Basis for the fine grid

In addition, we define a set of fine grid basis functions with twice the cut-off frequency as the $D(\mathbf{r})$:

$$B_{XYZ}(\mathbf{r}) = \frac{1}{8N_{1}N_{2}N_{3}}\sum_{p=-N_{... ...{B}_{2}+s\mathbf{B}_{3})\cdot(\mathbf{r} - \mathbf{r}_{XYZ})},$$ (53)

that correspond to a plane-wave representation that has twice the wavevector cutoff of the standard basis functions. $\{\mathbf{r}_{XYZ}\}$ are the points of the fine grid. This basis is localised on the grid points of the fine grid and is also orthogonal:

$$B_{XYZ}(\mathbf{r}_{ABC}) = \delta_{XA}\delta_{YB}\delta_{ZC}$$ (54)

$$\int_{V} d\mathbf{r} \: B_{XYZ}^{\ast}(\mathbf{r})B_{ABC}(\mathbf{r}) = \frac{\Omega}{8} \delta_{AX}\delta_{BY}\delta_{CZ},$$ (55)

and satisfies a property analogous to equation (52):

$$\int_{V} d\mathbf{r} \: f^{\ast}(\mathbf{r})B_{XYZ}(\mathbf{r}) = \int_{V} d\mathbf{r} \: f_{B}^{\ast}(\mathbf{r})B_{XYZ}(\mathbf{r})$$

$$= \frac{\Omega}{8}f_{B}(\mathbf{r}_{XYZ}),$$ (56)

where $f_{B}(\mathbf{r})$ is a version of the function $f(\mathbf{r})$ that is bandwidth limited to the same frequencies as the basis functions $B(\mathbf{r})$.