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关于复平面中条上的全纯函数

On holomorphic functions on a strip in the complex plane

Konrad Schmuedgen

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摘要：

设$f$是带$\{z\in C: -α<Im z<α\}, α> 0$上的全纯函数，属于下面定义的类$H(α,-α;ε)$。证明了存在$\{z\in C: 0<Im z <2 α\}$上的全纯函数$w_1$和$\{z\in C: -2 α<Im z<2 α\}$上的$w_2$，使得$w_1$和$w_2$在实轴上具有模1的边界值，并且对于$0<Im z<2$满足关系$w_1(z)=f(z-αi)w_2(z-2 αi)$和$w_2(z+2 αi)= \bar{f}(z+αi)w_1(z)$，其中$\bar{f}(z):=\bar{f(\bar{z})}$ .这导致函数$f(z)$的“极分解“$f(z)=u_f(z+αi)g_f(z)$，其中$u_f(z+αi)$和$g_f(z)$是$-α<Im z<α$的全纯函数，使得$|u_f(x)|=1$和$g_f(x)\ge 0$ a.e.在实轴上。作为一个副产品，$q$变形海森堡代数的运营商表示的发展。

英文摘要：

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 "polar decomposition" $f(z)=u_f(z+αi)g_f(z)$ 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.

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