Weighted hyperbolic cross polynomial approximation

TL;DR

This paper investigates optimal polynomial approximation in weighted Sobolev spaces with Freud weights, establishing asymptotic optimality of certain hyperbolic cross methods and providing error bounds for various dimensions and parameters.

Contribution

It introduces hyperbolic cross polynomial approximation methods for weighted Sobolev spaces and proves their asymptotic optimality in terms of Kolmogorov and linear n-widths.

Findings

Asymptotic optimality of de la Vallée Poussin sums for 1D cases.

Construction of hyperbolic cross methods for higher dimensions.

Error bounds for specific weights and dimensions.

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 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.