I found myself trying to solve an equation of that kind :
$$ H f= R f, $$ where $f$ has to be found in $L^2(\mathbb{R})$, $H$ is the Hilbert transform and $R$ is a rational function having no poles on $\mathbb{R}$, and such that $|R(X)|\rightarrow +\infty$ as $|X|\rightarrow +\infty$. Since $H$ is an isometry on $L^2(\mathbb{R})$, I thought I could extract a necessary condition on the growth of $f$, but I am starting to think that evrything could happen for $f$. Are they any ideas or reference for these kind of equations ?
Thanks in advance !
Ayman
