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).