<p>In this paper we study best <i>m</i>-term trigonometric approximation in weighted Wiener spaces and its consequences for Besov and Sobolev spaces with bounded mixed derivative/difference. We obtain several sharp asymptotic bounds for weighted Wiener spaces including the quasi-Banach case. It has recently been observed that best <i>m</i>-term trigonometric widths in the uniform norm together with recovery algorithms stemming from compressed sensing serve to control the optimal sampling recovery error in various relevant spaces of multivariate functions. We use a collection of old and new tools as well as novel findings to extend the recovery bounds to classical multivariate smoothness spaces. It turns out that embeddings into Wiener spaces serve as a powerful tool to improve certain recent bounds.</p>