On separated Carleson sequences in the unit disc of ${\mathbb{C}}.$

TL;DR

This paper extends Hartman's recent result by using Carleson's Corona theorem proof techniques to characterize separated Carleson sequences in the unit disc, providing new insights into interpolation for bounded holomorphic functions.

Contribution

It generalizes Hartman's characterization of interpolating sequences using a single function, applying Carleson's Corona theorem methods to separated sequences.

Findings

Extended Hartman's result to broader classes of sequences.

Used Carleson's Corona theorem proof techniques.

Provided new characterizations of separated Carleson sequences.

Abstract

The interpolating sequences for $H^{\infty}({\mathbb{D}}),$ the bounded holomorphic function in the unit disc ${\mathbb{D}}$ of the complex plane ${\mathbb{C}},$ {\small where characterised by L. Carleson by metric conditions on the points. They are also characterised by "dual boundedness" conditions which imply an infinity of functions. A. Hartmann proved recently 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 the proof of Carleson for the Corona theorem, to extend Hartman's result and answer a question he asked in his paper.}\ \par