  Let $X$ be a symmetric space of the noncompact type. The goal of the paper is
to construct in the space $L_{2}(X)$ nearly Parseval frames consisting of
functions which simultaneously belong to Paley-Wiener spaces and to Schwartz
space on $X$. We call them Paley-Wiener-Schwartz frames in $L_{2}(X)$. These
frames are used to characterize a family of Besov spaces on $X$. As a part of
our construction we develop on $X$ the so-called average Shannon-type sampling.
