Dinh Dũng, Weighted hyperbolic cross polynomial approximation, Vol. 2026 (2026), No. 24, pp. 1-15

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DOI: 10.23952/cot.2026.24

Received September 1, 2024; Accepted March 28, 2025; Published online February 21, 2026

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 ${\bf W}^r_{p,w}(\mathbb{R}^d)$ 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 $\lambda_n\big(W^r_{p,w}(\mathbb{R}), L_{q,w}(\mathbb{R})\big)$ and Kolmogorov n-widths $d_n\big(W^r_{p,w}(\mathbb{R}), L_{q,w}(\mathbb{R})\big)$ 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 $\lambda_n\big(W^r_{2,w}(\mathbb{R}^d), L_{2,w}(\mathbb{R}^d)\big)$ and $d_n\big(W^r_{2,w}(\mathbb{R}^d), L_{2,w}(\mathbb{R}^d)\big)$ which is performed by a constructive hyperbolic cross polynomial approximation.

How to Cite this Article:
D. Dũng, Weighted hyperbolic cross polynomial approximation, Commun. Optim. Theory 2026 (2026) 24.