Weighted hyperbolic cross polynomial approximation
Dinh Dũng
TL;DR
This work addresses the problem of optimally approximating functions from weighted Sobolev spaces with mixed smoothness on ${\mathbb R}^d$ using weighted polynomial schemes, and quantifies optimality through Kolmogorov $n$-widths and linear $n$-widths in $L_{q,w}({\mathbb R}^d)$. The authors develop univariate approximation via de la Vallée Poussin sums relative to the weight $w^2$ and extend to multivariate hyperbolic cross constructions using Smolyak-type tensor products, establishing upper bounds for approximation errors and resulting rates for $d_n$ and $\lambda_n$ across regimes of $p,q$ and dimension. In particular, for Freud-type weights with $\lambda \in \{2,4\}$, they prove that in $L_{2,w}$ the widths decay at the right rate $n^{- r_\lambda} (\log n)^{ r_\lambda (d-1) }$, by connecting to the norm-equivalent space ${\mathcal{H}}^{r_\lambda}_w({\mathbb R^d})$ and constructing constructive hyperbolic cross operators. These results supply sharp, constructive guidance for high-dimensional weighted polynomial approximation under exponential-type weights, informing both theory and practical schemes.
Abstract
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 Vallée 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 $λ_n\big(\boldsymbol{W}^r_{p,w}(\mathbb{R}, L_{q,w}(\mathbb{R})\big)$ 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 Vallée 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 $λ_n\big(\boldsymbol{W}^r_{2,w}(\mathbb{R}^d), L_{2,w}(\mathbb{R}^d)\big)$ 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.