Let $G$ be an locally compact abelian group with $\Gamma$ a discrete cocompact subgroup. I'm looking for precise conditions by which Poisson summation formula holds. That is, for some function $f$ on $G$, possibly discountinuous, we have $$ \sum_{g \in \Gamma} f(g) = \sum_{g \in \Gamma} \hat{f}(g) $$ where $\hat{f}$ denotes the abelian Fourier transform of $f$ with respect to a fixed additive character of $G$ and self-dual with respect to the measure $dx$ on $G$. Most of the time one requires (1) $f$ to be in $L^1(G)$, and (2) for the dual sum to converge (absolutely, I think, because an interchange of integral and sum is involved).

Examples such as this answer suggest that we need (3) the sum $$ F(x) = \sum_{g\in\Gamma}f(xg) $$ to be continuous, at least at the identity $g = $ Id.

Most texts tend to assume $f$ is Schwartz or at least continuous. I haven't been able to find a good reference for these conditions for Poisson summation *without* this condition. (The closest I found was in Bourbaki XXXII.)

Also, if a second question is permitted, the continuity of $F(x)$ does not seem obvious how to prove, in particulr when $f$ itself is not continuous.

on$G$, and $\hat{f}$ on $\hat{G}$. I've conveniently chosen the character so that $\Gamma^\perp$ can be identified with $\Gamma$. $\endgroup$ – TA Wong Jun 22 '18 at 14:28