(DFS)-spaces of holomorphic functions invariant under differentiation

Let be a bounded convex domain, A^−∞(G) be the (DFS)-space of all holomorphic functions of polynomial growth on G and A^∞(G) be the Fréchet space of C^∞-functions on closure of G which are holomorphic on G. With the help of the Laplace transform we describe the strong dual of A^−∞(G) and prove that A^−∞(G) is the unique (DFS)-space H such that the space A^∞(G) is contained in H, H is embedded continuously in A^−∞(G) and H is invariant under differentiation.