Free interpolation by nonvanishing analytic functions

Abstract

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 {ε j } (i.e., a sequence of positive numbers bounded by 1 and tending to 0) such that every interpolation problem f (z j ) = a j has a nonvanishing solution f ∈ H ∞ whenever 1 ≥ |a j | ≥ ε 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.

Thin interpolating sequences in the disk

“…The main goal of this survey is to show in Theorem 3.3 that every asymptotic interpolation problem |f (a n ) − w n | → 0, sup n |w n | ≤ 1, can be solved by a thin Blaschke product itself whenever (a n ) is thin. That result will follow from the Sundberg-Wolff Lemma mentioned above combined with several statements in [3]. The result was known to Dyakonov and Nicolau; though they don't present the details in [3], since this was not the goal of their paper.…”

“…That result will follow from the Sundberg-Wolff Lemma mentioned above combined with several statements in [3]. The result was known to Dyakonov and Nicolau; though they don't present the details in [3], since this was not the goal of their paper. In Section 3 we will provide those details.…”

“…We note that most parts of the proof of the following theorem are the same as those given in [3] by Dyakonov and Nicolau. They were merely interested in a norm one solution, that is not necessarily a thin Blaschke product.…”

“…Later on, Dyakonov and Nicolau [3] proved the next result Theorem 0.2 (Dyakonov, Nicolau). Given an H ∞ -interpolating sequence (z n ) in D, the following are equivalent:…”

“…In Section 3 we will provide those details. We proceed according to their suggestions in [3] that the solution F in Theorem 0.2 above can be chosen to be thin.…”