  In this paper we give a simpler proof of the $L^p$-Schwartz space isomorphism
$(0< p\leq 2)$ under the Fourier transform for the class of functions of left
$\delta$-type on a Riemannian symmetric space of rank one. Our treatment rests
on Anker's \cite{A} proof of the corresponding result in the case of left
$K$-invariant functions on $X$. Thus we give a proof which relies only on the
Paley--Wiener theorem.
