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

Paley-Wiener-Schwartz nearly Parseval frames and Besov spaces on noncompact symmetric spaces