Functional equations on groups

Remarks

Example: for $G$ a group and $Y = \C$, the equation $f(xy) + f(xy^{-1}) = 2 f(x) f(y)$ is d'Alembert's equation on $G$. It has the solution $f(x) = (\chi(x) + \chi(x)^{-1})/2$ for every homomorphism $\chi \colon G \to \C^{\times}$, as a direct expansion shows. Non-example: the equation $f(f(x)) = x$ for $f \colon G \to G$ does not fit the definition. There the unknown function is evaluated at a value of itself, not at a word in the variables, so it is an iterative equation. For $G = (\R, +)$, this setting includes the classical equations of Cauchy, d'Alembert and Pexider, and Jensen's equation in the equivalent form $f(x + y) + f(x - y) = 2 f(x)$. Solutions on $G$ are often required to be continuous or measurable when $G$ is a topological group.