Piecewise linear interpolation via kernels
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866915827507265536 |
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| author | Karvonen, Toni Santin, Gabriele Wenzel, Tizian |
| author_facet | Karvonen, Toni Santin, Gabriele Wenzel, Tizian |
| contents | We consider piecewise linear interpolation from the perspective of kernel interpolation and quadrature. If the Sobolev space $W_2^1(0, 1)$ is equipped with a suitable inner product, its reproducing kernel is piecewise linear and gives rise to piecewise linear interpolation. We show that such kernels are Green kernels for certain second-order partial differential equations and use kernel-based superconvergence theory to obtain rates of convergence for approximation of functions lying in $W_2^s(0, 1)$ for $s \in [1, 2]$. The rates coincide with classical rates for linear splines. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_01555 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Piecewise linear interpolation via kernels Karvonen, Toni Santin, Gabriele Wenzel, Tizian Numerical Analysis We consider piecewise linear interpolation from the perspective of kernel interpolation and quadrature. If the Sobolev space $W_2^1(0, 1)$ is equipped with a suitable inner product, its reproducing kernel is piecewise linear and gives rise to piecewise linear interpolation. We show that such kernels are Green kernels for certain second-order partial differential equations and use kernel-based superconvergence theory to obtain rates of convergence for approximation of functions lying in $W_2^s(0, 1)$ for $s \in [1, 2]$. The rates coincide with classical rates for linear splines. |
| title | Piecewise linear interpolation via kernels |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2603.01555 |