Projection from the spaces $E^p$ on a convex polygon onto subspaces of periodic functions

Notation: $D$ is a convex polygon with vertices $a_1,\dots,a_m$, $P_k$ is the half-plane bounded by the extension of the side $a_k$, $a_{k+1}$ and containing $D$, $E^p$ is the Hardy–Smirnov space on $D$, and $Q_s$ is the subspace of $E^p$ consisting of the analytic functions on $P_k$ that are periodic with period $a_{k+1}-a_k$ and that vanish at $\infty$. For suitable $s$ the subspaces $Q_s$ and $H_1^p,\dots,H_m^p$ generate $E^p$. Is $E^p$ ($1<p<\infty$) decomposable into their direct sum? If $m$ is odd, then the answer is positive for $p\ne2$ and negative for $p=2$.
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