Resumen de D'Alembert's functional equation on topological groups

Thomas M. K. Davison

• D�Alembert�s equation f(xy) + f(xy -1) = 2f(x)f(y) is solved for all compact groups: all solutions come from continuous homomorphisms from the group to . We introduce the notion of a basic d�Alembert function: a continuous solution for which f(xy) = f(y) for all y implies that x = e. It is shown that every d�Alembert function factors through a basic d�Alembert function. We show that the only compact groups that support a basic d�Alembert function are topologically isomorphic to compact subgroups of . Each subgroup (compact or not) of supports a basic d�Alembert function.