Lets say I have a function $f(x,y)$ and its corresponding Fourier transform $F(u,v)$. I'm trying to show that $f(x+by,dx + y) \leftrightarrow \frac{1}{|\Delta|}F(\frac{u-dv}{\Delta},\frac{-bu+v}{\Delta} )$ where $\Delta = 1 - bd$ but plugging into the formula for the Fourier transform is not helping because I do not know what the function $f(x,y)$ is.
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You can make the change of variables in that formula, as is done here. You don't need to know what a function is to change variables in the integral. The number $\Delta$ is the Jacobian determinant, which you see in the linked answer.
Also, it seems to me that the argument of $F$ should have $\frac{1}{\Delta}(u-bv, -du+v)$, because this is the inverse of linear transformation $(x,y)\mapsto (x+by,dx+y)$.
