275
Internat. J. Math & Math. Sci.
VOL. 14 NO. 2 (1991) 275-282
A NOTE ON THE CONVOLUTION AND THE PRODUCT
D’ AND S’
A. KAMINSKI and R. RUDNICKI
Institute of Mal.hemat[,’s
Polish Academy of Sciences
8, 40-03
W[eczorka
Katow[ce
Poland
(Received November 27, 1989 and in revised form March 3, 1990)
Examples of tempered d[stibutions are shown such that_ the convo[ut[on and
On
and are tempered distributions, but they do not exist in S.
product exist in I)
ABSTRACT.
the other hand, there exist tempered d[stributions whose convolution or
product exist in )
KEY
WORDS
AND
but are not tempered distributions.
PHRASES.
Product
and
Convolution
Distributions,
Tempered
Distributions.
46FI0.
1980 AMS SUBJECT CLASSIFICATION CODE.
INTRODUCTION.
the convolution and the product have been considered in
Various definitions of
I
the space
of distributions and in the space S
to the spaces
D
and S
respectively (see Schwartz [i], pp. 21, 24, 233, 2B7).
In the case of the convolution f*g in
’
resp.),
[4] (p. 62), Dierolf and Voigt [5], are equivalent.
will consider the
sequences,
IP’ (and
which
convolution in
are
equivalent
most
of the known
Schwartz [I] (p. 153), Cheval[ey [2], Shiraishi [3],
definitions, e.g. those given by
Vladimirov
of tempered distributions, the duals
i.
IP
and
in
to
the
mentioned
classes of unit-sequences as shown in Kamiski
For our purposes, we
defined by means of so-called unitdefinitions
[6].
the
for
respective
By changing classes of unit-
IP
sequences one can obtain, however more general definitions of the convolution in
and in
S-
The
definitions
of
the
in
IP
(in
$
resp.)
considered
’
For instance, the definition for
literature are not equivalent.
and Ogata
f.g
product
[7] and by Mikusiski
in
the
given by Hi rata
[8] by means of so-called delta-sequences depends
essentially on the class of delta-sequences taken into consideration.
Notice
that
given two tempered distributions it makes sense to consider the two operations of the
convolution and product: in
(I) Let f, g
resp.) belong to $
and in
S
Therefore the following questions appear:
IP
Let f, g e $
Suppose that f’g, (f.g, resp.) exists in
Does f*g (f.g, resp.) exist in S
(II)
S
IP
Suppose that f*g (f.g, resp.) exists in
Does f*g f.g,
IP
and belongs to
A. KAMINSKI AND R. RUDNICKI
276
The first question in the case of the convolut.[o,1 was posed by Shiralshl [3] and
answered negatively by Kamifskl [9] (see also Kamiski [I0]) and by Dierolf and Voigt
[5] (see also Ortner and Wagner ill]).
For the case of the product see Section 3.
A respective counter-example
The answer to the second question is also negative.
for the convolution in R 2 was given by Wagner [12]. He also showed that the existence
of the convolution f*g in ) of homogeneous tempered distributions f, g implies the
existence of f*g in
/
Since
Wagner’s counter-example
cannot be transferred into the
one-dimenslonal case, we give a suitable example in Section 4.
can be automatically extended to R d.
This counter-example
Let us underline that in our example
g and f
is a real-valued d[stributlon.
In the case of the product, a respective counter-example could be obtained from
This
the example constructed for the convolution by means of the Fourier transform.
requires, however, careful considerations on relations between various definitions of
Let us notice that the tempered distribution
the product of tempered distributions.
obtained
way
this
in
complex-valued.
is
Therefore
we
give
in
5
Section
a
straightforward example with real-valued tempered distributions.
2.
DEFINITIONS.
By E we denote the class of all unit-sequences, i.e. sequences kn n) of functions
of the class
Dk(n-l)
(i)
k
D satisfying
(<I
e N
<d
the conditions:
0 almost uniformly on R d for every multl-index
d
Nt){0}.
o’ where N o
d
For every k e N there is a constant M
k
o
(ll)
D
k
n(X)l
<
$d
e R
(i
for all x
M
d
>
0 such that
k
and n e N.
By E m we denote the class of all model unit-sequences, i.e. sequences
(qn)
of the
form
n(X)
(a x),
n
fromD
where o is a function
(in
S’, resp.)
(in S’).
such that 0(O)
0 and (a) is a sequence of real numbers
e
S’, resp.).
We say that the convolution f*g exists in
if the limit
*
D’ (in S’, resp.)
lim
exists
n e N,
m
D’ (f, g
lira (O f)
n
exists in
d,
Evidently, we have E C E.
converging to 0.
Let f, g e
x e R
O
n
f) *
(n g)
(2.1)
for all (),
n
(ng)
()
n
e E.
If
(2 2)
all (n) e E, we say that the symmetric convolution f*g exists in
In both cases the limits do not depend on unlt-sequences of the class E.
for
In
D’ AND S’
CONVOLUTION AND PRODUCT IN
the above definitions
by E m,
class E can be replaced
the
27"
but
then we additionally
assume that the limits (2.1) and (2.2) do not depend on model unlt-sequences and
symmetric) convolution in )’ kin
is called the model
As shown in [6], the convolution in D’ in S’) as above is equivalent to the
convolution in )’ (in S ’) in the sense of the other known definitions mentioned in
Section I.
(in S ), then the symmetric convolution f*g
If the convolution f*g exists in
If the ksymmetric) convolution f*g exists
exists in ) (in S ), but not conversely.
[n )
in S ), but
(ins ), then the model (symmetric) convolutLon f*g exists in
DI
not conversely.
By AO we denote the class of all delta-sequences (),
n
functi.ons satisfying the conditions:
(iii) there exists a sequence of positive numbers
1/2
f n(t)
(iv)
(v)
>
I1
n
i.e.
sequences of smooth
0 such that
i
and n e N"
for n e N;
dt
>
there exists an M
0 such that
n
By
k
we denote the class of all model delta-sequences of the form
m
n(X)
where o e
l),
f
ot
-d
o(ot
n
o
1 x),
and (a
n
R
x
is
d
n
of
sequence
a
N,
real
numbers
to
converging
0.
Evidently, k C ko
m
f, g e S
Let f, g
S
exists in
D
(in S
lira
exists
If
We say that the product f.g exists
in
tin
resp.) if the limit
lim
One
resp.).
O
<f *
n
<g *
k2.3)
n
k n)
resp.) for all k n
(f *
n
kg
*
e
ao"
If
(2.4)
n
p
for all () e b o, we say that the symmetric product exists in
n
is replaced by Am, f.g is called the model symmetric) product in
can
from b.
O
show
tha
the
limits
(2.3)
We assume this for the class
and
k2.4)
do
not
depend
on
kin S ).
,in S ).
delta-sequences
m
and their
and in -q
For these and other definitions of the product, tn /)
equivalence see Nikusirski [8 and [13], Hirata and Ogata [7], Shraishi and Itano
[14], Antosik, N+/-kusirsk[ and Sikorski [15] ,p. 242), Kami[ski [6], Oberguggeaberger
Note that if the product f.g exists n ) ,In S ), then
[16], and Wawak [17, 18].
If the
(in 3 ), but not conversely.
the symmetric product f.g exists in
A. KAMINSKI AND R. RUDNICKI
278
(symmetric) poduct f.g exists in D in
exists in ) (ins ), but not_ conversely.
S,),
then the model symmetric) product f.g
QUESTION ([) FOR PRODUCT.
It is very easy to give the negative answer to the question (I) for the product
in R in case complex-valued distributions (functions) are admitted. Namely, given an
(in particular, F can be chosen so
arbitrary real-valued continuous function F on R
that F
S’)
put
G<x)
f
fx
exp [i
O
G’(x)
l(x)
F(t)dt]
iG(x)V(x)
and
f2(x)
-iG
-1(x
I.
Then f l,
are continuous complex-valued functions on R such that
because
and
F In the usual sense, so the more in )
for x e R
f[’f2
f2
fl’ f2 ES’
IG(x)l
I.
We are going to show that there exists also a real-valued smooth function # which
(which also exists in the sense of the product
belongs to S such that the square
-j
-j
[j-? 3
j+3 -j] J
[j-3
First denote
in )’) does not belong to
j
-j
Let
for j e N.
j+2.3
be a smooth function such that
2
S’
(x)
0
2
lj
j,
((x)
(
x e
Ij,
(3.1a)
x e
Ji’
(3.1b)
2j,
and
,(x)
0,
x
By (3.!) we have
N.
for j
e S
so
exists in )
re(X)
(3.1c)
Jj
J=l
n)
Ao, the product #.
Since
almost uniformly for any (6
* 6n
of functions. Taking
and coincides with the usual product
2
exp(-x/4), we have
f ,2(x)o(x)
where e
exp(-I/4).
Another
dx
)
2
Consequently
example,
given
I (4=/3)J
",
j=l
2
explicitly
by
(x)
eXcos
e
x
but
based
on
similar
calculations, was communicated to us by P. Wagner.
4.
QUESTION (II) FOR CONVOLUTION.
Now,
we
shall
construct
convolution square f*f exist in
an
’
example
of
a
distribution f e
S’ such that the
and is of the class S’, but does not exist in S’.
CONVOLUTION AND PRODUCT IN
Let f
+
2
D’ AND S’
279
with
fl(x)
(-I)
2
[tog2J]
6(x
2
j=2
and
6[x + 2 j
[log2j]),
where the symbol ix] means the integer part of x.
It is clear that
Note that
Each
of
A
supp
fl
{2k
B
supp
f2
{-2
the
sets
k e
k
+
fl’ f2’
f e
S’-
N\[I}},
[log2k]:
k
N\
{I}}.
fl and supp f2 is bounded from one slde, so it is
polynomlally compatible in the sense of Kamldskl [I0]. Therefore the convolutions
and
*
exists in S
(cf. [9] and [I0]).
*
It suffices to show that the
fl fl
supp
f2 f2
fl * f2 f2 * fl exists in P’ and does not exist in S j.
exists in P" it is enough to verify that for every bounded
convolution
To prove that
fl
interval I in R
f
2
the set
{(x,y): x c A, y
is bounded in R
2.
B, x + y e I}
But if
12j -2k + [lg2k
m
N, then we have
for some m
2 +
max(j,k)
log2m
for j
k
for j
k,
and
2
j, k
m+l
.
In addition, we have
which proves the required property.
[
* f2
f
k=2
J--2
*
In order to prove that f
2
2k+
]
[log
f2
$’
+ 2k
log 2 k] ).
fix m e Z and consider the pairs
(J k) for
m.
k then
If j
J
If j
Iml
j
(-l)J 2 [lgzj] 6(x-2J
)
k e {2
m,
2
m
+ I,
...,
2
m+l
I}.
k then there exists at most one pair (J,k) such that (4.1) holds and
m
we obtain
j
{j e N: 2
Setting J
2 j-I ) j.
2m+l-l},
which
.
280
A. KAMINSKI AND R. RUONICKI
2m6(x-m)=
(-1)
O.
(4.2)
jc
Hence
where
[ am6(X-m),
(f[* f2 (x)
(-l)j 2[lg2J] if
am
In
otherwise.
both
meZ
am
cases
m, which
be a real-valued function of the class
let
(x)
0 for
Ixl
)
3/2.
2
I (-l)J
2
From
the
last
2 [log2j] 6(x-2
6(x+2
k
k=2
it follows
formulae
expressed as a sum of two distributions
jk
and l
2[Ig2j]6(x
(-I) j
such that (x)
summation over all j,k
lira
for all # e
S.
<
h
2
j, k
that
gm
2
m
,>
2
and
j
j
the
distribution
ll"kjj-
and h
[log2k]m
+ 2
k.
2
j
m
(nnf I)
E2
k
(nnf 2)
*
can
be
where
12 denotes the summation
E3j k’ where 13 denotes the
and
Setting h
k, it is easy to check that
2 with j
)
2
m
2
Ixl
[log2k]).
denotes the summation over all j with 2
over all j and k with 2
for
0
ow,
$’
m
[
nn(X)f2(x)
*f
fl
that
m
j=2
and
D
implies
(x/n) we have, for n
n(X)
Setting
n(X)fl(x)
am--
there exists a pair (j,k) such that (4.1) holds and
<h,#>
eS
On the other hand, taking
such that (x)
(2/3) x for x
>
0, we
have, in view of (4.2),
<gm’ >
<2m (x-k), (x >
2m#(k)
(4/3) m
=.
Therefore the convolutions
means that f*f does not exist in S
as m
REMARK 4.1.
fl* f2 f2 * fl
g
S’
k =2
gl + g2
(-l)k(x- 2k), g2(x)
(n), (n)
convolution f
* f
with
exists in
S
D"
and does not exist
[.
(x+2
k
k =2
in the sense of
commonly bounded
a
even in
ins
can be
with
In this example f*f exists in S
seqaences
and this
Another example of a distribution
such that the convolution f*f exists in
obtained by taking f
gl(x)
ins
Note that in the construction above f*f does not exist in S
the sense of the symmetric model convolution.
f
do not exist
[log2k]).
the limit
variation.
(2.3) for all unit-
In paricular,
in contrast to the example presented before.
the
model
CONVOLUTION AND PRODUCT IN
REMARK 4.2.
Put F
f
*
,
AND S’
281
is the distribution constructed In the example
where
above and m is a function of
D’
the class
D
Then F is a slowly increasing (i.e.
bounded by a polynomial) smooth function such that the convolution F*F exists in
and
slowly
a
represents
that f
space
is
ins
function
while
(model
the
symmetric)
The example can also be modified in such a way
(even a smooth function) from an arbitrary Gelfand-Shilov
a distribution
K’(Mp)
smooth
increasing
convolution F*F does not exist
D
with the property: the convolution f*f exists in
(smooth function) of the class
K’(Mp),
but f*f does not exist in
and is a distribution
K’(Mp).
QUESTION (II) FOR PRODUCT.
5.
Now we are going to give a
.
and belongs to S
such that the product f.g exists in
S,.
Let
f(x)
example of two tempered distributions f,g on R
slmple
(x-k-2-k).
g(x)
(x-k),
k=l
k=l
Obviously, f,g e S
a) (x
Since the product g(x
for ab, it follows that the product f.g exists in
symmetric model product f.g does not exist in l)
n
(x)
D
b) exists in I)
and f.g
and equals 0
To prove that the
0.
define
(5.1)
nm(nx),
is a non-negatlve function of the class
where
but f.g does not exist in
D
such that
re(x)
for x e [-I/4, 114]
(5.2)
f
1.
(53)
and
o<x) dx
of the class S such that
Moreover, consider a positive function
>
0.
(x)
(2/3) x for x
Then we have
<(f *
n
f
where j
n + 2.
not exist in
(g *
n
n )’ >
(x- j)
Since
2J -4
n
(x
"
f n(x-m)n(x
m=l
j
2
-j
2/ 3 )J +1* -as
(x)
J
-,
dx
-m- 2 -m)
2
j-4
(x)
dx
(2/3) j+1
the symmetric model product f.g does
S..
Finally, let us pose the following open question:
PROBLEM.
Does there exist a real-valued tempered distribution f such that the
product f.g exists in
not exist in
S/?
D
and belongs to
but the (symmetric model) product f.f does
A. KAMINSKI AND R. RUDNICKI
282
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on