On holomorphic functions on a strip in the complex plane

Abstract

Let $f$ be a holomorphic function on the strip $\{z\in C: -\alpha<Im z<\alpha\}, \alpha > 0$, belonging to the class $H(\alpha,-\alpha;\epsilon)$ defined below. It is shown that there exist holomorphic functions $w_1$ on $\{z\in C: 0<Im z <2 \alpha\}$ and $w_2$ on $\{z\in C: -2 \alpha<Im z<2 \alpha\}$ such that $w_1$ and $w_2$ have boundary values of modulus one on the real axis and satisfy the relation $w_1(z)=f(z-\alpha i)w_2(z-2 \alpha i)$ and $w_2(z+2 \alpha i)= \bar{f}(z+\alpha i)w_1(z)$ for $0<Im z<2$, where $\bar{f}(z):=\bar{f(\bar{z})}$. This leads to a "polar decomposition" $f(z)=u_f(z+\alpha i)g_f(z)$ of the function $f(z)$, where $u_f(z+\alpha i)$ and $g_f(z)$ are holomorphic functions for $-\alpha<Im z<\alpha$ such that $|u_f(x)|=1$ and $g_f(x)\ge 0$ a.e. on the real axis. As a byproduct, an operator representation of a $q$-deformed Heisenberg algebra is developed.