Free interpolation by nonvanishing analytic functions

Abstract

We are concerned with interpolation problems in H ∞ where the values prescribed and the function to be found are both zero-free. More precisely, given a sequence {z j } in the unit disk, we ask whether there exists a nontrivial minorant {e j } (i.e., a sequence of positive numbers bounded by 1 and tending to 0) such that every interpolation problem f(zj) = aj has a nonvanishing solution f ∈ H ∞ whenever 1 ≥ |a j | ≥ e j for all j. The sequences {z j } with this property are completely characterized. Namely, we identify them as thin sequences, a class that arose earlier in Wolff's work on free interpolation in H ∞ ∩ VMO.