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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.

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Just a comment: the Fourier method is most applicable when the evolution equations are linear, so this puts some limit on the level of complexity you can have. You may want to look into some applications in waveguides, or perhaps something like solving the heat equation on the torus. –  Willie Wong Feb 20 '11 at 15:03
An interesting application is deriving series for $\pi$ - try plugging in values like $0$, $\pi/4$, to the Fourier series of $x$ or $\lvert x\rvert$, for example. Or you can use them in a similar fashion to find the value of infinite sums, such as $\sum\frac{1}{n^2}=\frac{\pi^2}{6}$. –  Thomas Bloom Feb 20 '11 at 19:23
@Thomas: yes, those are interesting results and I may include some of those... but I was looking for a more real-world example –  nareto Feb 22 '11 at 10:40
@Willie: the torus thing sounds very interesting to me... unfortunately I know nothing of thermodynamics (which is bad because it would have really fitted well in my thesis given Fourier's treatise on heat), but OTOH I've taken a course in electromagnetism - where could I find examples of Fourier series applied to linear em equations? I'm afraid to say something terribly wrong as I seem to remember that em equations generally aren't linear –  nareto Feb 22 '11 at 10:41
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4 Answers

up vote 3 down vote accepted

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.

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I was looking for a real-world example but wow, this looks really cool. I'll certainly look into this! thanks –  nareto Feb 22 '11 at 10:50
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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.

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Yes, this is a nice book. By the way, there is the companion "Exercises in Fourier Analysis" by the same author. –  Andrey Rekalo Feb 21 '11 at 10:10
Yes, how could I forget of this :) I had it some time ago from the library, actually used it for Fejer's proof. I'll look again into it, thanks for the suggestion. –  nareto Feb 22 '11 at 10:31
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I strongly suggest you look at Dym and McKean's "Fourier Series and Integrals". They have lots of really nice applications (of both).

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+1. That's a great book! Unfortunately it appears to be out of print. –  Andrey Rekalo Feb 20 '11 at 21:46
You can get it on amazon.com, so I don't think it is actually out of print. If you email me, I can send you an electronic version. I am also guessing that the OP's university has it in the library. –  Igor Rivin Feb 20 '11 at 23:46
@Igor Rivin: Thanks a lot! I do have a copy of the book. –  Andrey Rekalo Feb 21 '11 at 9:49
thanks, seems good. I'll have it in a few days from the library. –  nareto Feb 22 '11 at 10:29
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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"...

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I especially back up the planar isoperimetric inequality: the proof is really simple and plainly relies on Parceval's equality, plus a little bit of Stokes theorem. It can for example be found here: cornellmath.wordpress.com/2008/05/16/… –  Benoît Kloeckner Feb 21 '11 at 10:03
although I think I won't go for this in the end, I want to thank you for your suggestions because they indeed seem interesting applications; the proof in Benoit's link does really seem fairly straightforward. –  nareto Feb 22 '11 at 11:05
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