  We study \emph{property RD} in terms of rapid decay of matrix coefficients.
We give new formulations of property RD in terms of a $L^{1}$-integra\-bility
condition of a Banach representation. Combining this new definition with the
existence of cyclic subgroups of exponential growth in non-uniform lattices in
semisimple Lie groups, we deduce that there exist matrix coefficients
associated to several kinds of quasi-regular representations which satisfy a
"non-RD condition" for non-uniform lattices. We obtain also that such
coefficients can not satisfy \emph{the weak inequality} of Harish-Chandra.
