Some quite careful math for Fourier series
is in Rudin, Principles of Mathematical
Analysis. Fourier series applies to a
function on the whole real line that is
periodic, that is, repeats exactly once
each some number of seconds. For the
math, need only one period so in effect
throw away the rest of the function and,
indeed, really need the function defined
only on the interval of some one period.

For some intuition, consider music,
especially on a violin. Fourier series
applies to a periodic function (wave), and
represents the whole wave as sine waves
that fit the one period exactly. So, get
sine waves at frequency 1, 2, ... that of
the period. In music, these waves are
called overtones.

Playing with a violin, the overtones are
fully real and even important! E.g., get
a tuning fork and tune the A string
(second from the right as the violinist
sees them) to 440 cycles per second (440
Hertz, 440 Hz). Then the D string, the
next to the left, is supposed to have
frequency 2/3rds that of the A string.
So, bow the two strings together and
listen for the pitch 880 Hz, that is, 3
times the desired frequency of the D
string and twice that of the A string. So
are listening to the second overtone of
the D string and the first overtone of the
A string; are hearing the third Fourier
series term of the D string and the second
Fourier series term of the A string.
Adjust the tuning peg of the D string
until don't hear beats. If the D string
is at, say, 881 Hz, then will get 1 beat a
second -- so this is an accurate method of
tuning. Similarly for tuning the E string
from the A string and the G string from
the D string -- on a violin, the
frequencies of adjacent strings are in the
ratio of 3:2, that is, a perfect fifth.
That's how violinists tune their violin --
which is needed often since violins are
just wood and glue and less stable than,
say, the cast iron frame of a piano.

For one more, hold a finger lightly
against a string at 1/2 the length of the
string and hear a note one octave, twice
the frequency, higher. That's often done
in the music, e.g., playing harmonics.
And it's a good way to get the left hand
where it belongs at the start of the
famous Bach Preludio in E-major that
starts on the E half way up the E string.
Lightly touch one third of the way up the
string and get three times the
fundamental frequency, sometimes done in
music to give a special tone color. Net,
Fourier series, harmonics, and overtones
are real everyday for violinists.

E.g., on a piano, hold down a key and then
play and release the key one octave lower
and notice that the strings of the key
held down still vibrate. The key
vibrating was stimulated by the first
overtone of the key struck and released.

The Fourier integral applies to functions
on the whole real line. Very careful math
is in Rudin, Real and Complex Analysis.

Yes, Fourier series and integrals can be
looked at as all about perpendicular
projections of rank 1 as emphasized in
Halmos, Finite Dimensional Vector
Spaces, written in 1942 when Halmos was
an assistant to John von Neumann at the
Institute for Advanced Study. That Halmos
book is a finite dimensional (linear
algebra) introduction to Hilbert space
apparently at least partly due to von
Neumann. So, right, Fourier theory can be
done in Hilbert space.

Fourier integrals and series are very
close both intuitively and mathematically,
one often an approximation to the other.
E.g., if multiply in one (time, frequency)
domain, then convolve in the other
(frequency, time) domain. E.g., take a
function on the whole real line, call it a
box, that is 0 everywhere but 1 on, say,
[-1,1]. Well the Fourier transform of the
box is a wave, roughly a bell curve, that
goes to zero quickly away from 0. A
convolution is just a moving weighted
average, usually a smoothing. Then given
a function on the whole real line, regard
that line as the time domain and
multiply by the box. Now can regard the
result as one period, under the box, of a
periodic function to which can apply
Fourier series. And in the frequency
domain, the Fourier transform of the
product is the smoothing with the Fourier
transform of the box of the Fourier
transform of the function and, then, an
approximation of Fourier series
coefficients of a periodic function with
the one period under the box. Nice. That
is partly why the fast Fourier transform
algorithm is presented as applying both to
Fourier series and the Fourier transform.

Mostly Fourier theory is done with an
L^2, that is, finite square integral,
assumption, but somewhere in my grad
school notes I have some of the theory
with just an L^1 assumption. Nice notes!

Essentially the Fourier transform of a
Gaussian bell curve is a Gaussian bell
curve -- if the curve is wide in one
domain, then it is narrow in the other.

The uncertainty principle in quantum
mechanics is just Plancherel's theorem
from Fourier theory.

Can do a lot with Fourier theory just with
little pictures such as for that box --
can get a lot of intuition for what is
actually correct.

I got all wound up with this Fourier stuff
when working on US Navy sonar signal
processing.

Then at one point I moved on to power
spectral analysis of wave forms, signals,
sample paths of stochastic processes, as
in

Blackman and Tukey, The Measurement of
Power Spectra: From the Point of View of
Communications Engineering.

Can get more on the relevant wave forms,
signals, stochastic processes from

Athanasuis Papoulis, Probability, Random
Variables, and Stochastic Processes, ISBN
07-048448-1.

with more on the math of the relevant
stochastic processes in a chapter of

J. L. Doob, Stochastic Processes.

Doob was long a leader in stochastic
processes in the US and the professor of
Halmos.

At one time a hot area for applications of
Fourier theory and the fast Fourier
transform was to looking for oil, that is,
mapping underground layers, as in

Enders A. Robinson, Multichannel Time
Series Analysis with Digital Computer
Programs.

Quickly antenna theory depends deeply on
Fourier theory so can do beam forming,
etc.

Can also see

Ron Bracewell, The Fourier Transform and
its Applications.

Of course, one application is to
holography. So, that's why can cut a
hologram in half and still get the whole
image, except with less resolution: The
cutting in half is like applying that box,
and the resulting Fourier transform is
just the same as before except smoothed
some by the Fourier transform of the box.

As I recall, in

David R. Brillinger, Time Series
Analysis: Data Analysis and Theory,
Expanded Edition, ISBN 0-8162-1150-7,

every time-invariant linear system
(maybe with some meager additional
assumptions) has sine waves as
eigenvectors. That it, feed in a sine
wave and, then, will get out a sine wave
with the same frequency but maybe with
amplitude and phase adjusted.

So, in a concert hall, the orchestra plays
and up in the cheap seats what hear is the
wave filtered by a convolution, that is,
with the amplitudes and phases of the
Fourier transform of the signal adjusted
by the characteristics of the concert
hall.

In particular, the usual audio tone
controls are essentially just such
adjustments of Fourier transform
amplitudes and phases.

Since there a lot of systems that are
time-invariant and linear or nearly so,
there is no shortage of applications of
Fourier theory.