Consider the integral equation $$f=g^2+H[g]^2$$ where $f\colon\mathbf R\to \mathbf R$ is an even and integrable function, $g$ is the function to be solved for, and $H[g]$ is the Hilbert transform of $g$. Furthermore, $g$ is (should be) even and real-valued.

The equation can be rewritten in several ways, for example, $$P\int_{-\infty}^\infty g(y)\frac1{\pi(x-y)}dy = \sqrt{f(x)-g(x)^2}$$ whose general form is $$P\int_{-\infty}^\infty g(y)h(x-y)dy = \phi(f(x),g(x))$$ where $P\int$ denotes the principal value. Although this form it resembles a Fredholm equiation, I have not been able to find any litterature that covers it (integral equations are not my specialty). Therefore, I would greatly appreciate any pointers to litterature on the solution (especially numerical) of this kind of equations.

Best regards, Emil Hedevang