Admissible Majorants for Model Subspaces, and Arguments of Inner Functions

Let $\Theta$ be an inner function in the upper half-plane $\mathbb{C}^+$ and let $K_\Theta$ denote the model subspace $H^2\ominus\Theta H^2$ of the Hardy space $H^2=H^2(\mathbb{C}^+)$. A nonnegative function $w$ on the real line is said to be an admissible majorant for $K_\Theta$ if there exists a nonzero function $f\in K_\Theta$ such that $|f|\le w$ a.e. on $\mathbb{R}$. We prove a refined version of the parametrization formula for $K_\Theta$-admissible majorants and simplify the admissibility criterion (in terms of $\arg\Theta$) obtained in [V. P. Havin and J. Mashreghi, "Admissible majorants for model subspaces of $H^2$. Part I: slow winding of the generating inner function", Canad. J. Math., 55, 6 (2003), 1231–1263]. We show that, for every inner function $\Theta$, there exist minimal $K_\Theta$-admissible majorants. The relationship between admissibility and some weighted approximation problems is considered.