Thursday, 11 April 2013

analysis - Smart tricks used to prove formulas by approximation

I've recently seen a proof of the inversion formula for the Fourier transform for f,ˆfL1(Rd).
The main idea of the proof is this




We have
Rde2πixξˆf(ξ)dξ=Rde2πixξ(Rde2πiyξdy)dξ. But at this point Fubini doesn't apply thus we can't change the two integrals.
Nevertheless, here is the trick (which I find really smart) for t>0 we consider this modified version of the integral
It(x):=Rdˆf(ξ)eπt2|ξ|2e2πixξdξ.

Now we can use Fubini and compute the integral in two ways and for t0 we get the result. By the way everything works because the damping factor is an eigenvector of the Fourier transform of eigenvalue 1 (so it is far from being a random choice).




I was astonished by the proof, we considered an apparently more complicated expression which allowed us to use a particular property and then passed to the limit and obtained the result.
My question is: what are other examples of the same idea?
I know that basically anything which involves approximation with smooth functions falls in this class but I'm asking for some particular example/application (of this or a similar idea) that made you think 'cool'.
(details are welcome).

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