Fourier postulated around 1807 that any periodic signal (equivalently finite length signal) can be built up as an infinite linear combination of harmonic sinusoidal waves.
i.e. Given the collection
B
=
{
e
j
2
π
T
n
t
}
n
=

∞
∞
B
=
{
e
j
2
π
T
n
t
}
n
=

∞
∞
(1)any
f
(
t
)
∈
L
2
[
0
,
T
)
f
(
t
)
∈
L
2
[
0
,
T
)
(2)can be approximated arbitrarily closely by
f
(
t
)
=
∑
n
=

∞
∞
C
n
e
j
2
π
T
n
t
.
f
(
t
)
=
∑
n
=

∞
∞
C
n
e
j
2
π
T
n
t
.
(3)Now, The issue of exact convergence did bring
Fourier
much criticism from the French Academy of Science (Laplace,
Lagrange, Monge and LaCroix comprised the review committee) for
several years after its presentation on 1807. It was not
resolved for also a century, and its resolution is interesting
and important to understand from a practical viewpoint. See more in the section on Gibbs Phenomena.
Fourier analysis is fundamental to understanding the behavior of
signals and systems. This is a result of the fact that
sinusoids are Eigenfunctions of linear, timeinvariant (LTI) systems.
This is to say that if we pass any particular sinusoid through a
LTI system, we get a scaled version of that same sinusoid on the
output. Then, since Fourier analysis allows us to redefine the
signals in terms of sinusoids, all we need to do is determine
how any given system effects all possible sinusoids (its transfer function) and we
have a complete understanding of the system. Furthermore, since
we are able to define the passage of sinusoids through a system
as multiplication of that sinusoid by the transfer function at
the same frequency, we can convert the passage of any signal
through a system from convolution (in time) to multiplication
(in frequency). These ideas are what give Fourier analysis its
power.
Now, after hopefully having sold you on the value of this method
of analysis, we must examine exactly what we mean by Fourier
analysis. The four Fourier transforms that comprise this
analysis are the Fourier
Series, ContinuousTime Fourier Transform,
DiscreteTime Fourier
Transform and Discrete Fourier Transform. For this
document, we will view the Laplace Transform and ZTransform as simply
extensions of the CTFT and DTFT respectively. All of these
transforms act essentially the same way, by converting a signal
in time to an equivalent signal in frequency (sinusoids).
However, depending on the nature of a specific signal
i.e. whether it is finite or infinitelength
and whether it is discrete or continuoustime) there is an
appropriate transform to convert the signal into the frequency
domain. Below is a table of the four Fourier transforms and
when each is appropriate. It also includes the relevant
convolution for the specified space.
Table 1: Table of Fourier Representations
Transform 
Time Domain 
Frequency Domain 
Convolution 
ContinuousTime Fourier Series 
L
2
0
T
L
2
0
T

l
2
Z
l
2

ContinuousTime Circular 
ContinuousTime Fourier Transform 
L
2
R
L
2

L
2
R
L
2

ContinuousTime Linear 
DiscreteTime Fourier Transform 
l
2
Z
l
2

L
2
0
2π
L
2
0
2

DiscreteTime Linear 
Discrete Fourier Transform 
l
2
0
N−1
l
2
0
N
1

l
2
0
N−1
l
2
0
N
1

DiscreteTime Circular 
"My introduction to signal processing course at Rice University."