Majorants of meromorphic functions with fixed poles

Abstract

Let $B$ be a meromorphic Blaschke product in the upper half-plane with zeros $z_n$ and let $K_B=H^2\ominus BH^2$ be the associated model subspace of the Hardy class. In other words, $K_B$ is the space of square summable meromorphic functions with the poles at the points $\bar z_n$. A nonnegative function $w$ on the real line is said to be an admissible majorant for $K_B$ if there is a non-zero function $f\in K_B$ such that $|f|\le w$ a.e. on $\mathbb{R}$. We study the relations between the distribution of the zeros of a Blaschke product $B$ and the class of admissible majorants for the space $K_B$.