Sampling recovery on some function classes is studied in this paper. Typically, function classes are defined by imposing smoothness conditions. It was understood in nonlinear approximation that structural conditions in the form of control of the number of big coefficients of an expansion of a function with respect to a given system of functions plays an important role. Sampling recovery on smoothness classes is an area of active research, some problems, especially in the case of mixed smoothness classes, are still open. It was discovered recently that universal sampling discretization and nonlinear sparse approximations are useful in the sampling recovery problem. This motivated us to systematically study sampling recovery on function classes with a structural condition. Some results in this direction are already known. In particular, the classes defined by conditions on coefficients with indices from the domains, which are differences of two dyadic cubes are studied in the recent author's papers. In this paper we concentrate on studying function classes defined by conditions on coefficients with indices from the domains, which are differences of two dyadic hyperbolic crosses.

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