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

Abstract

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 : Cn → W to Vn, 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 Vn 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.