I've recently seen a proof of the inversion formula for the Fourier transform for f,ˆf∈L1(Rd).
The main idea of the proof is this
We have
∫Rde2πix⋅ξˆf(ξ)dξ=∫Rde2πix⋅ξ(∫Rde−2πiy⋅ξdy)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)e−πt2|ξ|2e2πix⋅ξdξ.
Now we can use Fubini and compute the integral in two ways and for t→0 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).
No comments:
Post a Comment