Weighted hyperbolic cross polynomial approximation

We study linear polynomial approximation of functions in weighted Sobolev spaces Wʳp,w(Rᵈ) of mixed smoothness r ∈ N, and their optimality in terms of Kolmogorov and linear n-widths of the unit ball Wʳp,w(Rᵈ) in these spaces. The approximation error is measured by the norm of the weighted Lebesgue space Lq,w(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 λₙ(Wʳp,w(R, Lq,w(R)) and Kolmogorov n-widths dₙ(Wʳp,w(R), Lq,w(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 λₙ(Wʳ2,w(Rᵈ), L2,w(Rᵈ)) and dₙ(Wʳ2,w(Rᵈ), L2,w(Rᵈ)) which is performed by a constructive hyperbolic cross polynomial approximation.