I also don't understand if the double sum screws things up.
2.
The Direct Calculation of the 2-D DFT is simply the evaluation of the double Sum
3.
For large N, for each of these N planes, there are two neighbours m planes away, so the above double sum becomes a single sum over pairs of neighbours either side of an atom, at positions-m and m lattice spacings away, times N.
4.
The DFT double sum can be decomposed into four separate summations, one over those samples of x for which both n _ 1 and n _ 2 are even, one for which n _ 1 is even and n _ 2 is odd, one for which n _ 1 is odd and n _ 2 is even and the last one for which n _ 1 and n _ 2 are odd.
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