# Decoupling of double discrete Fourier transform

I have a problem with a double Fourier transform I encountered:

$$\sum_{j=1}^L \sum_{l=1}^L e^{-i\pi \frac{n_1}{L} (j+l)}e^{-i\pi \frac{n_2}{L} (j-l)}V(j-l)$$

where $n_1,n_2$ are integer. If the sums over $j$ and $l$ were integrals, so not discrete but continuous Fourier transforms, it would easily be possible to decouple the two integrations by a change of variable and obtain a delta function times the Fourier transform of $V$. Would this be possible in the case of the discrete version as well?

I'm not sure if this will make the Fourier transform easier, but nevertheless you should be able to do a change of variables in the vein of $n = j+l$ and $m = j-l$. My advice would be to create a table of $j$,$l$,$j+l$, and $j-l$. What you will find is that there is a weighting function associated with each variable. For instance, the $(j,l)$ pairs $(1,1)$, $(2,2)$, $(3,3)$, etc.. all correspond to $m = 0$. and there are $L$ number of these. For $m=1$, there is $L-1$ ways, and so on. The range of $m$ will then be between $-L$ to $L$.
You can do a similar argument for $j+l$. The weighting function will be different however and range will go from $2$ to $2L$.
Start out simple with say an $L=3$ case and you should get the methodology. Then it is easy to expand to the general case. Just keep in mind that there are $L^2$ terms and all need to be accounted for during the change in variable.