Approximation by entire functions on a countable set of continuums

We consider a problem of approximation by entire functions of exponential type of functions defined on a countable set $E$ of continuums $G_n$, $E = \bigcup_{n\in\mathbb{Z}} G_n$. We assume that all $G_n$ are pairwise disjoint and are situated near the real axis. We assume too that all $G_n$ are commensurable in a sense and have uniformly smooth boundaries. A function f is defined independantly on each $G_n$ and is bounded on $E$ and $f^{(r)}$ has a module of continuity $\omega$ which satisfies a condition $\int_0^x\omega(t)/t dt+x\int_x^\infty\omega(t)/t^2dt\leqslant c\omega(x)$. Then we construct an entire function $F_\sigma$ of exponential type $\leqslant\sigma$ such that we have the following estimate of approximation of the function $f$ by functions $F_\sigma$: $|f(z) - F_\sigma(z)| \leqslant c_f\sigma^{-r} \omega(\sigma^{-r}), z \in \mathbb{Z}, \sigma \leqslant 1$.