Weighted hyperbolic cross polynomial approximation
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929653062565888 |
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| author | Dũng, Dinh |
| author_facet | Dũng, Dinh |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_19442 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weighted hyperbolic cross polynomial approximation Dũng, Dinh Numerical Analysis 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. |
| title | Weighted hyperbolic cross polynomial approximation |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2407.19442 |