This was Fourier's big theorem, that every periodic, non-pathological function is "simply" the sum of phase-shifted sine waves of all necessary frequencies. You can remove the "periodic" if you allow infinitely many frequencies, and so on. The phase-shifting is taken care of by having a sine and a cosine at the same frequency, but (possibly) different amplitudes.

So start with a periodic wave and look at how much sin(x) is in it. Do this by integrating:

    Integral from 0 to 2.pi of f(x)*sin(x) dx

That's a dot product of your function f(x) with the sine wave, and that tells you how much of the first frequency you need. Subtract off the result, and then go again with sin(2x). You find the residuals get less and less. More, for something nice like a square wave or triangular wave the coefficients you get form a predictable sequence.

Interesting note:

    integral sin(k.x)*sin(m.x)

is 0 if k != m, and 1 otherwise, so the basis elements have dot-product 0, and hence are thought of as being at right angles. They also have "length" 1, since the dot-product with themselves is 1. So they are an ortho-normal basis.

So the question is: what functions can be reached by adding and subtracting multiples of sine (and cosine) functions of different frequencies? In Linear Algebra terms, what is the space of functions spanned by these basis functions?

That's harder, but it turns out to be "everything non-pathological".

http://en.wikipedia.org/wiki/Fourier_series#Hilbert_space_in...

http://en.wikipedia.org/wiki/Hilbert_space#Fourier_analysis

Edited for typos