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May 30, 2022 at 13:57 comment added Peter Humphries @asrxiiviii I wrote this proof up in Appendix A of my masters thesis, which is available on my website. From that, I see that the reference that I used for the Poisson summation formula was Theorem 3.6.3 of "Principles of Harmonic Analysis" by Deitmer and Echterhoff.
May 30, 2022 at 10:46 comment added asrxiiviii @Peter Humphries Thank you for the nicely explained answer. Could you kindly provide a reference for the Poisson summation formula for closed subgroups of $\mathbb T^n$ that you are using here? I am not very comfortable with the theory of topological groups and have been unable to find a reference. Thank you for your time.
Apr 15, 2014 at 19:01 vote accept Greg Martin
Apr 15, 2014 at 19:01 comment added Greg Martin I don't think at this point that there is a suitable reference in the literature, so I'm accepting this answer in recognition of all the trouble Peter went to.
Apr 9, 2014 at 18:59 history answered Peter Humphries CC BY-SA 3.0