)
x
-r-a i
x
R
(-N)
.
The similar assertion can be formulated for a periodic distribution from
ACKNOWLEDGEMENT.
683
S+
([4, p. 386])
This material is based on work supported by the U.S. Yugoslav Joint
Fund for Scientific and Technical Co-operation in co-operation with NSF under Grant JFP
544 and JF 838.
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