Extremely weak interpolation in $H^{\infty}$

TL;DR

This paper demonstrates that in the Hardy space, having just one interpolating function for a sequence of points suffices to establish the sequence's interpolating property, highlighting a weaker condition than classical results.

Contribution

It shows that a single interpolating function for a sequence implies the sequence is interpolating in the Hardy space, extending classical interpolation results.

Findings

One interpolating function implies the sequence is interpolating.

Weak interpolation condition suffices for Hardy space interpolation.

Results extend to other function spaces.

Abstract

Given a sequence of points in the unit disk, a well known result due to Carleson states that if given any point of the sequence it is possible to interpolate the value one in that point and zero in all the other points of the sequence, with uniform control of the norm in the Hardy space of bounded analytic functions on the disk, then the sequence is an interpolating sequence (i.e.\ every bounded sequence of values can be interpolated by functions in the Hardy space). It turns out that such a result holds in other spaces. In this short note we would like to show that for a given sequence it is sufficient to find just {\bf one} function interpolating suitably zeros and ones to deduce interpolation in the Hardy space.