On holomorphic functions on a strip in the complex plane

Abstract

Let $f$ be a holomorphic function on the strip $\{z\in C: -α<Im z<α\}, α> 0$, belonging to the class $H(α,-α;ε)$ defined below. It is shown that there exist holomorphic functions $w_1$ on $\{z\in C: 0<Im z <2 α\}$ and $w_2$ on $\{z\in C: -2 α<Im z<2 α\}$ such that $w_1$ and $w_2$ have boundary values of modulus one on the real axis and satisfy the relation $w_1(z)=f(z-αi)w_2(z-2 αi)$ and $w_2(z+2 αi)= \bar{f}(z+αi)w_1(z)$ for $0<Im z<2$, where $\bar{f}(z):=\bar{f(\bar{z})}$. This leads to a &#34;polar decomposition&#34; $f(z)=u_f(z+αi)g_f(z)$ of the function $f(z)$, where $u_f(z+αi)$ and $g_f(z)$ are holomorphic functions for $-α<Im z<α$ 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.