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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?

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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.

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