[Patient0]

This is a great explanation.

The only thing that I still can't get my head around or visualize is how it is that we know that the set of all cos and sin functions is sufficient to span this set of "non-pathological" functions.

So assuming some vector v is reachable by a linear combination of vectors u1,u2,u3... we know that each component should be v.u1, v.u2 etc. But how do you know that v is in fact "reachable" by some combination of vectors in the basis u?

That is, suppose I describe some (infinite) set of a functions. How do I determine the set of functions that it spans? Is there an intuitive way to picture why the set of sin and cos functions forms a suitable basis?

[ColinWright]

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

[Patient0]

ah thanks for answering the other questions I hadn't gotten around to asking (the reason why it's an orthonormal basis).

I think I need to go home and have a play with this now!

[JadeNB]

For an orthonormal set, one also gets another answer to your question: how can you see how much such a set spans?

Theorem: If S is an orthonormal (or even orthogonal) set in a Hilbert space V (here, L^2 functions on R/Z), then put S^\perp = {v in V : v is orthogonal to every s in S}. Then the span of S is (S^\perp)^\perp. (The containment of the span in the double-orthocomplement is formal; the other direction requires a supplemental theorem on the geometry of Hilbert spaces.)

With this in mind, we see that S is a topological basis (spans everything) when S^\perp = 0; so, to show that the sine and cosine functions span, it suffices to show that nothing (non-0) is orthogonal to all of them. This still requires computation, but at least it's easier to imagine doing this than somehow finding a Fourier series for any L^2 function.

[VLM]

"That is, suppose I describe some (infinite) set of a functions.... Is there an intuitive way to picture why the set of sin and cos functions forms a suitable basis?"

There are infinite expansions of trig functions to solve the finite vs infinite apparent mismatch, so don't worry about that.

An intuitive way to look at it, is on a 2-d graph there's no spot on the graph that can't be hit by a point of a triangle aka trig function collection, so it doesn't particularly matter how you pick any one spot, a triangle can always hit that one spot because it can hit all spots.

Comp sci analogy would be something like x=x+1 starting at x=0 will hit all the positive integers eventually...

[ColinWright]

That doesn't really work, because you have to be able to hit them all simultaneously, and in fact, you can't. The theorem says that for any given epsilon you can choose enough to ensure that you are within epsilon everywhere except in a very small section of the real line, and you can make the parts where it's not within epsilon very small.

The problem with your comment is that yes, for any given point you can get as close as you like, but the hard part is getting close nearly everywhere all at once. More, you then have to show that there's some sort of convergence, and that where you are close now is a superset of where you are close when you demand to be closer.

It's not straight forward.

[mnl]

http://en.wikipedia.org/wiki/Riesz%E2%80%93Fischer_theorem