OK, three people have mentioned some more-or-less "general" properties
of Fourier analysis that make it special.
By Fourier "analysis" I mean to include Fourier series, Fourier
transforms, and suchlike.
I mentioned periodicity, and said that the periodicity need not
be wavelike. Well, that was misleading. That was at best only a
half-step in the right direction. The task need not be periodic
(let alone wavelike) to be amenable to Fourier analysis. Sorry
about that.
David B. mentioned derivatives and translational invariance. In
classical mechanics, the derivative operator is the generator of
translations. (In QM, we say the momentum operator is the generator
of translations, which comes to the same thing.) That's true in
free space, and tremendously important. But unless I'm missing
something, the fact that the Fourier transform has a nice way of
dealing with translations is only one of its nice properties.
This is not "the" key to understanding Fourier analysis, especially
in non-free space (such as waves on a finite string, where we do
not have true translational invariance).
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Let me add another way of looking at things. Here is another important
property of Fourier transforms: The sinusoidal waves are a complete
set of orthonormal basis functions. They form a basis in function-space.
I'm not going to fall into the trap of suggesting this is "the"
crucial nice property. It is merely /another/ nice property. It
has the advantage of applying in situations that are:
-- wavelike or not
-- periodic or not
-- translationally invariant or not
-- finite domain or not
(although the normalization is tricky in the infinite case)
The Fourier basis is by no means the only orthonormal basis in
function-space. The Legendre polynomials are another well-known
basis (on a finite interval).
Of course your notion of "orthonormal" depends on your notion of
inner product.
-- If the inner product is ∫...dx you get Fourier sinusoids or
Legendre polynomials.
-- If the inner product is ∫...x dx you get Bessel functions.
-- Et cetera............
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It is important to be able to look at things from more than one
point of view. For waves on a string, each of the following is
true and nontrivial:
-- The Fourier transform is linear and the sinusoids provide
an orthonormal basis. (This implies Parseval's theorem.)
-- The sinusoids are periodic (unlike, say, Legendre polynomials).
-- The transform provides a simple representation of d/dt and d/dx.
-- The transform provides a nice representation of convolution and
correlation.
None of those statements implies any of the others. They are
different views of the same elephant. There are probably other
valuable views that I haven't mentioned.
Depending on the application, you might find that only a small
subset of these properties is sufficient to make Fourier analysis
worthwhile.