Let $g(x,y)$ have Fourier transform $G(k_x, k_y)$.
I am interested in the Fourier transform of $g(x - \frac{xy}{f}, y)$ for some $f > 0$.
I've gotten part of the way there using the separability and dilation properties of the Fourier transform, but am stuck past a certain point:
\begin{align} & \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} g(x - \frac{xy}{f}, y) e^{-i2\pi(xk_x + yk_y)} \mathrm d x \mathrm dy \\ &= \int_{-\infty}^{+\infty} \left[\int_{-\infty}^{+\infty} g\left(x(1-\frac{y}{f}), y\right) e^{-i2\pi xk_x} \mathrm dx \right] e^{-i2\pi y k_y} \mathrm dy \\ &= \int_{-\infty}^{+\infty} \frac{1}{\left| 1 - \frac{y}{f} \right|} G\left(\frac{k_x}{1-\frac{y}{f}}, y\right) e^{-i2\pi y k_y} \mathrm dy \end{align}
I'm unsure how to proceed from here — I could try integration by substitution with $u = 1 - \frac{y}{f}$, but this gives an integral that I'm unsure how to tackle further:
\begin{align} & \int_{+\infty}^{-\infty} \frac{1}{|u|} G\left(\frac{k_x}{u}, f(1-u)\right) e^{-i2\pi f(1-u)k_y} (-f) \mathrm du \\ &= f e^{-i2\pi f} \int_{-\infty}^{+\infty} \frac{1}{|u|} G\left(\frac{k_x}{u}, f(1-u)\right) e^{-i2\pi u (-fk_y)} \mathrm du \end{align}
EDIT: It's also interesting to simply consider the Fourier transform of $g(x - \frac{xy}{f}, y)$ at $k_y=0$. Perhaps this integral is easier to solve?
\begin{align} & \int_{-\infty}^{+\infty} \frac{1}{\left| 1 - \frac{y}{f} \right|} G\left(\frac{k_x}{1-\frac{y}{f}}, y\right) \mathrm dy \\ &= f \int_{-\infty}^{\infty} \frac{1}{|u|} G\left( \frac{k_x}{u}, f(1-u) \right) \mathrm du \end{align}