1Some preliminary analysis

III Modular Forms and L-functions

1.2 Fourier transforms
Equipped with the notion of characters, we can return to our original goal of
understand Fourier transforms. We shall first recap the familiar definitions of
Fourier transforms in specific cases, and then come up with the definition of
Fourier transforms in full generality. In the mean time, we will get some pesky
analysis out of the way.
Definition
(Fourier transform)
.
Let
f
:
R C
be an
L
1
function, i.e.
R
|f|
d
x <
. The Fourier transform is
ˆ
f(y) =
Z
−∞
e
2πixy
f(x) dx =
Z
−∞
χ
y
(x)
1
f(x) dx.
This is a bounded and continuous function on R.
We will see that the “correct” way to think about the Fourier transform is
to view it as a function on
ˆ
R instead of R.
In general, there is not much we can say about how well-behaved
ˆ
f
will
be. In particular, we cannot expect the “Fourier inversion theorem” to hold for
general
L
1
functions. If we like analysis, then we can figure out exactly how
much we need to assume about
ˆ
f
. But we don’t. We chicken out and only
consider functions that decay rely fast at infinity. This makes our life much
easier.
Definition (Schwarz space). The Schwarz space is defined by
S(R) = {f C
(R) : x
n
f
(k)
(x) 0 as x ±∞ for all k, n 0}.
Example. The function
f(x) = e
πx
2
.
is in the Schwarz space.
One can prove the following:
Proposition. If f S(R), then
ˆ
f S(R), and the Fourier inversion formula
ˆ
ˆ
f = f(x)
holds.
Everything carries over when we replace
R
with
R
n
, as long as we promote
both x and y into vectors.
We can also take the Fourier transform of functions defined on
G
=
R/Z
.
For n Z, we let χ
n
ˆ
G by
χ
n
(x) = e
2πinx
.
These are exactly all the elements of
ˆ
G
, and
ˆ
G
=
Z
. We then define the Fourier
coefficients of a periodic function f : R/Z C by
c
n
(f) =
Z
1
0
e
2πinx
f(x) dx =
Z
R/Z
χ
n
(x)
1
f(x) dx.
Again, under suitable regularity conditions on
f
, e.g. if
f C
(
R/Z
), we have
Proposition.
f(x) =
X
nZ
c
n
(f)e
2πinx
=
X
nZ
=
ˆ
G
c
n
(f)χ
n
(x).
This is the Fourier inversion formula for G = R/Z.
Finally, in the case when G = Z/NZ, we can define
Definition
(Discrete Fourier transform)
.
Given a function
f
:
Z/NZ C
, we
define the Fourier transform
ˆ
f : µ
N
C by
ˆ
f(ζ) =
X
aZ/NZ
ζ
a
f(a).
This time there aren’t convergence problems to worry with, so we can quickly
prove this result:
Proposition. For a function f : Z/N Z C, we have
f(x) =
1
N
X
ζµ
N
ζ
x
ˆ
f(ζ).
Proof.
We see that both sides are linear in
f
, and we can write each function
f
as
f =
X
aZ/NZ
f(a)δ
a
,
where
δ
a
(x) =
(
1 x = a
0 x 6= a
.
So we wlog f = δ
a
. Thus we have
ˆ
f(ζ) = ζ
a
,
and the RHS is
1
N
X
ζµ
N
ζ
xa
.
We now note the fact that
X
ζµ
N
ζ
k
=
(
N k 0 (mod N)
0 otherwise
.
So we know that the RHS is equal to δ
a
, as desired.
It is now relatively clear what the general picture should be, except that we
need a way to integrate functions defined on an abelian group. Since we are not
doing analysis, we shall not be very precise about what we mean:
Definition
(Haar measure)
.
Let
G
be a topological group. A Haar measure
is a left translation-invariant Borel measure on
G
satisfying some regularity
conditions (e.g. being finite on compact sets).
Theorem. Let G be a locally compact abelian group G. Then there is a Haar
measure on G, unique up to scaling.
Example. On G = R, the Haar measure is the usual Lebesgue measure.
Example.
If
G
is discrete, then the Haar measure is the counting measure, so
that
Z
f dg =
X
gG
f(g).
Example. If G = R
×
>0
, then the integral given by the Haar measure is
Z
f(x)
dx
x
,
since
dx
x
is invariant under multiplication of x by a constant.
Now we can define the general Fourier transform.
Definition
(Fourier transform)
.
Let
G
be a locally compact abelian group with
a Haar measure d
g
, and let
f
:
G C
be a continuous
L
1
function. The Fourier
transform
ˆ
f :
ˆ
G C is given by
ˆ
f(χ) =
Z
G
χ(g)
1
f(g) dg.
It is possible to prove the following theorem:
Theorem
(Fourier inversion theorem)
.
Given a locally compact abelian group
G
with a fixed Haar measure, there is some constant
C
such that for “suitable”
f : G C, we have
ˆ
ˆ
f(g) = Cf(g),
using the canonical isomorphism G
ˆ
ˆ
G.
This constant is necessary, because the measure is only defined up to a
multiplicative constant.
One of the most important results of this investigation is the following result:
Theorem (Poisson summation formula). Let f S(R
n
). Then
X
aZ
n
f(a) =
X
bZ
n
ˆ
f(b).
Proof. Let
g(x) =
X
aZ
n
f(x + a).
This is now a function that is invariant under translation of
Z
n
. It is easy to
check this is a well-defined
C
function on
R
n
/Z
n
, and so has a Fourier series.
We write
g(x) =
X
bZ
n
c
b
(g)e
2πib·x
,
with
c
b
(g) =
Z
R
n
/Z
n
e
2πib·x
g(x) dx =
X
aZ
n
Z
[0,1]
n
e
2πib·x
f(x + a) dx.
We can then do a change of variables
x 7→ x a
, which does not change the
exponential term, and get that
c
b
(g) =
Z
R
n
e
2πib·x
f(x) dx =
ˆ
f(b).
Finally, we have
X
aZ
n
f(a) = g(0) =
X
bZ
n
c
b
(x) =
X
bZ
n
ˆ
f(b).