Decomposition of an analytic function into a sum of periodic functions

Let $D$ be a convex polygon in $\mathbf C$ with vertices $a_1,\dots,a_m$, $m$ odd, and let $P_k$ be the half-plane bounded by an extension of the side $(a_k,a_{k+1})$ and containing $D$. A necessary and sufficient condition is found for a function analytic in $D$ and continuous on $\overline D$ to split into a sum of functions $f_k(z)$, $k=1,\dots,m$, where $f_k(z)$ is analytic in $P_k$, continuous in $\overline P_k$ and periodic with period $a_{k+1}-a_k$.
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