CHAPTER 1

The L1 Fourier transform

If f ∈

L1(Rn)

then its Fourier transform is

ˆ

f :

Rn

→ C defined by

ˆ(ξ)

f =

e−2πix·ξf(x)dx.

More generally, let

M(Rn)

be the space of finite complex-valued mea-

sures on

Rn

with the norm

μ =

|μ|(Rn),

where |μ| is the total variation. Thus

L1(Rn)

is contained in

M(Rn)

via the

identification f → μ, dμ = fdx. We can generalize the definition of Fourier

transform via

ˆ(ξ) μ =

e−2πix·ξdμ(x).

Example 1 Let a ∈

Rn

and let δa be the Dirac measure at a, δa(E) = 1

if a ∈ E and δa(E) = 0 if a ∈ E. Then δa(ξ) =

e−2πia·ξ.

Example 2 Let Γ(x) =

e−π|x|2

. Then

(1)

ˆ

Γ( ξ) =

e−π|ξ|2

.

Proof. The integral in question is

ˆ

Γ( ξ) =

e−2πix·ξe−π|x|2

dx.

Notice that this factors as a product of one variable integrals. So it suﬃces

to prove (1) when n = 1. For this we use the formula for the integral of a

Gaussian:

∞

−∞

e−πx2

dx = 1. It follows that

∞

−∞

e−2πixξe−πx2

dx =

∞

−∞

e−π(x+iξ)2

dx ·

e−πξ2

=

∞+iξ

−∞+iξ

e−πx2

dx ·

e−πξ2

=

∞

−∞

e−πx2

dx ·

e−πξ2

=

e−πξ2

,

where we used contour integration at the next to last line.

1

http://dx.doi.org/10.1090/ulect/029/01