Proving the Poisson Summation Formula: A Formal Approach

[stripes]

Homework Statement

Prove the Poisson summation formula.

Homework Equations

The Attempt at a Solution

Correction to image below: the very last line of the theorem (italicized) should say f hat is the Fourier transform, not f(n).

Does this proof make sense and is it complete? I mean there are a million proofs online that are identical to mine but I am presenting this in a seminar tomorrow for my class so I wanted to make sure it was descriptive enough.

 

[CompuChip]

I have a few small remarks.

First of all, I'd make the change of variables explicit (saying that you change to y = x + n).

Secondly, if you make that change, you should also change it in the exponent, so $e^{2\pi i m x}$ becomes $e^{2 \pi i m (y - n)}$. You should then argue that the n can be dropped and replace the x with y in the next lines so that in the last line you get $e^{2 \pi i m y}$.

Finally, I wouldn't say "... the RHS is simple" in a formal proof. It's not an objective statement and doesn't add anything, just write "For the right hand side, notice that ...".

Also, what is F? And where are you using its periodicity?