5Fourier transform

II Probability and Measure

5.1 The Fourier transform

When talking about Fourier transforms, we will mostly want to talk about

functions

R

d

→ C

. So from now on, we will write

L

p

for complex valued Borel

functions on R

d

with

kfk

p

=

Z

R

d

|f|

p

1/p

< ∞.

The integrals of complex-valued function are defined on the real and imaginary

parts separately, and satisfy the properties we would expect them to. The details

are on the first example sheet.

Definition

(Fourier transform)

.

The Fourier transform

ˆ

f

:

R

d

→ C

of

f ∈

L

1

(R

d

) is given by

ˆ

f(u) =

Z

R

d

f(x)e

i(u,x)

dx,

where u ∈ R

d

and (u, x) denotes the inner product, i.e.

(u, x) = u

1

x

1

+ ··· + u

d

x

d

.

Why do we care about Fourier transforms? Many computations are easier

with

ˆ

f

in place of

f

, especially computations that involve differentiation and

convolutions (which are relevant to sums of independent random variables). In

particular, we will use it to prove the central limit theorem.

More generally, we can define the Fourier transform of a measure:

Definition

(Fourier transform of measure)

.

The Fourier transform of a finite

measure µ on R

d

is the function ˆµ : R

d

→ C given by

ˆµ(u) =

Z

R

d

e

i(u,x)

µ(dx).

In the context of probability, we give these things a different name:

Definition

(Characteristic function)

.

Let

X

be a random variable. Then the

characteristic function of X is the Fourier transform of its law, i.e.

φ

X

(u) = E[e

i(u,X)

] = ˆµ

X

(u),

where µ

X

is the law of X.

We now make the following (trivial) observations:

Proposition.

k

ˆ

fk

∞

≤ kfk

1

, kˆµk

∞

≤ µ(R

d

).

Less trivially, we have the following result:

Proposition. The functions

ˆ

f, ˆµ are continuous.

Proof. If u

n

→ u, then

f(x)e

i(u

n

,x)

→ f(x)e

i(u,x)

.

Also, we know that

|f(x)e

i(u

n

,x)

| = |f(x)|.

So we can apply dominated convergence theorem with |f| as the bound.