6Fourier transforms
IB Methods
6.3 Parseval’s theorem for Fourier transforms
Theorem
(Parseval’s theorem (again))
.
Suppose
f, g
:
R → C
are sufficiently
well-behaved that
˜
f
and
˜g
exist and we indeed have
F
−1
[
˜
f
] =
f, F
−1
[
˜g
] =
g
.
Then
(f, g) =
Z
R
f
∗
(x)g(x) dx =
1
2π
(
˜
f, ˜g).
In particular, if f = g, then
kfk
2
=
1
2π
k
˜
fk
2
.
So taking the Fourier transform preserves the
L
2
norm (up to a constant
factor of
1
2π
).
Proof.
(f, g) =
Z
R
f
∗
(x)g(x) dx
=
Z
∞
−∞
f
∗
(x)
1
2π
Z
∞
−∞
e
ikx
˜g(x) dk
dx
=
1
2π
Z
∞
−∞
Z
∞
−∞
f
∗
(x)e
ikx
dx
˜g(k) dk
=
1
2π
Z
∞
−∞
Z
∞
−∞
f(x)e
−ikx
dx
∗
˜g(k) dk
=
1
2π
Z
∞
−∞
˜
f
∗
(k)˜g(k) dk
=
1
2π
(
˜
f, ˜g).