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Perhaps surprisingly, I don't use Poisson summation per se all that often, but I do repeatedly use the more general principle that Poisson summation exemplifies, namely that the Fourier transform intertwines restriction and projection. Restricting a function $f: G \to {\bf C}$ on a group G to a subgroup H corresponds to projecting out the Fourier transform $\hat f: \hat G \to {\bf C}$ along the orthogonal complement $H^\perp$ (all the characters in $\hat G$ that annihilate H). Conversely, averaging out f along H corresponds to restricting $\hat f$ to $H^\perp$. Plugging in $G = {\bf R}$ and $H = {\bf Z}$ more or less gives the classical Poisson summation formula.

The same principle also applies to approximate groups, such as balls: localising a function in physical space to a ball of scale r corresponds to averaging the Fourier transform at scale 1/r, and conversely; this already explains the uncertainty principle to a large extent, and also is the starting point for Littlewood-Paley theory. Or, localising the zeroes of the Riemann zeta function to a strip of height T corresponds to understanding the distribution of the (logarithm of the) primes at scales 1/T and above; understanding the evolution of the wave equation up to time T controls the eigenvalues of the associated Laplacian at scales 1/T and above; and so on and so forth. So, in general, I know that fine scale behaviour of physical space is tied to coarse scale behaviour of frequency space and vice versa; and whenever I need to formalise this sort of intuition, I have to reach for something like a Poisson summation formula (though, as I said above, I rarely use that formula directly, but usually some variant that is jury-rigged from convolutions and cutoff operators).

[To interpret the classical Poisson summation formula as a variant of the uncertainty principle, one has to use an "adelic" topology and view the integers in physical space as "small", and the integers in frequency space as "low frequencies". I personally find this perspective helpful, but this may be because I come from a real-variable harmonic analysis background.]

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Perhaps surprisingly, I don't use Poisson summation per se all that often, but I do repeatedly use the more general principle that Poisson summation exemplifies, namely that the Fourier transform intertwines restriction and projection. Restricting a function $f: G \to {\bf C}$ on a group G to a subgroup H corresponds to projecting out the Fourier transform $\hat f: \hat G \to {\bf C}$ along the orthogonal complement $H^\perp$ (all the characters in $\hat G$ that annihilate H). Conversely, averaging out f along H corresponds to restricting $\hat f$ to $H^\perp$. Plugging in $G = {\bf R}$ and $H = {\bf Z}$ more or less gives the classical Poisson summation formula.

The same principle also applies to approximate groups, such as balls: localising a function in physical space to a ball of scale r corresponds to averaging the Fourier transform at scale 1/r, and conversely; this already explains the uncertainty principle to a large extent, and also is the starting point for Littlewood-Paley theory. Or, localising the zeroes of the Riemann zeta function to a strip of height T corresponds to understanding the distribution of the (logarithm of the) primes at scales 1/T and above; understanding the evolution of the wave equation up to time T controls the eigenvalues of the associated Laplacian at scales 1/T and above; and so on and so forth. So, in general, I know that fine scale behaviour of physical space is tied to coarse scale behaviour of frequency space and vice versa; and whenever I need to formalise this sort of intuition, I have to reach for something like a Poisson summation formula (though, as I said above, I rarely use that formula directly, but usually some variant that is jury-rigged from convolutions and cutoff operators).