Some extension theorems for n-Jensen functions and functional equations characterizing generalized polynomials

Summary.

The object of this paper is to investigate possibilities of extending the solution F : C → W of certain functional equations to V and of n-Jensen functions f : C ⁿ → W to V ⁿ, where C is a ${\mathbb{Q}}$ -convex subset of V and V, W are ${\mathbb{Q}}$ -vector spaces. Solutions of the considered functional equations defined on V and n-Jensen functions defined on V ⁿ have been utilized in recent papers (e.g. [2], [4]) in order to characterize generalized polynomials P : V →W of degree  ≤ n. Therefore these extension theorems may present a point of view for defining generalized polynomials on a ${\mathbb{Q}}$ -convex subset of a ${\mathbb{Q}}$ -vector space.

Even more generally, we consider certain functional equations for functions f : C →W, where C is an L-convex subset of an L-vector space V and L is an arbitrary subfield of ${\mathbb{R}}$ , possibly different from ${\mathbb{Q}}$ .