Internat. J. Math. & Math. Sci.
VOL. 15 NO. 4 (1992) 631-634
631
ASYMPTOTICS OF REGULAR CONVOLUTION QUOTIENTS
BOGOLJUB STANKOVI(
Institute of Mathematics
University of Novi Sad
Yugoslavia
(Received October I0, 1991)
generalized functions, named regular
convolution quotients, has been defined and analysed. Some properties of such asymptotics, which can be useful in applications, have been proved.
ABSTRACT. The asymptotic behaviour of a
class, of
KEY WORDS AND PHRASES. S-asymptotics, convolution quotient, regular convolution quotients, distributions, generalized functions.
1991 AMS SUBJECT CLASSIFICATION CODE 44A40
I. INTRODUCTION.
T.K. Boehme in [I defined and investigated a class of generalized functions
This class is a geq u o t i e n t s
c o n v o i u t i o n
named r e g u i a r
Mikusinski operathe
and
regular
of
also
distributions
Schwartz
neralization of the
tors (see [2]
[3] and [4] ). On the other hand, for the Schwartz distributions a
S-asymptotics, has been developed (see for examtheory of the asymptotic behaviour
ple
[5], [6], [7]
and
dels. A distribution
ction
c
iff lim
[8] ),
which can be applied in solving a lot of mathematical mo-
T has S-asymptotics related to a positive and measurable func(h).S,
(S’w)(0) for every w D We write" T
(T*w)(h)/c(h)
s
In thls paper we shall enlarge the definition of S-asymptotics of distribu-
h
tions to the regular convolution quotients having in view the application of this
class of generalized functions.
2. REGULAR CONVOLUTION QUOTIENTS.
By Boehme [I]
an
a p p r o x i m a t e
identity
c
conditions"
(u
L(R)
the
following
functions
satisfying
n)
i)
5R
Un(X)
dx
ii) there is an M0
nN
such that
iii) there exists a sequence
supp u
nc[-kn,kn
is a sequence of
R lu(x)
(kn)CR+
nN;
dxM
such that
neN
k
n
-
0 as
n
and
,
/ will be the set of all approximate identities and
ung C n
N}. A d e f i n i n g s e q u e n c e for a regular convolution quotient is a
sequence of pairs
where (fn) C Lloc(R),
and for all m,ngN the foliowing convolution products are equal-
(Un)e
((fn,Un)),
iv)
fn*Um fm*Un
(Un)e
the asterisk is the sign of the convolution).
632
B. STANKOVIC
Two defining sequences ((fn,Un))
v) f *v
gm*Un for n,mN
n m
((gn’Vn)
and
are said to be equivalent if"
By
fn/Un we
((fn,Un)).A r e g
shall denote the equivalence class containing the difining sequence
u 1 a r c o n v o 1 u t i o n
q u o t i e n t
X is an equivalence class of defining sequences
The regular convolution quotients are a vector
space when the usual multiplication by scalars and addition of fractions is used; we
).The space
denote it by
A
contains D’ (space of Schwartz’s distributions) under the isomorphism: D’9 T
), where
n
contains the class of all regular Mikusinski operators. Both of
Moreover,
these containments are proper.
Let (h be any continuously differentiable approximate identity. By D
B(Lloe,
B(oe,
B(Lloc,
e
&
n)
B(Lloc,6 is
B(LIoc i) is
-
(T*Vn)/V
B(Lloe,
(Vn)E
defined the differentiation operator. The derivative of an X
now, defined to be DX
&
(fn hn)/(Un*hn)
n/hn
fn/hn %
S(hoc’)"
For a distribution T e D’
and w&D we shall write
use the following properties of elements belonging to
< t,w > We shall
and distributions defined
T(w)
by local integrable functions
where
I. For
(x)
vn(-x)
2. If (un) and
3. If
n
(fn)Lloc
,
(Vn) e
we have
(vn)
(fn*Vn)(O)
all x
and
<fn(x+h),%(x)> (fn*Vn)(h)
belong to
then
0 ,neN for every (v
(Un*Vn) e
n) e Zi
as well.
then
R
fn (x)
0 for almost
-
3. S-ASYMPTOTICS OF REGULAR CONVOLUTION QUOTIENTS
Let E be the set of all real valued, positive and measurable functions: R
+
DEFINITION i. A regular convolution quotient X has S-asymptotics at infinity,
related to c e r. and with the limit U
Fn/Un E B(oc,) if there exists ((fn,Un))
belonging to the class X such that
fn*Vn
lira
h-
for every
h
(Fn*Vn)(0)
write X ,s c(h).U
c(h)
(vn) e Zi
nN
-
We shall
h
This definition does not depend on the defining sequence
((fn,Un)) in the equivalence class X
Let ((gn,Jn)) e fn/Un and let Gn/J & B(Lloc,/i) such that
n
(gn’Vn)()
im
hThen
((Fn,un))
and
c(h)
((Gn,Jn))
(Gn*vn) (0)
h-,
(Vn)
PROPOSITION I.
(vn)
e
belong to the same class because of"
< (Fn*Jm) ,n > ((Fn*Jm)*Vn)(O)
fn" Jm’Vn ).(h)
lira
e(h)
for every
nS and
lira
(gin* (Un.Vn)
(h)
< %’Us, % >
and
m,n,N
Hence,
Fn*Jm
Gin*
u
n
for m,n S
If a distribution T has
S-asymptotics, T
c(h).S, hc 7.
then the regular convolution quotient
X
which
corresponds to
n
has S-asymptotics, as well and X
h
-
s c(h).(S,Un)/Un(T*Un)/U
ASYMPTOTICS OF REGULAR CONVOLUTION QUOTIENTS
n) e we have"
((T*Un)*Vn) (h)
Proof. For every
633
v
lim
C(h)
(T* (Un*Vn) )(h)
c(h)
lim
h-
(S*(Un*Vn))(0) ((S*un)*vn)(O) neN
(S*un )/un belongs to B(Lloc,)because of (S*Un)*U
(S*Um)*Un
m
Hence X s, c(h).(S.Un)/Un hLet u’remark that (S*Un)/Un
.
g
m,n
for every
corPesponds to Se
D’ by the mentioned isomorphism. In such a way
S-asymptotics of regular convolution quotients, defined by Definition
generalizes S-asymptotics of distributions.
PROPOSITION 2. If X has S-asymptotics X
c E
then
c(h).U
h-
Dnx
Dnx s c(h).DnU
has S-asymptotics,as well and
tion operator in B(Lloc,
Proof. It is enough to prove for n=1
lim
By definition, DX
where
Now, the following
(h)
fn*h )*vn
,
fn hn) /(Un,hn)
Hence
U Fn/Un
s
(hn)
and let for every
(Vn)e
ne N
is any continuously differentiable
relation is true"
fn* (h*vn) )(h)
lim
h-
c(h)
fn/Un
(Fn*Vn)(0)
(h)
(fn*h)/(Un*hn)
approximate identity.
lim
h-
(fn*Vn)(h)
Let X
D is the differentia-
h
((F
c(h)
c(h).(Fn*h)/(Un*hn)
and
n
DX
hn)*Vn)(O)
s, c(h).DU,
nN
h-
where
This proposition can be useful in applying regular convolution quotients to differential equations. The next proposition precises the analytical form of the function
c
g
E
which measures the asymptotical behaviour of a regular convolution quotient and
U, the limit in Definition I.
PROPOSITION 3. Suppose hat X B(Lloc,
c(h).U
where
and X
hc. and U Fn/Un If Fn 0 for one n then c(h) exp(ah) L(exph)
>0
and F n(x)
C exp(ax)
C
where a{R
C
0 and L is a slowly
n
n
varying func t ion.
Proof. L is a slowly varying function by definition iff LeE and lim L(ux)!
the form of the regular convolution quotient
s
neB
X--
/L(x)
(For slowly varying functions see, for example
u>0
((fn,Un)) e X
there exists
h
Now,
-lim
(fn*Vn)(h)
)- By Definition
such that
(Fn*Vn)(0)
c(h)
[9]
ne N
for every
(Vn)e /i
the proof of Proposition 3 follows directly from Proposition
or propositions 9
and 10
in
4.3
in
[5]
[7].
B(Lloc,),
then X has a compact support if and only if
PROPOSITION 4. If X
c(h).O
X
lhl-> for any eel
has compact support if and only if
Proof. We know (see [10]) that X e
and
N
n
e
that
such
a
is
’neN has compact support
there
ne N
Moreover, if X has compact support, then this is true for every gn
0
0,
and supp v c
((gn,Jn)) X Suppose that supp C
n
Hence
+k
for
we
Then
(v
and
neN
have’(fn*Vn)(h):0
n)
n
,s
B(Lloc,)
(Un)e
lira
h-=
e’.
fn* Vn
c(h)
UnX fn
fn
fn I-an’an]
XJn
[-kn,kn], an>
hl>an
h
0
neN
for any
ceE and any
(vn) e
kn>
B. STANKOVIC
634
Suppose ,pow, that
and suppose that for every
8.1
then by Proposition
c(h).O
lhl-w for every c e Z where X
(vn) eZi we have"
(fn*Vn) (h) 0 n(N
lim
c(h)
h..
or by Proposition 12 in 71,
p. 98 in
’neN
X
151
fn
fn/Un
has a
compact support
The S-asymptotic behaviour of a regular convolution quotient is a local property.
This property precises the following proposition.
c(h).U
and X
PROPOSITION 5. Suppose that X and Y belong to
B(Lloc,i)
Y on an interval (a,)
h-, c(.. If X
a eR
then
y
s. c(h).U
h,
as well.
X
Proof. Let
Y
fn/Un
gn/jn
(fn*Vn)(h)
and for every (v
(Fn*Vn)(O)
lim
c(h)
hBy properties of the convolution it follows:
lim
h--
If X
((fn*Jn)*Vn)(h)
Y
exists a sequence
c(h)
((Fn*Jn)*Vn)(0)
n) e Z-
n
(Vn)e /i
ne
(fn*Jn gn*Un)/(Jn*Un).
(fn*Jn gn*Un) C (bn,).
((fn*Jn gn*Un)*Vn)(h) 0 ne (v e
n)
c(h)
X-Y O, where X-Y
(bn)e R such that supp
then
lim
h--
.
Hence, there
Now,
,
Therefore,
((gn*Un)*Vn) (h)
limc(h)
hThe equivalence class
*(Jn*Un)
Fn/Un
((Fn*Jn)*Vn)(O)
ne M
(Vn)e
Y because of (gn*Un)*Jm gin*
(gn*Un)/(Jn*Un)
y
and
c(h).(Fn, Jn)/(Jn, Un).i t remains only to see that (Fn*Jn)/(Jn*Un)=
Fm *(Jn*Un)
m,n M
This follows from the relation (Fn*Jn)*Um
is just
.s
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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