Peter Conlon
Frequent poster

Post Number: 78

Posted on Sunday, 21 November, 2004 - 12:26 pm:

That

\sin (n π x / L)

\sin (m π x / L)

and

\cos (n π x / L)

are orthogonal can be easily shown by integration over the interval

L

But how do we know that the set of functions (say, $\sin (n π x / L)$ ) is sufficient to express every odd function as a linear combination? How can we show that the $\sin$ and $\cos$ functions as they are used in Fourier series actually span the function space they are used to describe? (Hope that makes sense).

Peter

David Loeffler
Veteran poster

Post Number: 1089

Posted on Sunday, 21 November, 2004 - 02:31 pm:

That's a very good question. Answering it requires some quite advanced techniques (depending on what generality of functions you want to work with). Look up the Stone-Weierstrass theorem if you want to know the standard approach.

Morally the reason it works is that the Dirichlet kernel
Latex image

satisfies

(just plug the integral definition of c_n into the sum) and D_n has integral 2pi and eventually becomes more and more concentrated in a short interval around 0, so the interval "picks out" the value of f(x). Plot a graph of D_n (x) for a few values of n to see what I mean.

(You can think of D_n (x) to be the first few terms of the series for the Dirac delta function if you like. Just don't regard this as a rigorous proof of anything: that way lies madness, or at least applied mathematics

)

You can in fact use this argument to get a direct formal proof of the validity of the Fourier expansion but it's rather fiddly, which is why most books on the subject do it indirectly via Stone-Weierstrass.

David