Internat. J. Math. & Math. Sci.
VOL. Ii NO. 3 (1988) 449-456
449
S-ASYMPTOTIC EXPANSION OF DISTRIBUTIONS
BOGOLJUB
STANKOVlC
Institute of Mathematics
University of Novi Sad
Yugoslavia
(Received December 2, 1986 and in revised form April 28, 1987)
ABSTRACT.
This paper contains first a definition of the asymptotic expansion at
infinity of distributions belonging to
’(Rn),
named S-asymptotic expansion, as also
its properties and application to partial differential equations.
KEYS WORDS AND PHRASES. Convex cone, distribution, behaviour of a distribution at
infinity, asymptotic expansion.
1980 AMS SUBJECT CLASSIFICATION CODE.
i.
Primary 41A60, Secondary 46F99.
INTRODUCTION.
The basic idea of the asymptotic behaviour at infinity of a distribution one can
find already in the book of L. Schwartz [I].
To these days many mathematicians tried
to find a good definition of the asymptotic behaviour of a distribution.
We shall
infinity" explored by Lavoine and Misra [2] and the
"quasiasymptotic" elaborated by Vladimlrov and his pupils [3]. Brichkov [4] intro-
mention only "equivalence at
duced the asymptotic expansion of tempered distributions as a useful mathematical
tool in quantum field theory.
His investigations and definitions were turned just
In [4] one can find cited literature in which asymptotic
towards these applications.
expansion technique, introduced by Brichkov, was used in the quantum field theory.
This is a reason to study S-asymptotlc expansion.
2.
DEFINITION OF THE S-ASYMPTOTIC EXPANSION.
In the classical analysis we say that the sequence
is asymptotic if and only if
n+l(t)
O(n(t)),
t
=.
{n(t)}
of numerical functions
The formal series
nl un (t)
is an asymptotic expansion of the function u(t) related to the asymptotic sequence
{n(t)
if
k
u(t)
[ Un(t)
n=l
for every k
.
O(@k(t)),
t
(2.1)
{*n(t) },
t
(2.2)
N and we write
u(t)
u
n=l
When for every n
N
n(t)
Un(t)
is unique, that means the numbers c
Cn n(t) Cn
n
are complex numbers, expansion (2.2)
can be determined in only one way.
STANKOVI6
B.
450
n
In this text F will be a convex cone with vertex at zero belonging to R and
F. Notations for
Z(F) the set of all real valued and positive functions c(h), h
the spaces of distributions are as in the books of Schwartz [I].
DEFINITION I.
the asymptotic sequence {c (h)}
T(t+h)
"
n
for n
Z
<T(t+h), 0(t) >
<
n=4
Z(F), we write it
{c (h)}
N and h
r,
llh[l
n
n
Bn(t,h),
0
,
{Cn(h)}, l[hll
Un(t)Cn(h),
Un(t,h)
(2.3)
F
h
if for every
p(t) >
i) In the special case
REMARK.
has the S-asymptotic expansion related to
U (t,h)
n=l
where U (t,h)
"
The distribution T
u
,
h
n
F
(2.4)
n
N, we shall
write
s
T(t+h)
Z
n=l
[lh[l
Un(t) Cn(h)
(2.5)
h
and the given S-asymptotic expansion is unique.
inRn),
2) To define the S-asymptotic expansion
relation (2.4) T and U
n
are
in"
and p
we have only to suppose that in
in.
Brichkov’s general definition is slightly different [5].
DEFINITION l’.
The distribution g
{@n(t)}
the asymptotic sequence
g(ho-t)
where c (t X)
n
"
Z
n=l
for X
X
has the asymptotic expansion related to
{ho, > 0}
n(t,)
{0)_n..}
> 0
if for every
o
E
(t) >
<g(kho-t),
"
on the ray
<
Cn(t,%),
n=l
h
n
R
o
(2.6)
R,
{n(%) },
(t) >
(2.7)
%
Relation 2.6 can be transformed in
i
f(x) e
<< x
ho>>
Z
n=l
en(X,%)
by the Fourier transform, if we take
F[n(t,%)]
F-l[g]
(2)
n
Cn(X,).
f(x)
{@n(%)),
F-l[g(t)];
(2.8)
%
0(x)
F-l[(t)]
and
We denote by F[0] the Fourier transform of 0 and by
n
n
E
Also, for x,t
R x,t
the inverse Fourier transform of g.
xit i.
i=l
In his papers Brihkov considered only the asymptotic expansinons (2.8) and in
one dimensional case.
the
whole’(Rn),
We shall study the asymptotic expansion not
n.
not only on a ray but on a cone in R
im"(R)
but in
Our results enlarge
Brlchkov’s to be value for the elements of o’(Rn) (Corollary I), they are proved with
less suppositions (Propositions 5 and 6) or give new properties of the S-asymptotic.
A distribution belonging
can have S-asymptotic expansion in" without
to"
having the same S-asymptotic expansion in
distribution
f
’.
Such an example is the regular
defined by the function
f(t)
H(t)
exp(I/(l+t2))
exp(-t)
t
R
S-ASYMPTOTIC EXPANSION OF DISTRIBUTIONS
451
where
I, t
H(t)
0
H(t)
and
R+
It is easy to prove that for h
(t+h)
t < O.
0,
7.
(I+ (t+h)
(n-l)’
n=l
2 l-n
exp(-t-h)
l{ e-hh 2 (l-n) },
.
h
But
U (t h)
(I+
n
do not belong
(t+h)2) l-n
to’.
exp(l+
(t+h)
7.
.
(l+t2))
but it is not in
belongs to
(n-l)’
h > 0
N,
defined by the function
The regular distribution
g(t)
n
exp(-t-h),
exp(t),
R
t
in’:
It has S-asymptotic expansion
(I + (t+h)
ehh2(l-n) },
2)l-n exp(t+h)
h
R+.
where F
PROPERTIES OF THE S-ASYMPTOTIC EXPANSION.
3.
Let S
PROPOSITION i.
T (t+h)
s
7.
n=l
’
U
n(t,h)
l
(S *
and T
’.
{c n(h)},
If
,h
llh[l
F
then the convolution
(S,T)(t+N)
n=l
Un)(t,h)
{Cn(h)} [[h[[
h
F
k
7.
n=l
Un(t,h)],
(3.1)
We know that
PROOF.
k
<(S*T)(t+h), 0(t)>
7.
n=l
<S*[T(t+h)
<(S*Un)(t,h), 0(t)>
0(t) >
It remains only to use the continuity of the convolution.
COROLLARY I.
T(t+h)
If
_s
[[h[[
{Cn(h)}
U (t h)
n
7.
n=l
F
h
then
T
(k)’" (t+h)
7.
n:l
where
T
kl
(k)
(Dtl
D
n)
T
untk)’" (t,h)
k
REMARK.
kn)
(k
n
We have only to take S
PROOF
{Cn(h)
6<kjZ
]]h]]
N
n
o
,
(3.2)
F
h
N :NU{ 0
o
in Proposition I.
Proposition i. is valued as well if we suppose that T
PROPOSITION 2.
Let f
U (t,h) and V (t), n
N and h
n
n
n
integrable functions such that for every compact set K & R
f(t+h)
l
n=l
Un(t,h)
{Cn(h)}, JJhJj
h
r,
F,
t
"
and S
be the local
K
.
452
STANKOVI
B.
and
k
Z
n=l
f(t+h)
llhll
and
/Ck(h)
Un(t,h)
Vk(t),
r(k,K), then for the regular distribution
(t+h)
Z
n
n=l
PROOF.
r
h
The proof is a consequence of the Lebesgue’s theorem.
PROPOSITION 3.
"
Suppose that T
and T belong to
and equal over the open set
2
for every r > 0 there exists a B such that the ball
o
r} is in {a-h, h g
B }. If
which has the property:
n, llxll
llhll
=<
{x e R
B(0,r)
defined by f we have
llhll
{c (h)}
n
(t,h)
r
K, h
t
Z
n=l
T l(t+h
,
llhll
{c n(h)},
Un(t,h)
o
r
h
then
T2(t+h)
Z
,
{Cn(h)}, llhll
Un(t,h)
n=l
r
h
as well.
PROOF.
We have only to prove that for every
<
lira
/
[Tl(t+h)-T2(t+h)]
Let supp p =B(0,r).
Ck(h
O(t)>
Ck(h)
O, 0
(3.3)
Tl(t+h)-T2(t+h
The distribution
equals zero over -h.
By the supposition there exists a B such that the ball B(0,r) is in {f-h, h
o
-> S }. This proves out relation (3.3).
llhll
o
PROPOSITION 4.
D
t
r
N, h
Let S
S(t+h)
.s
m
l
i=l
(R),
f
where
m
i--4
(t) --f
O(t
PROOF.
f
R
t
m
V i(t,h)
l
i=l
then O(t)
If p
t
t
t
n)
dt
n
F
h
Vi(t,h)]/Ck(h), o(tm)m(t)
m
R
S(t+h)
(t
E
< [S(t+h)
lira
,
dt
m
D
I, and for every
o()d
R
k
n
I’} has the properties:
N, h
i
<
m
{el(h)}, llhll
Ui(t,h
and for a 0
llhll+,h F
<
and for 1
{Vi(t,h),
If the family
i
I’,
>
t
,
Vi(t,h)
0
k
Ui(t,h),
N
0
then
{c i(h)},
l}hll
Oo(tm)%m(t)
,
r
h
and
+ (t) where
O.
m
Now we have the following equality
<[S(t+h)
k
l
i=l
Vi(t,h)], p(t)>
t
k
<[D t S(t+h)
m
F.
i=l
k
l
<[S(t+h)
Ui(t+h)]
f
i=l
Vi(t,h)] Polm(t)>
m
(t
It remains only to use the limit in it and Corollary I.
u
m
t
n)
du >
m
S-ASYMPTOTIC EXPANSION OF DISTRIBUTIONS
m is fixed, 1 _-< m
-<-
’, r
Suppose that S
PROPOSITION 5.
n,
{h
R
m,...,0)},
(0,...,h
h
where
n and
(D t S)(t+h)
l
i=l
m
Vi(t,h),
If there exists
i(h)
Ui(t,h),
DhmVi(t,h)
i(u)
r
N and if
i
ci(h),
i
N are
m
,
{i(h)}, llhll
Vi(t,h)
h
h
as
then
l
i=l
,
{ci(h)}, llhll
Ui(t,h)
local integrable in h m and such that
hm
du
c
f
m
S(t+h)
453
r
h
By L’Hospital’s rule with the Stolz’s improvement we have for every
PROOF.
N
and k
k
< I V
i=l
<S(t+h), p(t)>
lim
h/, h r
p(t)>
i(t,h),
k (h)
k
<(DtmS)
< I
(t+h), O(t)>
U i(t,h), 0(t)>
i=l
lim
h+,hF
Ck(h)
These five propositions give how is related the S-asymptotic with convolution,
derivative, classical expansion and the primitive of a distribution.
proposition gives the analytical expression of
PROPOSITION 6.
Suppose that T
Z
T(t+h)
u (t)
n
n=l
If u
m
# 0,
N, then u
m
m
u (t)
m
where a
(a k
k
every t i, i
PROOF.
m
P(t
tn)
l(t+ho)
h
exp(<<a
k
m
N
(3.4)
the power of them less of k in
and our supposition
T(t+h)/c l(h)
u
d(h o) u l(t)
l(t)
0
lim
I[ h ]].+=o,h F
c
(3.5)
satisfies the equation
h
O
F
(3 6)
where
d(h o
interior,
r
t>>), m
Pkn are polynomials,
<<x,t >>
=i xit i.
From relation (3.5) follows that u
u
Un(t) Cn(h).
has the form
By Definition
lim
Un(t,h)
F with nonempty
llhll
Cn(h)
Z
k=l
k
n
,a
f R and
n
l,...,n:
’,
The next
l(h+ho)/c
(h)
454
B. STANKOVIC
n
is such element from R
k
coordinates equal zero except the k-th which is 1. Then
If
ho
is an interior point of F and e
u
l(t+ho+eek)
Hence the existence of
D
t
k
u
[d(ee k)
l(t+ho)
a
and
Ul(t+ho),
k--
Dhkd(h)h=
Ul(t+h o)
a
k
d(0)]u l(t+ho).
(3.7)
n.
We know that all the solutions of equation (3.7) are of the form
where C
The following limit gives u
2
<u l(t),O(t)> c l(h)
<T(t+h),0(t)>
lim
Ilhll+,hF
By Corollary
follows for i
lim
,n
ali)T(t+h),0(t)>
i
llhil/-,hr
2(t)
C
2
(Dtl-al)
a) If
exp(<<al,t>>).
b) If (D
t
<(Dr i
c2(h)
Two cases are possible,
u
<u2,0>
c2(h)
<(D t
# 0 for some
-ai)u 2
Ul(t)=Clexp(<<al,t>>),
an).
(a
is a constant and a
for which all the
then (D
i
(<<al,t>>)
u
O, i=l
2
t
ali)u2(t),0(t)
>
n, then
-ai)u2(t)
c
exp(<<a2,t >>)
and u
2
has
ex
is a polynomial of
exp(<<a ,t>>) where
+ P (t
2
the power less of 2 in every t i, i=l,...,n.
In the same way we prove for every um.
n
PROPOSITION 7. Let T ’and fl
R be an open set with the property: for every
the form C
r > 0 there exists a 8
r
tn)
P2
such that the ball B(h,r)=
for all h
F,
llhll
>
8r
Suppose
m
T(t+h)
7.
n=l
{Cl(h),...,Cm(h)},
Un(t+h)
m
for any function c (h) from E(F), then T
m
PROOF.
n=l
n
The statement of this Proposition can be obtained from a proposition
proved in [6].
However, for completeness, we shall give the proof on the whole.
First we shall prove that if for every c (h)
m
m
n__E1 Un(t+h)
T(t+h)
<
lim
Ilhll+,hCF
C
m
O(t)>
(3.8)
0
(h)
then there exists a 8(0) such that
m
<[T(t+h)
Z U (t+h)]
n
n=l
Suppose the opposite.
<[T(t+hn
then we choose
m
l
n=l
Cm(h)
O(t)>
0
h
We would have a sequence h
Un(t+hn)],
0(t)>
in such a way that
Pn
# 0,
Cm(hn)
n
Pn
n
F
llhn I
such that
N
and relation (3.8) would be false.
455
S-ASYMPTOTIC EXPANSION OF DISTRIBUTIONS
We denote by
8o(p)
sequence
{hk},
h
, llhkl
k
P (t)>
<T(t+hk),
8o(#k)
there exist {h
k}
{ak#0
Ak, p
p=k
for every
Z
T=T-
k}
and
such that
n=l
U
n
Let
can be the following.
#k
k6
is a strict monotone sequence which tends to infinity, then
F and e
k
N such that
> 0, k
8o(#k_ I) +ek
{p(t)}, p
0,
p#k
la
p=k
k,
,
in such a way that
It is easy to see that
<T(t+h k)
ek"
The numbers
l
we can find
<T(t+hk), p(t)>
a
Ak,p-l Ak, l-’"
p(t)>
8o(#k)
0, k > p.
and
k
i=l,...,p-I so that for k=l,...,p-i
<T(t+hk), k(t)>
For a fixed p and k < p we can find
llhkll
for which we have
p(t) %l(t)-...-IPp_l#p_l(t), p > i.
p(t) satisfies the sought property.
p(t)
4}
m
p < k
0
<T(t+hk) p(t)>
0
{#k(t)}, #k 6
and the sequence
Now, we shall construct the sequence
Let
p
Let us suppose the opposite; then there exists a
The constructio of the sequence {h
be such that
{8o(p),
We shall prove that the set
infB(p).
n
compact set K C R is bounded.
Ak,p-i
p-I
Hence
...+xPp-i ,p-I
+
As
k=l,...,p-l, p > I.
Ak,p
# 0 for every k, this system has always a solution.
We introduce now a sequence of numbers {b
b
k
,k
sup{2klo k(i) (t)
k}
i < k}
Then the function
(t)
K
p(t)/bp
Z
p=l
and this series converges
<T(t+hk),@(t)>
inK,
Z
p=l
thus in
<T(t+hk), p(t)/bp>
If we choose c(h) such that c(h
llhll
converge to zero when
/, h 6
k)
F.
for every compact set K there exists a
h
F, #
(t)
4.
K"
That means that
JJhJJ
0 over B(h,r),
as well.
ak/b k
akJb
then <[T(t+h)/c(h)l,(t) does not
k
This is in contradiction with (3.8). Hence,
8o(K)
(t+h)
8(r), h
With this
<(t+h),(t)>
such that
0 over B(0,r),
llhll
8(r), h
0,11hll a 8o(K),
6
F and
F.
APPLICATION OF THE S-ASYMPTOTIC EXPANSION TO PARTIAL DIFFERENTIAL EQUATIONS.
As we mentioned in [4], one can find cited literature in which asymptotic expansion technique
theory.
(in"
and in one dimensional case) was used in the quantum field
We show how the S-asymptotic expansion
in
can be applied to solutions of
partial differential equations.
PROPOSITION 8.
L(D)
such that
Suppose that E is a fundamental solution of the operator
l
aaD a, aa
R,
(NQ0)
n;
L(D)
0
456
B.
E(t+h)
Z
n=l
u (t,h)
n
STANKOVI
Ilhll
{c (h)}
n
(4.2)
ic
h
Then there exists a solution X of the equation
L(D) X
G, G
"
(4.3)
which has S-asymptotic expansion
X(t+h)
.
Z
n=l
(G * un(t,h))
{c n(h)},
[[h[[
,
r.
h
The well-known Malgrange-Ehrenpreis theorem (see for example [7], p. 212)
PROOF.
asserts that there exists a fundamental solution of the operator (4.1) belonging to
X
The solution of equation (4.3) exists and can be expressed by the formula
E
G.
To find the S-asymptotic of X we have only to apply Propostition I.
REMARKS.
convolution E
in ,
If we denote by A(L(D),E) the collection of those T
* T and L(D)6
E * T exist
"
then the solution X
for which the
E * G is
unique in the class A(L(D),E) ([7], p. 87).
We can enlarge the space to which belongs G ([7], p. 216).
The fundamental solutions are known for the most important operators L(D).
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 (JFP) 544.
REFERENCES
Thorie des distributions, T. I, II, Herman, Paris, 1957-1951.
I.
SCHWARTZ, L.
2.
LAVOINE, J. and MISRA, O.Po Thormes Ablians pour la transformation de
Stieltjes des distributions, C,R. Acad. Sci. Paris Sdr. I Math. T. 279 (1974),
99-102.
3.
VLADIMIROV, V.S., DROZZINOV, Yu.N., and ZAVJALOV, B.I. Multidimensional Theorems
of Tauberian Type for Generalized Functions, Moskva, "Nauka", 1986.
4.
BRICHKOV, Yu.A. Asymptotical Expansions of the Generalized Functions I, Theoret.
Mat. Fiz. T. 5 (1970), 98-109.
5.
ZEMANIAN, A.H.
6.
STANKOVI,
7.
VLADIMIROV, V.S. Generalized Functions in Mathematical Physics, Mir Publishers,
Moscow, 1979.
Generalized Integral Transformations, Appendix I in the Russian
translation, Moskva "Nauka", 1974.
B. Characterization of Some Subspaces
Inst. Mat. Beorad 41 (1987), in print.
of"
by S-asymptotic, Publ.