Chapter 7: Fourier transform
7.1. Preliminaries
The goal of this chapter is to provide tools on how to solve some PDE using
the Fourier transform. The idea is the following: the Fourier transform allows
us to replace a given PDE by a simplier, merely algebraic problem. After
solving that problem, the Inverse Fourier transform gives us the solution of
the original PDE problem. In contrast to the previous chapters, we will deal
now with functions
f
: IR
→
IC which are Riemann integrable on bounded
intervals of IR. We will say that

f

p
is Riemann integrable on IR  in short

f

p
∈ R
(IR), (
p
≥
1)  if
Z
IR

f
(
x
)

p
dx <
∞
.
In the case
p
= 2 we work with the following interior product and norm:
h
f, g
i
:=
Z
R
f
(
x
)
g
(
x
)
dx,
k
f
k
:=
q
h
f, f
i
.
A certain rˆ
o
le, both in proofs and in applications of our results, will play the
convolution
of two functions
f, g
which is deﬁned as
(
f
*
g
)(
x
) :=
Z
IR
f
(
x

y
)
g
(
y
)
dy,
provided that the integral exists. Notice that this is the case, if both

f

2
and

g

2
are Riemann integrable on IR, by the CauchySchwarz inequality, or if
f
is bounded and

g
 ∈ R
(IR). For instance, if
f
is piecewise continuous and
g
is bounded and vanishes on a ﬁnite interval, then
f
*
g
(
x
) exists for all
x
. The
convolution obeys the following operations, (
f, g, h
: IR
→
IC,
a, b
∈
IR):
f
*
(
ag
+
bh
) =
a
(
f
*
g
) +
b
(
f
*
h
)
,
(1)
f
*
g
=
g
*
h,
(2)
f
*
(
g
*
h
) = (
f
*
g
)
*
h,
and
(3)
(
f
*
g
)
0
=
f
0
*
g,
if
f
is diﬀerentiable .
(4)
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