Let *G*^{+} be the group of real points of a possibly disconnected linear reductive algebraic group defined over , which is generated by the real points of a connected component *G'*. Let *K* be a maximal compact subgroup of the group of real points of the connected component of this algebraic group. We characterize the space of maps π —> tr(π( *f* )), where π is an irreducible tempered representation of *G*^{+}, and *f* varies over the space of smooth, compactly supported functions on *G'*, which are left- and right-*K*-finite. This work is motivated by applications to the Arthur-Selberg trace formula.