Geodesic
A Discrete Analog of the Poisson Summation Formula
Matematičeskie zametki, Tome 73 (2003) no. 1, pp. 106-112

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The first part of this paper is concerned with the proof of a discrete analog of the Poisson summation formula. In the second part, we describe an elementary proof of a functional equation for the function $\theta(t)$, based on the summation formula derived in the paper. 
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     title = {A {Discrete} {Analog} of the {Poisson} {Summation} {Formula}},
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A. V. Ustinov. A Discrete Analog of the Poisson Summation Formula. Matematičeskie zametki, Tome 73 (2003) no. 1, pp. 106-112. http://geodesic.mathdoc.fr/item/MZM_2003_73_1_a8/