Weighted approximate sampling recovery and integration based on B-spline interpolation and quasi-interpolation
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arXiv
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| Format: | Preprint |
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2025
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| author | Dũng, Dinh |
| author_facet | Dũng, Dinh |
| contents | We propose novel methods for approximate sampling recovery and integration of functions in the Freud-weighted Sobolev space $W^r_{p,w}(\mathbb{R})$. The approximation error of sampling recovery is measured in the norm of the Freud-weighted Lebesgue space $L_{q,w}(\mathbb{R})$. Namely, we construct equidistant compact-supported B-spline quasi-interpolation and interpolation sampling algorithms $Q_{ρ,m}$ and $P_{ρ,m}$ which are asymptotically optimal in terms of the sampling $n$-widths $\varrho_n(\boldsymbol{W}^r_{p,w}(\mathbb{R}), L_{q,w}(\mathbb{R}))$ for every pair $p,q \in [1,\infty]$, and prove the right convergence rate of these sampling $n$-widths, where $\boldsymbol{W}^r_{p,w}(\mathbb{R})$ denotes the unit ball in $W^r_{p,w}(\mathbb{R})$. The algorithms $Q_{ρ,m}$ and $P_{ρ,m}$ are based on truncated scaled B-spline quasi-interpolation and interpolation, respectively. We also prove the asymptotical optimality and right convergence rate of the equidistant quadratures generated from $Q_{ρ,m}$ and $P_{ρ,m}$, for Freud-weighted numerical integration of functions in $W^r_{p,w}(\mathbb{R})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01167 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weighted approximate sampling recovery and integration based on B-spline interpolation and quasi-interpolation Dũng, Dinh Numerical Analysis 41A46, 41A25, 41A63, 41A81, 65D05 We propose novel methods for approximate sampling recovery and integration of functions in the Freud-weighted Sobolev space $W^r_{p,w}(\mathbb{R})$. The approximation error of sampling recovery is measured in the norm of the Freud-weighted Lebesgue space $L_{q,w}(\mathbb{R})$. Namely, we construct equidistant compact-supported B-spline quasi-interpolation and interpolation sampling algorithms $Q_{ρ,m}$ and $P_{ρ,m}$ which are asymptotically optimal in terms of the sampling $n$-widths $\varrho_n(\boldsymbol{W}^r_{p,w}(\mathbb{R}), L_{q,w}(\mathbb{R}))$ for every pair $p,q \in [1,\infty]$, and prove the right convergence rate of these sampling $n$-widths, where $\boldsymbol{W}^r_{p,w}(\mathbb{R})$ denotes the unit ball in $W^r_{p,w}(\mathbb{R})$. The algorithms $Q_{ρ,m}$ and $P_{ρ,m}$ are based on truncated scaled B-spline quasi-interpolation and interpolation, respectively. We also prove the asymptotical optimality and right convergence rate of the equidistant quadratures generated from $Q_{ρ,m}$ and $P_{ρ,m}$, for Freud-weighted numerical integration of functions in $W^r_{p,w}(\mathbb{R})$. |
| title | Weighted approximate sampling recovery and integration based on B-spline interpolation and quasi-interpolation |
| topic | Numerical Analysis 41A46, 41A25, 41A63, 41A81, 65D05 |
| url | https://arxiv.org/abs/2501.01167 |