Optimal $C^{1,1}$ and Quasi-Optimal $C^2$ Monotone Interpolation with Curvature Control
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| author | Jiang, Fushuai Luli, Garving K. |
| author_facet | Jiang, Fushuai Luli, Garving K. |
| contents | We study monotone Hermite interpolation on an interval, where both function values and first derivatives are prescribed at the nodes. Among all $C^{1,1}$ interpolants, we seek one with optimal curvature, measured by $\|F''\|_{L^\infty}$. In this paper, we analyze the limitations of some classical techniques, and provide an explicit optimal construction in $C^{1,1}$ given by quadratic splines by studying the optimal velocity profile. Moreover, given $E = \{x_1,\cdots,x_N\}$ and $f: E\to \mathbb{R}$ (without derivatives), we also provide a formula to compute the corresponding trace seminorm \[ \inf\Bigl\{ \|F''\|_{L^\infty} :
F(x)=f(x) \text{ on $E$ and } F'\ge 0 \text{ everywhere} \Bigr\}. \] In addition, we also describe how to mollify $C^{1,1}$ solutions to $C^2$ while preserving monotonicity and sacrificing a controlled amount of optimality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_14302 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimal $C^{1,1}$ and Quasi-Optimal $C^2$ Monotone Interpolation with Curvature Control Jiang, Fushuai Luli, Garving K. Classical Analysis and ODEs 41A29, 41A15, 41A44 We study monotone Hermite interpolation on an interval, where both function values and first derivatives are prescribed at the nodes. Among all $C^{1,1}$ interpolants, we seek one with optimal curvature, measured by $\|F''\|_{L^\infty}$. In this paper, we analyze the limitations of some classical techniques, and provide an explicit optimal construction in $C^{1,1}$ given by quadratic splines by studying the optimal velocity profile. Moreover, given $E = \{x_1,\cdots,x_N\}$ and $f: E\to \mathbb{R}$ (without derivatives), we also provide a formula to compute the corresponding trace seminorm \[ \inf\Bigl\{ \|F''\|_{L^\infty} : F(x)=f(x) \text{ on $E$ and } F'\ge 0 \text{ everywhere} \Bigr\}. \] In addition, we also describe how to mollify $C^{1,1}$ solutions to $C^2$ while preserving monotonicity and sacrificing a controlled amount of optimality. |
| title | Optimal $C^{1,1}$ and Quasi-Optimal $C^2$ Monotone Interpolation with Curvature Control |
| topic | Classical Analysis and ODEs 41A29, 41A15, 41A44 |
| url | https://arxiv.org/abs/2605.14302 |