provided the second sum is absolutely convergent. And if the Fourier transform is defined as .
The left-hand side is exactly what you want, so you have to check that the right-hand side matches the formula you're given. In other words, you have to compute the Fourier transform of .
This can be tedious, but there are tricks:
first, notice that , hence . You have to be careful here, since is not integrable, but only square-integrable.
Then you can notice that (by easy integration) where is the Heavyside function ( if and else). As a consequence, using Fourier inversion formula (in ), .
We deduce in (hence for almost every ), and hence for every because both functions are continuous (even for because ).
As a consequence, if and 0 otherwise.
The series is absolutely convergent because and . That's why we can apply the Poisson sommation formula. And it gives exactly what is expected.
By the way, if , the series would diverge.