Math 8602
Spring ‘94
Some notes on tempered distributions.
1
Assigning a topology to the Schwartz class
The Schwartz class is interesting, and applicable, with the Fourier transform and the tools of
Lebesgue measure.
We will define open sets for
S
,
and with this topology, introduce the space
S
′
of continuous linear functionals on
S
.
These objects are also called
tempered distributions
.
They
“include” all the
L
p
spaces,
S
itself, and much more besides.
The key ideas are these:
regarding
the action of a linear functional as a “formal” integral, and the “transpose” of a linear mapping.
We
have seen this in two places recently: the equations
⌡
⌠
∂
f
∂
x
k
g dx = –
⌡
⌠
f
∂
g
∂
x
k
dx
for integration by parts, and
⌡
⌠
^
f(
ξ
)
g(
ξ
) d
ξ
=
⌡
⌠
f(x)
^
g(x)
dx,
that
says we can “move the hat.”
The first equation is valid with
f
in
S
and
g
in
M
.
In that equation,
the left side makes sense for
any
measurable function
g
bounded by a polynomial, if
f
is in
S
.
So
we
define
a linear functional on
S
by the equation
T
g
(f) :=
⌡
⌠
f(x) g(x) dx.
The left side of the first
equation can then be expressed as
T
g
(
∂
f
∂
x
k
).
If
g
is in
M,
then we can write the equation as
T
g
(
∂
f
∂
x
k
) = –T
∂
g/
∂
x
k
(f) .
The right side may
not
make sense, if
g
is not in
M
,
so we
define
∂
g
∂
x
k
to be the
linear functional
, defined on
S
,
given by
the left side.
The same is true in the second
equation.
The second equation is valid with
f
in
S
and
g
in
S
.
The left side makes sense if
f
is
in
S
and
g
is in
M
.
But the right side does not — how do we define the Fourier transform of
x
2
,
for example?
The point is, we define it by the equation, not as a function, but as a
linear
functional
.
In fact, we have
^
ξ
)
ξ
k
2
= –
^
ξ
) (i
ξ
k
)
2
= –
∂
2
f
∂
x
k
2
^
(
ξ
),
so
^
ξ
)

ξ

2
= – (
∆
f)^(
ξ
),
where
∆
denotes the Laplace operator.
Now the equation reads
–
⌡
⌠
(
∆
f) ^(
ξ
) d
ξ
= (x
2
)^(f).
The
left side contains the factor
1 = e
–i
ξ•
0
.
By the inversion theorem we recognize this as
–(2
π
)
n
(
∆
f)(0).
Thus we will be able to say
(x
2
)^ = –(2
π
)
n
(
∆
•)(0).
two possible topologies for
S
We have infinitely many norms on
S
,
namely the
 f 
αβ
,
with
α
and
β
in
N
n
.
Each of these
gives us a collection of open sets, i.e. a topology on
S
.
There are two immediate possibilities:
we
can define our topology to be the intersection of all these norm topologies.
This will certainly yield a
topology.
In fact, we can use this fact to talk about the smallest topology containing a given collection
of sets.
The other possibility is the one
we will use:
the smallest topology that contains
all
these norm topologies
.
It is larger than the first one.
Thus it is “easier” for a function that has
S
as its domain to be continuous.
Let
X
be a set that has a topology.
Then,
f:
S
→
X
is continuous with respect to our first topology if and only if
f
is continuous with respect
to every topology in the collection.
On the other hand,
f:
S
→
X
is continuous with respect to our
second topology if
f
is continuous with respect to just one topology in the collection.