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.

翻译：本文研究了某些函数类上的采样恢复问题。通常，函数类通过施加光滑性条件来定义。在非线性逼近理论中已认识到，以控制函数在给定函数系展开下的大系数数量形式呈现的结构条件具有重要作用。光滑性类上的采样恢复是一个活跃的研究领域，其中一些问题，尤其是混合光滑性类情形下的问题，至今尚未解决。近期研究发现，通用采样离散化和非线性稀疏逼近在采样恢复问题中具有应用价值。这促使我们系统研究具有结构条件的函数类上的采样恢复问题。该方向已有部分研究成果，特别是作者近期论文中研究了由指标属于两个二进立方体差集的系数条件所定义的函数类。本文重点研究由指标属于两个二进双曲十字差集的系数条件所定义的函数类。