This is actually an example of what is known as an inverse problem. You want to find a function $f$ that reproduces known behavior, rather than asking what behavior is produced by $f$.

One approach for doing this is to use an approach like reverse Monte Carlo, developed by Lyubartsev and Laaksonen. What they did was to try to find a potential $u(r)$ that reproduced a known radial distribution function $g(r)$. What you are looking to do is to find a function $f$ that reproduces ${\mathcal A}f = f$, so while the entirety of the approach may not be suitable, you will want to pay special attention to the iteration procedure they use—that is, how do they go from guess $u^{(k)}(r)$ to guess $u^{(k+1)}(r)$.

This is a classic example of what is known as an inverse problem. You want to find a function $f$ that reproduces known behavior, rather than asking what behavior is produced by $f$.

One approach for doing this is to use an approach like reverse Monte Carlo, developed by Lyubartsev and Laaksonen. What they did was to try to find a potential $u(r)$ that reproduced a known radial distribution function $g(r)$. What you are looking to do is to find a function $f$ that reproduces ${\mathcal A}f = f$, so while the entirety of the approach may not be suitable, you will want to pay special attention to the iteration procedure they use—that is, how do they go from guess $u^{(k)}(r)$ to guess $u^{(k+1)}(r)$.