I have just completed an introductory course on analysis, and have been looking over my notes for the year.  For me, although it was certainly not the most powerful or important theorem which we covered, the most striking application was the Fourier analytic proof of the isoperimetric inequality.  I understand the proof, but I still have no feeling for why anyone would think to use Fourier analysis to approach this problem. I am looking for a clear reason why someone would look at this and think "a Fourier transform would simplify things here".  Even better would be a physical interpretation.  Could this somehow be related to "hearing the shape of a drum"? Is there any larger story here?
Experience with Fourier analysis and representation theory has shown that every time a problem is invariant with respect to a group symmetry, the representation theory of that group is likely to be relevant.  If the group is abelian, the representation theory is given by the Fourier transform on that group.

In this case, the relevant symmetry group is that of reparameterising the arclength parameterisation of the perimeter by translation.  This operation does not change the area or the perimeter.  When combined with the observation (from Stokes theorem) that both the area and perimeter of a body can be easily recovered from the arclength parameterisation, this naturally suggests to use Fourier analysis in the arclength variable.
