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.