Cauchy's functional equation in restricted complex domains

Under various assumptions on A and αj s, Pisot and Schoenberg [8] solved for monotone solutions of (1.1). In a subsequent paper [9], they treated the case where the domain of solutions is a subset of Rn with the following main result. Let α1,α2, . . . ,αA be elements of Rn (n < A) satisfying the following conditions: (1) every set of n elements among the αi is linearly independent over R, (2) the elements α1,α2, . . . ,αA are rationally independent, that is, linearly independent overQ. Let S = {Aj=1ujαj | uj ∈N0, the set of nonnegative integers}, B a Banach space and f : S→ B. If f is a uniformly continuous solution of the CFE (1.1), then f (x) admits a unique representation of the form f (x) = λ(x) +Am=1φm(x), where λ is a linear function from Rn into B, and each φm (m= 1,2, . . . ,A) is a function from Rm = {umαm + ∑A j=1, j =mkjαj | um ∈N0, kj ∈ Z} into B satisfying (1) φm(0)= 0, (2) φm(x+αj)= φm(x) ( j =m, x ∈ Rm), (3) φm is a uniformly continuous function on Rm into B. Studying both of Pisot’s and Schoenberg’s works, we observe two significant features, first, that their method of proof in [9], both beautiful and powerful, can be extended to