Figure 3.

Numerical and analytical representations for sampling. A representation in Fourier space of a single projection with Fourier magnitudes less than L. Lattice sites, shown as small black spheres in the k_z = 0 plane are considered to be sampled. Our coarse-grained approximation is that the entire slab of unit width containing those points are sampled: $\text{sp}(\overrightarrow{k})=\Theta \left(|k_z|\le \frac{1}{2}\right)$ where Θ is the indicator function. Our numerical algorithm hinges on finding lattice sites that satisfy the generalization of this condition: $\text{sp}(\overrightarrow{k})=\Theta \left(|\widehat{n}\cdot \overrightarrow{k}|\le \frac{1}{2}\right)$ for any arbitrary projection direction given by $\widehat{n}$. For analytical calculations, we further approximate this as a delta function $\text{sp}(\overrightarrow{k})=\delta (\widehat{n}\cdot \overrightarrow{k})$, which satisfies the proper normalization.