Weighted sampling recovery of functions with mixed smoothness

We study sparse-grid linear sampling algorithms and their optimality for approximate recovery of functions with mixed smoothness on Rᵈ from a set of n their sampled values in two different settings: (i) functions to be recovered are in weighted Sobolev spaces Wʳ₏, ₖ (Rᵈ) of mixed smoothness and the approximation error is measured by the norm of the weighted Lebesgue space Lₐ, ₖ (Rᵈ), and (ii) functions to be recovered are in Sobolev spaces with measure Wʳₚ (Rᵈ; w) of mixed smoothness and the approximation error is measured by the norm of the Lebesgue space with measure Lq (Rᵈ; w). Here, the function w, a tensor-product Freud-type weight is the weight in the setting (i), and the density function of the measure w in the setting (ii). The optimality of linear sampling algorithms is investigated in terms of the relevant sampling n-widths. We construct sparse-grid linear sampling algorithms which are completely different for the settings (i) and (ii) and which give upper bounds of the corresponding sampling n-widths. We prove that in the one-dimensional case, these algorithms realize the right convergence rate of the sampling widths. In the setting (ii) for the high dimensional case (d 2), we also achieve the right convergence rate of the sampling n-widths for 1 q 2 p through a non-constructive method.