Rudin’s orthogonality problem and the Nevanlinna counting function

Abstract:

Let $\phi$ be a holomorphic function taking the open unit disk $U$ into itself. We show that the set of nonnegative powers of $\phi$ is orthogonal in $L^2(\partial U)$ if and only if the Nevanlinna counting function of $\phi$, $N_\phi$, is essentially radial. As a corollary, we obtain that the orthogonality of $\{\phi ^n: n=0,1,2,\ldots \}$ for a univalent $\phi$ implies $\phi (z) = \alpha z$ for some constant $\alpha$. We also show that if $\{\phi ^n: n=0,1,2,\ldots \}$ is orthogonal, then the closure of $\phi (U)$ must be a disk.