International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 4243

Let , so that . Let R be the lattice distribution associated to lattice Λ, with period matrix A, and let be associated to the reciprocal lattice . Then we may write: i.e.
This identity, which also holds for , is called the Poisson summation formula. Its usefulness follows from the fact that the speed of decrease at infinity of ϕ and are inversely related (Section 1.3.2.4.4.3), so that if one of the series (say, the lefthand side) is slowly convergent, the other (say, the righthand side) will be rapidly convergent. This procedure has been used by Ewald (1921) [see also Bertaut (1952), Born & Huang (1954)] to evaluate lattice sums (Madelung constants) involved in the calculation of the internal electrostatic energy of crystals (see Chapter 3.4 in this volume on convergence acceleration techniques for crystallographic lattice sums).
When ϕ is a multivariate Gaussian then and Poisson's summation formula for a lattice with period matrix A reads: or equivalently with
References
Bertaut, E. F. (1952). L'énergie électrostatique de réseaux ioniques. J. Phys. Radium, 13, 499–505.Google ScholarBorn, M. & Huang, K. (1954). Dynamical theory of crystal lattices. Oxford University Press.Google Scholar
Ewald, P. P. (1921). Die Berechnung optischer und electrostatischer Gitterpotentiale. Ann. Phys. Leipzig, 64, 253–287.Google Scholar