On the representation of an analytic function as a sum of periodic functions

Let $D$ be any convex polygon with vertices $\gamma_1,\gamma_2,\dots,\gamma_p$; let $D_k$ be the half-plane containing $D$ bounded by the line through $\gamma_k$ and $ \gamma_{k+1}$. We show that any function $F(z)$ analytic in $D$ can be represented in the form
$$ F(z)=\sum_{k=1}^pF_k(z),\qquad z\in D, $$
where $F_k(z)$ is regular and periodic in $D_k$, with period $\gamma_{k+1}-\gamma_k$. If $F(z)$ is regular in $D$ and if $F(z)$ and its first $s$ derivatives are continuous in $\overline D$, then
$$ F(z)=\sum_{k=1}^pF_k(z)+p(z),\qquad z\in\overline D. $$
Here for even $p$ we have that $F_k(z)$ is regular in $D_k$ and is continuous, together with its first $s-2$ derivatives, on $\overline D_k$ (we assume $ s\geqslant2$), $F_k(z)$ is periodic with period $\gamma_{k+1}-\gamma_k$, and $p(z)$ is a polynomial of degree at most $s+p/2-2$. For odd $p$, $F_k(z)$ is continuous, together with its first $s-4$ derivatives, in $\overline D_k$ (we assume $s\geqslant4$), and $p(z)$ is a polynomial of degree at most $s+(p-1)/2-2$.
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