Admissible Majorants for Model Subspaces of H^2, Part I: Slow Winding of the Generating Inner Function

A model subspace K(theta) of the Hardy space H^2 = H^2 (c)+ for the upper half plane (C)+ is H^2(C+) (circled minus) (theta) H^2 (C+) where (theta) is an inner function in (C)+. A function (omega) : (R) - [0,infinity) is called an admissible majorant for K(theta) if there exists an f (element of) K(theta), f (not identical to) 0, |f(x)| <= (omega)(x) almost everywhere on (R). For some (mainly meromorphic) (theta)ʹs some parts of (theta) (the set of all admissible majorants for K(theta) are explicitly described. These descriptions depend on the rate of growth of arg (theta) along (R). This paper is about slowly growing arguments (slower than x). Our results exhibit the dependence of Adm B on the geometry of the zeros of the Blaschke product B. A complete description of Adm B is obtained for Bʹs with purely imaginary ("vertical") zeros. We show that in this case a unique minimal admissible majorant exists.