Interpolating with outer functions

Abstract

The classical theorems of Mittag-Leffler and Weierstrass show that when $(\lambda _n)_{n \geqslant 1}$ is a sequence of distinct points in the open unit disk $\mathbb {D}$, with no accumulation points in $\mathbb {D}$, and $(w_n)_{n \geqslant 1}$ is any sequence of complex numbers, there is an analytic function $\varphi$ on $\mathbb {D}$ for which $\varphi (\lambda _n) = w_n$. A celebrated theorem of Carleson [2] characterizes when, for a bounded sequence $(w_n)_{n \geqslant 1}$, this interpolating problem can be solved with a bounded analytic function. A theorem of Earl [5] goes further and shows that when Carleson’s condition is satisfied, the interpolating function $\varphi$ can be a constant multiple of a Blaschke product. Results from [4] determine when the interpolating function $\varphi$ can be taken to be zero free. In this paper we explore when $\varphi$ can be an outer function.