We study linear polynomial approximation of functions in weighted Sobolev spaces $W^r_{p,w}(\mathbb{R}^d)$ of mixed smoothness $r \in \mathbb{N}$, and their optimality in terms of Kolmogorov and linear $n$-widths of the unit ball $\boldsymbol{W}^r_{p,w}(\mathbb{R}^d)$ in these spaces. The approximation error is measured by the norm of the weighted Lebesgue space $L_{q,w}(\mathbb{R}^d)$. The weight $w$ is a tensor-product Freud weight. For $1\le p,q \le \infty$ 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 $w^2$, is asymptotically optimal in terms of relevant linear $n$-widths $\lambda_n\big(\boldsymbol{W}^r_{p,w}(\mathbb{R}, L_{q,w}(\mathbb{R})\big)$ and Kolmogorov $n$-widths $d_n\big(\boldsymbol{W}^r_{p,w}(\mathbb{R}), L_{q,w}(\mathbb{R})\big)$ for $1\le q \le p <\infty$. For $1\le p,q \le \infty$ and $d\ge 2$, 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 $1 \le p, q \le \infty$. For some particular weights $w$ and $d \ge 2$, we prove the right convergence rate of $\lambda_n\big(\boldsymbol{W}^r_{2,w}(\mathbb{R}^d), L_{2,w}(\mathbb{R}^d)\big)$ and $d_n\big(\boldsymbol{W}^r_{2,w}(\mathbb{R}^d), L_{2,w}(\mathbb{R}^d)\big)$ which is performed by a constructive hyperbolic cross polynomial approximation.

翻译：我们研究了混合光滑度 $r \in \mathbb{N}$ 的加权 Sobolev 空间 $W^r_{p,w}(\mathbb{R}^d)$ 中函数的线性多项式逼近，以及这些空间中单位球 $\boldsymbol{W}^r_{p,w}(\mathbb{R}^d)$ 在 Kolmogorov $n$-宽度和线性 $n$-宽度意义下的最优性。逼近误差由加权 Lebesgue 空间 $L_{q,w}(\mathbb{R}^d)$ 的范数度量。权重 $w$ 是张量积 Freud 权重。对于 $1\le p,q \le \infty$ 和 $d=1$，我们证明了关于权重 $w^2$ 的正交多项式展开的函数，其 de la Vallée Poussin 和多项式逼近，在相关线性 $n$-宽度 $\lambda_n\big(\boldsymbol{W}^r_{p,w}(\mathbb{R}, L_{q,w}(\mathbb{R})\big)$ 和 Kolmogorov $n$-宽度 $d_n\big(\boldsymbol{W}^r_{p,w}(\mathbb{R}), L_{q,w}(\mathbb{R})\big)$ 的意义下，对于 $1\le q \le p <\infty$ 是渐近最优的。对于 $1\le p,q \le \infty$ 和 $d\ge 2$，我们基于 dyadic 尺度 de la Vallée Poussin 和的逐次差分的张量积，构造了双曲交叉多项式逼近的线性方法，这些方法是双曲交叉三角线性多项式逼近的对应物，并给出了对于 $1 \le p, q \le \infty$ 的各种 $p,q$ 组合，这些逼近误差的一些上界。对于一些特定的权重 $w$ 和 $d \ge 2$，我们证明了 $\lambda_n\big(\boldsymbol{W}^r_{2,w}(\mathbb{R}^d), L_{2,w}(\mathbb{R}^d)\big)$ 和 $d_n\big(\boldsymbol{W}^r_{2,w}(\mathbb{R}^d), L_{2,w}(\mathbb{R}^d)\big)$ 的正确收敛速率，这是通过一个构造性的双曲交叉多项式逼近实现的。