The poisson resummation formula lets us re-write the following sums:
$$ \sum_{n \in \mathbb{Z}} f(n) = \sum_{n \in \mathbb{Z}} \hat{f}(n)$$
Where $\hat{f}(y) = \int_{-\infty}^{\infty} f(x)e^{-2\pi i xy} dx $ is the fourier transform.
The has been a lot of work in generalizing this summation formula to groups that are not the integers such as the topics discussed here. But I was interested in a more elementary question. Do we know any linear or non linear operators $O$ that are not the fourier transform or one of its 4 composites and if there are multiple is there a way to characterize all of them?
$$ \sum_{n \in \mathbb{Z}} f(n) = \sum_{n \in \mathbb{Z}} O[f](n)$$
A starting point might be to consider fractional fourier transforms. Although it seems like a lot of effort to check if this respects the summation formula.
From first principles another option is to consider a function $H(x,y)$ such that for each $y \in \mathbb{Z}$, $H(x,y)$ forms an orthonormal basis with the inner product of $\langle a, b \rangle = \int_{-q}^{q} a*b$ for some choice of constant (possibly infinity) $q$ , and $H(0,y) =1$ if $y \in \mathbb{Z}$ and from here one can easily repeat the exact steps of this proof
So at the very least there is this family (which we haven't yet shown contains anything OTHER than the fourier transform) but most likely is not empty.
If we adjust our question to allow a weight term such as in the original poisson resummation formula of $\sum_{n \in \mathbb{Z}} f(x+n) = \sum_{n \in \mathbb{Z}} \hat{f}(n)e^{2\pi i nx}$ then the family above is definitely not EMPTY (we just dropped the $H(0,y)=1$ constraint).
With or without a weight term I don't believe this covers the entire linear operator case but at least it covers an uncountable set of such operators.
The non-linear case I haven't the faintest idea how to proceed since linearity seems quite essential to the proof of the poisson resummation formula.