Weighted hyperbolic cross polynomial approximation

We study linear polynomial approximation of functions in weighted Sobolev spaces Wʳ₏, ₖ (Rᵈ) of mixed smoothness r N, and their optimality in terms of Kolmogorov and linear n-widths of the unit ball Wʳ₏, ₖ (Rᵈ) in these spaces. The approximation error is measured by the norm of the weighted Lebesgue space Lₐ, ₖ (Rᵈ). The weight w is a tensor-product Freud weight. For 1 p, q and d=1, we prove that the polynomial approximation by de la Vall\'ee Poussin sums of the orthonormal polynomial expansion of functions with respect to the weight w², is asymptotically optimal in terms of relevant linear n-widths ₙ (Wʳ₏, ₖ (R, Lₐ, ₖ (R) ) and Kolmogorov n-widths dₙ (Wʳ₏, ₖ (R), Lₐ, ₖ (R) ) for 1 q p <. For 1 p, q and d 2, we construct linear methods of hyperbolic cross polynomial approximation based on tensor product of successive differences of dyadic-scaled de la Vall\'ee Poussin sums, which are counterparts of hyperbolic cross trigonometric linear polynomial approximation, and give some upper bounds of the error of these approximations for various pair p, q with 1 p, q. For some particular weights w and d 2, we prove the right convergence rate of ₙ (Wʳ₂, ₖ (Rᵈ), L₂, ₖ (Rᵈ) ) and dₙ (Wʳ₂, ₖ (Rᵈ), L₂, ₖ (Rᵈ) ) which is performed by a constructive hyperbolic cross polynomial approximation.