Fourier Series application for dissertation Hello, I'm writing my 3 years degree on Fourier Series. I give an historical introduction, then prove Dirichlet's convergence theorem, Fejer's and the Du-Bois Reymond counterexample of a continuos function with divergent Fourier series at one point. Then I'd like, for the last chapter, to give an application of Fourier Series. Do you have any suggestions for any such application?
The problem is all the (interesting) things I thought so far involve the Fourier Transform, which I know but since I don't have time/space to introduce it in my dissertation, I'd really like something using only the series.
Since in the first chapter I define the model of the string with fixed endpoints and give solutions for it (this was the subject of a controversy between Euler,d'Alambert and D.Bernoulli which somehow leads to F.series), it would be cool if the application could be something that's like an evolution or a more complex /real world version of this basic string model.
Any ideas/suggestions?
EDIT: asking here has proven to be very useful! Thanks to all your suggestions; even those that won't fit in my dissertation have been useful and I might come back to them in the future. Although I was asking for something real world/physical, I guess I've fallen in love with Weyl's equidistribution theorem, and I'll go for that. Again thanks.
 A: You can use Fourier series to prove Weyl equidistribution theorems. Take any irrational number $a$ and look at the fractional parts of $a,2a,3a,...$. Then this sequence is equidistributed in $[0,1]$. This is a special case of the ergodic theorem and is fairly straight forward to prove. Unless you have seen ergodic theory before it's a pretty darn surprising application of Fourier series. See for example Stein and Shakarchi's Fourier Analysis book for a reference. 
A: I strongly suggest you look at Dym and McKean's "Fourier Series and Integrals". They have lots of really nice applications (of both).
A: Fourier series are useful (and sometimes essential) for solving/understanding many problems involving periodic functions on $\mathbb{R}$ or, equivalently, functions $f$ on $[a,b]$ such that $f(a)=f(b)$. I was going to say almost all problems, but that's probably an exaggeration. Of course it helps if the problem is linear, and the properties of the functions you're considering can be easily expressed in terms of the Fourier coefficients -- but even then these restrictions are not always essential. 
e.g. the Heat Equation on a (physical) ring, where periodicity is assured by the shape of the space; (Willie Wong already mentioned this in the comments).
My favourite one: proving the Isoperimetric Inequality, that the circle has the largest area of all piecewise $C^1$ curves with given perimeter;
The functional equation for the Riemann Zeta Function $\zeta$: one proof involves the Fourier expansion of the sawtooth function $x - [x]$, which I think I saw in E. C. Titchmarsh's old book The Theory of the Riemann Zeta Function (although I'm sure many other books will give it also).
I think it was either Hardy or Littlewood (or maybe both?!) who said that a periodic function should always be expanded as a Fourier series; if you always follow this rule then it'll solve a lot of problems automatically!
Although one should be cautious; "if the only tool you have is a hammer, then everything looks like a nail"...
A: Körner's book "Fourier analysis" has a lot of interesting material. It's divided into 110 chapters, each of which is a few pages long and presents an aspect of the subject. For example, one is about ""Mathematical Brownian Motion" while another is about "Compass and Tides". At least a third of the book uses "only" Fourier series.
