Adaptive Sampling Recovery and Nonlinear Approximations of Multivariate Functions in Besov-type Spaces

We investigate nonlinear approximations by sets of finite cardinality or of finite pseudo-dimesion and the optimality in terms of entropy numbers and other characterizations for nonlinear approximations. Functions to be approximated are in Besov type spaces of functions having a certain mixed smoothness. We prove the asymptotic order of these quantities and explicitly construct asymptotically optimal methods of nonlinear approximation based on a trigonometric sampling representation in Besov type spaces.