On functionals which are orthogonally additive modulo \(\mathbb{Z}\)
The Cauchy difference \(C_f(x,y):=f(x+y)-f(x)-f(y)\) is studied for functions \(f\) mapping a real inner product space \(E\) \((\dim E>1)\) into the reals. The functions satisfying the congruence \(C_f(x,y)\in\mathbb{Z}\) (the set of integers) for all orthogonal \(x,y\in E\) are characterized by means of the square of the norm in \(E\) and the unique real linear functional acting on \(E\). This is an analogue, obtained both for sets with Baire property and for Christensen measurable sets, of a result by \textit{K. Baron} and \textit{G. L. Forti} [Result. Math. 26, No. 3-4, 205-210 (1994; Zbl 0828.39010)]. Following the pattern from the paper just quoted, the author also supplies solutions \(F:E\to \mathbb{C}\) of the conditional Cauchy equation \(F(x+y)=F(x)F(y)\) for all orthogonal \(x,y\in E\).
orthogonally additive linear functionals
scientific article; zbMATH DE number 933253