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In this talk I will report on recent trends and results for the multivariate recovery problem where we are only allowed to use a discrete set of point evaluations. We discuss optimality issues in terms of so-called sampling numbers and comment on recovery errors in integral and other norms based on purely random and (constructively) subsampled random points. We further apply recent techniques from the field of compressed sensing to show new upper bounds for general (non-linear) sampling numbers. In relevant cases such as mixed and isotropic weighted Wiener algebras or spaces with bounded mixed derivative sampling numbers can be upper bounded by best n-term trigonometric widths in the uniform norm.
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