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Poisson summation formula "trace in two ways" proof
See Sarnak course notes on SPECTRAL THEORY OF AUTOMORPHIC FORMS, written by Tony Feng and Gyujin Oh (within PDF, search "Poisson summation" to get the relevant parts).
Googling the title also turns up Rahul Krishna's NOTES ON THE TRACE FORMULA (within PDF, search "Poisson summation" to get the relevant parts).
Using the Poisson summation formula (PSF)
See MO: How does one use the Poisson summation formula?, and the previous links, for some nice examples and discussion.
Note that PSF can be the first step in approximating a discrete sum by an integral.
Uncertainty principle viewpoint of PSF
See e.g. Tao's blog post on the uncertainty principle.