Intuition behind Fourier coefficients

I'm glad you reposted this here; I saw it on MathOverflow earlier and wanted to answer, but it was closed (and it really is more appropriate here.)

As the answer there suggested, perhaps the best way to think about this is in terms of linear algebra.

Your sines and cosines form a complete orthonormal basis for your space of functions. The inner product on this space is $<f,g> = \int f(x)\bar{g(x)}dx$, and this gives a direct geometric interpretation of the integrals involved in calculating $a_n$ and $b_n$: you are simply projecting your function down onto a given basis vector to see "how much" of it is in "that direction." The $2/T$ factor out front comes from normalizing the basis vectors.

(For more details, see Bessel's inequality and Parseval's identity, or a good book like Stein and Shakarchi's 'Fourier Analysis: An introduction.' They have a wonderfully explicit visualization of this type on page 78, where they prove that partial sums $S_N(x)$ of Fourier series of a function f(x) are the best approximation possible with a trigonometric polynomial of order at most $N$.)

Your Fourier series, then, is equivalent to 'building up' f(x) from (an infinite linear combination of) basis vectors in the space, and the coefficients $a_n$ and $b_n$ tell you how much of each you need. The fact that you can do this at all is the content of Parseval's identity.

On mathoverflow.net Carlo Beenakker recommended http://techhouse.brown.edu/~dmorris/projects/tutorials/fourier_tutorial.pdf. He hit the nail (right) on the head.

I like to think of Fourier coefficients in terms of an equalizer in a stereo system. In an equalizer, you have a bunch of dials that provide an amplitude of a certain frequency. (Actually, I think it is the square of the amplitude, but never mind.) That is, you may have a $40$ Hz dial, a $100$ Hz dial, etc. The set of amplitudes of the dials provides a unique representation of a volume function in terms of frequency components (albeit, just a few). Some functions have more treble, some more bass, but the equalizer provides a representation of the volume function in such a way as to provide control over certain frequency amplitudes.

The Fourier coefficients of a function represent the settings of some function equalizer. That equalizer stresses certain frequency components and ignores others. Of course, there are an infinite (but countable) set of dials for a periodic function. What happens is that the dial settings for the highest frequencies tend toward zero as a necessity for being a Fourier representation. The zero frequency, on the other hand, may be the volume control (although perhaps that scales all of the coefficients.)