Recently, the first author of this paper, used the structure of finite dimensional translation invariant subspaces of C(\mathbb{R},\mathbb{C}) to give a new proof of classical Montel’s theorem, about continuous solutions of Fréchet’s functional equation \Delta_h^mf=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 \mathbb{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.

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Abu-Helaiel, Kh. F., Almira, J. M.