The Nevanlinna counting functions for Rudin’s orthogonal functions

Abstract:

$H^\infty$ and $H^2$ denote the Hardy spaces on the open unit disc $D$. Let $\phi$ be a function in $H^\infty$ and $\|\phi \|_\infty = 1$. If $\phi$ is an inner function and $\phi (0) = 0$, then $\{\phi ^n~;~n = 0,1,2,\cdots \}$ is orthogonal in $H^2$. W.Rudin asked if the converse is true and C. Sundberg and C. Bishop showed that the converse is not true. Therefore there exists a function $\phi$ such that $\phi$ is not an inner function and $\{\phi ^n\}$ is orthogonal in $H^2$. In this paper, the following is shown: $\{\phi ^n\}$ is orthogonal in $H^2$ if and only if there exists a unique probability measure $\nu _0$ on [0,1] with $1 \in$ supp $\nu _0$ such that $N_\phi (z) = {\int ^1_{|z|}} \log \frac {r}{|z|} d\nu _0(r)$ for nearly all $z$ in $D$ where $N_\phi$ is the Nevanlinna counting function of $\phi$. If $\phi$ is an inner function, then $\nu _0$ is a Dirac measure at $r = 1$.