Constructive sparse trigonometric approximation and other problems for functions with mixed smoothness

Our main interest in this paper is to study some approximation problems for classes of functions with mixed smoothness. We use technique, based on a combination of results from hyperbolic cross approximation, which were obtained in 1980s -- 1990s, and recent results on greedy approximation to obtain sharp estimates for best $m$-term approximation with respect to the trigonometric system. We give some observations on numerical integration and approximate recovery of functions with mixed smoothness. We prove lower bounds, which show that one cannot improve accuracy of sparse grids methods with $\asymp 2^nn^{d-1}$ points in the grid by adding $2^n$ arbitrary points. In case of numerical integration these lower bounds provide best known lower bounds for optimal cubature formulas and for sparse grids based cubature formulas.