  Let $ G $ be a real simple linear connected Lie group of real rank one. Then,
$ X := G/K $ is a Riemannian symmetric space with strictly negative sectional
curvature. By the classification of these spaces, $X$ is a
real/complex/quaternionic hyperbolic space or the Cayley hyperbolic plane. We
define the Schwartz space $ \mathscr{C}(\Gamma \backslash G) $ on $ \Gamma
\backslash G $ for torsion-free geometrically finite subgroups $ \Gamma $ of
$G$. We show that it has a Fr\'echet space structure, that the space of
compactly supported smooth functions is dense in this space, that it is
contained in $ L^2(\Gamma \backslash G) $ and that the right translation by
elements of $G$ defines a representation on $ \mathscr{C}(\Gamma \backslash G)
$. Moreover, we define the space of cusp forms $ \deg\mathscr{C}(\Gamma
\backslash G) $ on $ \Gamma \backslash G $, which is a geometrically defined
subspace of $ \mathscr{C}(\Gamma \backslash G) $. It consists of the Schwartz
functions which have vanishing "constant term" along the ordinary set
$\Omega_\Gamma \subset \partial X$ and along every cusp. We show that these two
constant terms are in fact related by a limit formula if the cusp is of smaller
rank (not of full rank). The main result of this thesis consists in proving a
direct sum decomposition of the closure of the space of cusp forms in $
L^2(\Gamma \backslash G) $ which respects the Plancherel decomposition in the
case where $ \Gamma $ is convex-cococompact and noncocompact. For technical
reasons, we exclude here that $X$ is the Cayley hyperbolic plane.
