Interpolating with outer functions

Abstract

The classical theorems of Mittag-Leffler and Weierstrass show that when $\{{\lambda }_n\}$ is a sequence of distinct points in the open unit disk $\D$, with no accumulation points in $\D$, and $\{w_n\}$ is any sequence of complex numbers, there is an analytic function $\varphi$ on $\D$ for which $\varphi ({\lambda }_n)=w_n$. A celebrated theorem of Carleson \cite{MR117349} characterizes when, for a bounded sequence $\{w_n\}$, this interpolating problem can be solved with a bounded analytic function. A theorem of Earl \cite{MR284588} goes further and shows that when Carleson's condition is satisfied, the interpolating function $\varphi$ can be a constant multiple of a Blaschke product. In this paper, we explore when the interpolating $\varphi$ can be an outer function. We then use our results to refine a result of McCarthy \cite{MR1065054} and explore the common range of the co-analytic Toeplitz operators on a model space.

27 pages