# Institut for Matematik

## Properties of d'Alembert functions

Publikation: Working paper/Preprint Working paperForskning

We study properties of solutions $~f$ of d'Alembert's functional equations on a topological group $~G$. For nilpotent groups and for connected, solvable Lie groups $~G$, we prove that $~f$ has the form $~f(x) = (\gamma(x) + \gamma (x^{-1}))/2$, $~x \in G$, where $~\gamma$ is a continuous homomorphism of $~G$ into the multiplicative group $~\mathbb{C}\setminus \{ 0\}$. We give conditions on $~G$ and/or $~f$ for equality in the inclusion $~\{ u \in G \mid f(xu) = f(x)$ for all $~x \in G\} \subseteq \{u \in G \mid f(u) =1\}$.