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.
Best Answer
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.