In this note we study questions of redundancy concerning the functional equation f(h(x)+k(x)) = f(h(x))+f(k(x)), where h and k are given functions and f is the unknown function.
In this paper we consider the Aleksandrov equation f(L + x) = f(L) + f(x) where L is contained in R and f: L --> R and we describe the class of solutions bounded from below, with zeros and assuming on the boundary of the set of zeros only values multiple of a fixed a > 0. This class is the natural generalization of that described by Aleksandrov itself in the one-dimensional case.
Let X be an arbitrary Abelian group and E a Banach space. We consider the difference-operators ∆n defined by induction: (∆f)(x;y) = f(x+y) - f(x), (∆nf)(x;y1,...,yn) = (∆n-1(∆f)(.;y1)) (x;y2,...,yn) (n = 2,3,4,..., ∆1=∆, x,yi belonging to X, i = 1,2,...,n; f: X --> E). ...
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