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 a 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 the numerical integration and approximate recovery of functions with mixed smoothness. We prove lower bounds, which show that one cannot improve the accuracy of sparse grids methods with $\asymp 2^nn^{d-1}$ points in the grid by adding $2^n$ arbitrary points. In the case of numerical integration these lower bounds provide the best available lower bounds for optimal cubature formulae and for sparse grids based cubature formulae.
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