  In \cite{Roe} Roe proved that if a doubly-infinite sequence $\{f_k\}$ of
functions on $\R$ satisfies $f_{k+1}=(df_{k}/dx)$ and $|f_{k}(x)|\leq M$ for
all $k=0,\pm 1,\pm 2,...$ and $x\in \R$, then $f_0(x)=a\sin(x+\varphi)$ where
$a$ and $\varphi$ are real constants. This result was extended to $\R^n$ by
Strichartz \cite{Str} where $d/dx$ is substituted by the Laplacian on $\R^n$.
While it is plausible to extend this theorem for other Riemannian manifolds or
Lie groups, Strichartz showed that the result holds true for Heisenberg groups,
but fails for hyperbolic 3-space. This negative result can be indeed extended
to any Riemannian symmetric space of noncompact type. We observe that this
failure is rooted in the $p$-dependance of the $L^p$-spectrum of the Laplacian
on the hyperbolic spaces. Taking this into account we shall prove that for all
rank one Riemannian symmetric spaces of noncompact type, or more generally for
the harmonic $NA$ groups, the theorem actually holds true when uniform
boundedness is replaced by uniform "almost $L^p$ boundedness". In addition we
shall see that for the symmetric spaces this theorem is capable of
characterizing the Poisson transforms of $L^p$ functions on the boundary, which
some what resembles the original theorem of Roe on $\R$.
