MATH 220: THE FOURIER TRANSFORM – TEMPERED
DISTRIBUTIONS
Beforehand we constructed distributions by taking the set
C
∞
c
(
R
n
) as the set
of ‘very nice’ functions, and defined distributions as continuous linear maps
u
:
C
∞
c
(
R
n
)
→
C
(or into reals). While this was an appropriate class when studying just
derivatives, we have seen that for the Fourier transform the set of very nice functions
is that of Schwartz functions. Thus, we expect that the set
S
′
(
R
n
) of ‘corresponding
distributions’ should consist of continuous linear maps
u
:
S
(
R
n
)
→
C
.
In order
to make this into a definition, we need a notion of convergence on
S
(
R
n
).
Since
C
∞
c
(
R
n
)
⊂ S
(
R
n
), it is not unreasonable to expect that if a sequence
{
φ
j
}
∞
j
=1
converges to some
φ
∈ C
∞
c
(
R
n
) inside the space
C
∞
c
(
R
n
) (i.e. in the sense of
C
∞
c
(
R
n
)
convergence), then it should also converge to
φ
in the sense of
S
convergence. We
shall see that this is the case, which implies that every
u
∈ S
′
lies in
D
′
as well: for
if
φ
j
→
φ
in
C
∞
c
(
R
n
) then
φ
j
→
φ
in
S
, hence
u
(
φ
j
)
→
u
(
φ
) by the continuity of
u
as an element of
S
′
. Thus,
S
′
⊂ D
′
, i.e.
S
′
is a special class of distributions.
In order to motivate the definition of
S
convergence, recall that
S
=
S
(
R
n
) is the
set of functions
φ
∈ C
∞
(
R
n
) with the property that for any multiindices
α,β
∈
N
n
,
x
α
∂
β
φ
is bounded. Here we wrote
x
α
=
x
α
1
1
x
α
2
2
...x
α
n
n
, and
∂
β
=
∂
β
1
x
1
...∂
β
n
x
n
; with
∂
x
j
=
∂
∂x
j
.
With this in mind, convergence of a sequence
φ
m
∈ S
,
m
∈
N
, to some
φ
∈ S
, in
S
is defined as follows. We say that
φ
m
converges to
φ
in
S
if for all multiindices
α
,
β
, sup

x
α
∂
β
(
φ
m
−
φ
)
 →
0 as
m
→ ∞
, i.e. if
x
α
∂
β
φ
m
converges to
x
α
∂
β
φ
uniformly.
A tempered distribution
u
is defined as a continuous linear functional on
S
(this
is written as
u
∈ S
′
), i.e. as a map
u
:
S →
C
which is linear:
u
(
aφ
+
bψ
) =
au
(
φ
) +
bu
(
ψ
) for all
a,b
∈
C
,
φ,ψ
∈ S
, and which is continuous: if
φ
m
converges
to
φ
in
S
then lim
m
→∞
u
(
φ
m
) =
u
(
φ
) (this is convergence of complex numbers).
As mentioned already, any tempered distribution is a distribution, since
φ
∈ C
∞
c
implies
φ
∈ S
, and convergence of a sequence in
C
∞
c
implies that in
S
(recall
that convergence of a sequence in
C
∞
c
means that the supports stay inside a fixed
compact set and the convergence of all derivatives is uniform).
The converse is
of course not true; e.g. any continuous function
f
on
R
n
defines a distribution,
but
integraltext
R
n
f
(
x
)
φ
(
x
)
dx
will not converge for all
φ
∈ S
if
f
grows too fast at infinity;
e.g.
f
(
x
) =
e

x

2
does not define a tempered distribution. On the other hand, any
continuous function
f
satisfying an estimate

f
(
x
)
 ≤
C
(1 +

x

)
N
for some
N
and
C
defines a tempered distribution
u
=
ι
f
via
u
(
ψ
) =
ι
f
(
ψ
) =
integraldisplay
R
n
f
(
x
)
ψ
(
x
)
dx, ψ
∈ S
.