Sampling Recovery on Function Classes with a Structural Condition

In this paper, we study recovery from values at points (sampling recovery) on some function classes. Function classes are usually defined by smoothness conditions. In the theory of nonlinear approximation, it has been noted that structural conditions in the form of controlling the number of large coefficients of the expansion of a function in a given system play an important role. The sampling recovery on classes of smooth functions is an actively developing area of research, and some problems, especially for classes with mixed smoothness, are still open. It has recently been found that universal sampling discretization and nonlinear sparse approximations are useful in the problem of sampling recovery. This motivated us to systematically study the sampling recovery on classes of functions with a structural condition. Some results in this direction are already known. In particular, classes defined by conditions on coefficients with indices from domains that are differences of two dyadic cubes have been studied in a recent paper by the second author. This paper studies function classes defined by conditions on coefficients with indices from domains that are differences of two hyperbolic crosses.