On Montel’s theorem in several variables

Recently, the first author of this paper, used the structure of finite dimensional translation invariant subspaces of C(R, C) to give a new proof of classical Montel’s theorem, about continuous solutions of Frechet’s functional equation ∆m h f = 0, for real functions (and complex functions) of one real variable. In this paper we use similar ideas to prove a Montel’s type theorem for the case of complex valued functions defined over the discrete group Z d. Furthermore, we also state and demonstrate an improved version of Montel’s Theorem for complex functions of several real variables and complex functions of several complex variables.