Extremely weak interpolation in 𝐻^{∞}

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 paper we would like to show that for a given sequence it is sufficient to find just one function suitably interpolating zeros as well as ones to deduce interpolation in the Hardy space. The result has an interesting interpretation in the context of model spaces.