On solutions of the Frechet functional equation

In this paper we give a new proof of a classical result by Frechet [M. Frechet, Une definition fonctionnelle des polynomes, Nouv. Ann. 9 (4) (1909) 145-162]. Concretely, we prove that, if Delta(k+1)(h) f = 0 and f is continuous at some point or bounded at some nonempty open set, then f is an element of P-k. Moreover, as a consequence of the technique developed for our proof, it is possible to give a description of the closure of the graph for the solutions of the equation. Finally, we characterize some spaces of polynomials of several variables by the use of adequate generalizations of the forward differences operator Delta(k+1)(h). (c) 2006 Elsevier Inc. All rights reserved.