# Conditions for Poisson summation (for discontinuous functions)

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.

• I have trouble parsing your first formula, if $f$ is a function in $G$, $\hat f \in \hat G$, how do you know that $\Gamma \leq \hat G$? – Adrián González-Pérez Jun 22 '18 at 14:01
• In the versions I know of Poisson summation formula in LCA groups, for example in Folland's Abstract Harmonic Anaysis Thm 4.42 what you do is to prove that the averaging operator $f \mapsto F$ and the restriction operator $f \mapsto f{|}_{\Gamma^\perp}$ are intertwined by the Fourier transform. In order to make sense of your result you will need the restriction $f{|}_{H^\perp} to be defined. – Adrián González-Pérez Jun 22 '18 at 14:19 • There are subgroup-restriction results that work for non-continuous functions, for example De Leeuw's original formulation of the restriction theorem required that every value$f(x)$shall be the limit of the averages on balls centered on$x$, but I do not known if you can prove Poisson's formula with that type of condition. – Adrián González-Pérez Jun 22 '18 at 14:20 • You mean that$f$is a function on$G$, and$\hat{f}$on$\hat{G}$. I've conveniently chosen the character so that$\Gamma^\perp$can be identified with$\Gamma\$. – TA Wong Jun 22 '18 at 14:28
• Yes, there was a typo in my first comment. – Adrián González-Pérez Jun 22 '18 at 14:40