Let $\rho(\vec{r})$ be the cosmic density field in a very large volume $V$. Then the fourier transformation of the field is:
$$ \delta(\vec{k}) = \frac{1}{V}\int_V \delta(\vec{r})e^{i \vec{k}\cdot \vec{r}}~d\vec{r} $$
However, in practice, we only know the discrete quantities. Suppose I have a discrete number density field defined as $\delta(\vec{r}) = \sum_j \delta^D(\vec{r} - \vec{r}_j)$, where $\vec{r}_j$ is the coordinate of object $j$, and $\delta^D(.)$ is the Dirac Delta Function.
I don't understand how the Fourier transform of this field is:
$$ \delta^d(\vec{k}) = \frac{1}{V \bar{n}}\int_V n(\vec{r})e^{i \vec{k}\cdot \vec{r}}~d\vec{r} - \delta^K_{k,0} $$ where, the superscript $d$ represents the discrete case of $\rho ( \vec{r})$ , and $\delta ^{\mathrm{K}\,}$ is the Kronecker delta, and $\bar{n}$ is the mean number density.
The related paper is here: https://iopscience.iop.org/article/10.1086/592079/fulltext/74517.text.html