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The Poisson summation formula says that for a Schwartz function $f : \mathbf R^d \to \mathbf R$ and its Fourier transform $\widehat f$, we have $$\sum_{n \in \mathbf Z^d} f(x) = \sum_{n \in \mathbf Z^d} \widehat f(x).$$ Can this formula be explained conceptually along the lines of the following:

  • we have a short exact sequence of groups $$0 \to \mathbf Z^d \to \mathbf R^d \stackrel \pi \to (\mathbf R/\mathbf Z)^d \to 0,$$
  • the Pontryagin duals of these groups is the "same" exact sequence, but with arrows reversed.
  • some sort of functoriality of Pontryagin duality.

Thanks!

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    $\begingroup$ Yes. That's Section 3.6 in Deitmar/Echterhoff: Principles of Harmonic Analysis. Springer 2014 $\endgroup$
    – user1688
    Commented Jan 21, 2016 at 16:16

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