On expansion of analytic functions in exponential series

Let $G$ be an arbitrary convex domain in the $p$-dimensional ($p\in\mathbf N$) complex space $\mathbf C^p$, and $H(G)$ the space of single-valued analytic functions on $G$, endowed with the topology $\tau_G$ of uniform convergence on compact subsets of $G$. In this paper the following assertion is obtained (as a corollary to a more general result proved here) for a bounded domain $G$: if a sequence $\{E_n\}_{n\in\mathbf N}$ of closed subspaces of $H(G)$ that are invariant under each partial differentiation $\frac{\partial}{\partial z_k}$ ($k=1,\dots,p$) has the property that every function locally analytic on $\overline G$ can be represented as a series
\begin{equation} \sum_{n=1}^\infty x_n(z),\qquad x_n(z)\in E_n,\quad\forall\,n\in\mathbf N, \end{equation}
convergent (absolutely convergent) in the topology $\tau_G$, then any function in $H(G)$ can be expanded in a series (1) convergent (absolutely convergent) in $\tau_G$.
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