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, = 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
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