D’Alembert’s functional equation on compact groups.

*(English)*Zbl 1131.39028In this paper, dedicated to Themistocles M. Rassias, the author offers an alternative proof to a theorem presented by T.M.K. Davison at the 44th International Symposium on Functional Equations 2006, Louisville KY on the general solution $f:G\to \u2102$ of the functional equation

$$f\left(xy\right)+f\left(x{y}^{-1}\right)=2f\left(x\right)f\left(y\right)\phantom{\rule{1.em}{0ex}}(x,y\in G),$$

where $G$ is a compact group. The author’s proof relies heavily on his paper “Spectral synthesis problems on locally compact groups”, that at the time of publication of the paper under review apparently existed only as preprint. In that paper the author offers extension of spectral synthesis from discrete abelian groups to not necessarily abelian compact groups.

Reviewer: János Aczél (Waterloo/Ontario)