Admissible majorants for model subspaces, and arguments of inner functions

Abstract

Let Θ be an inner function in the upper half-plane ℂ+ and let K Θ denote the model subspace H 2 ⊖ Θ H 2 of the Hardy space H 2 = H 2(ℂ+). A nonnegative function w on the real line is said to be an admissible majorant for K Θ if there exists a nonzero function f ∈ K Θ such that {f} ⩽ w a.e. on ℝ. We prove a refined version of the parametrization formula for K Θ-admissible majorants and simplify the admissibility criterion (in terms of arg Θ) obtained in [8]. We show that, for every inner function Θ, there exist minimal K Θ-admissible majorants. The relationship between admissibility and some weighted approximation problems is considered.