On separated Carleson sequences in the unit disc

Abstract

The interpolating sequences $S$ for $H^{\infty }(\mathbb{D})$, the bounded holomorphic functions in the unit disc $\mathbb{D}$ of the complex plane $\mathbb{C}$, were characterized by L. Carleson using metric conditions on $S$. Alternatively, to characterize interpolating sequences we can use the existence in $H^{\infty }(\mathbb{D})$ of an infinity of functions $\lbrace \rho _{a}\rbrace _{a\in S}$, uniformly bounded in $\mathbb{D}$, the function $\rho _{a}$ being $1$ at the point $a\in S$ and $0$ at any $b\in S\setminus \lbrace a\rbrace$. A. Hartmann recently proved that just one function in $H^{\infty }(\mathbb{D})$ was enough to characterize interpolating sequences for $H^{\infty }(\mathbb{D})$. In this work we use the "hard" part of Carleson's proof of the corona theorem to extend Hartmann's result and to answer a question he asked in his paper.