We have the Poisson summation formula for $f:\mathbb{R} \to \mathbb{R}$ with decent decay for both $f$, $\hat f$.
$\sum f(i) = \sum \hat f(i)$
There is an analogous each time fourier shows up.
For instance in boolean functions (i.e the group being $F_2^n$)-
if $U$ is a subspace, then
$\sum_{x\in U} f(x) = \sum_{x\in U^{\perp}} \hat f(x)$
The proof is clean- $<1_U, f> = <1_U^{\perp}, \hat f>$
I would love for a similiar proof to hold in $\mathbb{R}$, but I don't know enough math to formalize it. There doesn't seem to be a way to directly formalize it, since no inner product would give what I want, but if we think of inner product as a functional maybe there is hope.
I think we want to consider the functional of the sum of deltas over the integers (which is defined on say the schwartz functions). Is there a way to make this work? Sorry for being so vague, I feel like we want to define the fourier of a functional...