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.

  • $\begingroup$ I have trouble parsing your first formula, if $f$ is a function in $G$, $\hat f \in \hat G$, how do you know that $\Gamma \leq \hat G$? $\endgroup$ – Adrián González-Pérez Jun 22 '18 at 14:01
  • $\begingroup$ 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. $\endgroup$ – Adrián González-Pérez Jun 22 '18 at 14:19
  • $\begingroup$ 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. $\endgroup$ – Adrián González-Pérez Jun 22 '18 at 14:20
  • $\begingroup$ 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 $\Gamma$. $\endgroup$ – TA Wong Jun 22 '18 at 14:28
  • $\begingroup$ Yes, there was a typo in my first comment. $\endgroup$ – Adrián González-Pérez Jun 22 '18 at 14:40

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