Cauchy's functional equation in a restricted domain

Abstract

In 1964-1965, Pisot and Schoenberg made two important studies on solutions of Cauchy's functional equation over certain restricted domains, consisting of linear integral combinations of generating elements in the real n-dimensional space. Such generating elements are subject to two independence conditions, one linear and the other rational. It was discovered that under suitable regularity on the functions, the most general solutions can be written as a linear function plus a periodic part. In this thesis, we find uniformly continuous solutions of Cauchy's functional equation whose domain is a subset of the complex field comprising finite combinations over a subset of Gaussian integers. The solutions obtained are similar to those of Pisot and Schoenberg. The proofs employed come from a detailed analysis of those used by Pisot and Schoenberg (1965) with a number of modi?cation such as replacing some independence restriction by denseness.