On the expansions of analytic functions on convex locally closed sets in exponential series

Let $Q$ be a bounded, convex, locally closed subset of $\mathbb C^N$ with nonempty interior. For $N>1$ sufficient conditions are obtained that an operator of the representation of analytic functions on $Q$ by exponential series has a continuous linear right inverse. For $N=1$ the criterions for the existence of a continuous linear right inverse for the representation operator are proved.