A note on optimal Hermite interpolation in Sobolev spaces
Abstract This paper investigates the optimal Hermite interpolation of Sobolev spaces W ∞ n [ a , b ] $W_{\infty }^{n}[a,b]$ , n ∈ N $n\in \mathbb{N}$ in space L ∞ [ a , b ] $L_{\infty }[a,b]$ and weighted spaces L p , ω [ a , b ] $L_{p,\omega }[a,b]$ , 1 ≤ p < ∞ $1\le p< \infty $ with ω a cont...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2022-01-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s13660-021-02741-5 |
| _version_ | 1818979342814806016 |
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| author | Guiqiao Xu Xiaochen Yu |
| author_facet | Guiqiao Xu Xiaochen Yu |
| author_sort | Guiqiao Xu |
| collection | DOAJ |
| description | Abstract This paper investigates the optimal Hermite interpolation of Sobolev spaces W ∞ n [ a , b ] $W_{\infty }^{n}[a,b]$ , n ∈ N $n\in \mathbb{N}$ in space L ∞ [ a , b ] $L_{\infty }[a,b]$ and weighted spaces L p , ω [ a , b ] $L_{p,\omega }[a,b]$ , 1 ≤ p < ∞ $1\le p< \infty $ with ω a continuous-integrable weight function in ( a , b ) $(a,b)$ when the amount of Hermite data is n. We proved that the Lagrange interpolation algorithms based on the zeros of polynomial of degree n with the leading coefficient 1 of the least deviation from zero in L ∞ $L_{\infty }$ (or L p , ω [ a , b ] $L_{p,\omega }[a,b]$ , 1 ≤ p < ∞ $1\le p<\infty $ ) are optimal for W ∞ n [ a , b ] $W_{\infty }^{n}[a,b]$ in L ∞ [ a , b ] $L_{\infty }[a,b]$ (or L p , ω [ a , b ] $L_{p,\omega }[a,b]$ , 1 ≤ p < ∞ $1\le p<\infty $ ). We also give the optimal Hermite interpolation algorithms when we assume the endpoints are included in the interpolation systems. |
| first_indexed | 2024-12-20T16:58:01Z |
| format | Article |
| id | doaj.art-38f6061e488a478695b4453dbef2e4dc |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-12-20T16:58:01Z |
| publishDate | 2022-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-38f6061e488a478695b4453dbef2e4dc2022-12-21T19:32:39ZengSpringerOpenJournal of Inequalities and Applications1029-242X2022-01-012022111410.1186/s13660-021-02741-5A note on optimal Hermite interpolation in Sobolev spacesGuiqiao Xu0Xiaochen Yu1Department of Mathematics, Tianjin Normal UniversityDepartment of Mathematics, Tianjin Normal UniversityAbstract This paper investigates the optimal Hermite interpolation of Sobolev spaces W ∞ n [ a , b ] $W_{\infty }^{n}[a,b]$ , n ∈ N $n\in \mathbb{N}$ in space L ∞ [ a , b ] $L_{\infty }[a,b]$ and weighted spaces L p , ω [ a , b ] $L_{p,\omega }[a,b]$ , 1 ≤ p < ∞ $1\le p< \infty $ with ω a continuous-integrable weight function in ( a , b ) $(a,b)$ when the amount of Hermite data is n. We proved that the Lagrange interpolation algorithms based on the zeros of polynomial of degree n with the leading coefficient 1 of the least deviation from zero in L ∞ $L_{\infty }$ (or L p , ω [ a , b ] $L_{p,\omega }[a,b]$ , 1 ≤ p < ∞ $1\le p<\infty $ ) are optimal for W ∞ n [ a , b ] $W_{\infty }^{n}[a,b]$ in L ∞ [ a , b ] $L_{\infty }[a,b]$ (or L p , ω [ a , b ] $L_{p,\omega }[a,b]$ , 1 ≤ p < ∞ $1\le p<\infty $ ). We also give the optimal Hermite interpolation algorithms when we assume the endpoints are included in the interpolation systems.https://doi.org/10.1186/s13660-021-02741-5Optimal Hermite interpolationSobolev spaceWorst-case setting |
| spellingShingle | Guiqiao Xu Xiaochen Yu A note on optimal Hermite interpolation in Sobolev spaces Journal of Inequalities and Applications Optimal Hermite interpolation Sobolev space Worst-case setting |
| title | A note on optimal Hermite interpolation in Sobolev spaces |
| title_full | A note on optimal Hermite interpolation in Sobolev spaces |
| title_fullStr | A note on optimal Hermite interpolation in Sobolev spaces |
| title_full_unstemmed | A note on optimal Hermite interpolation in Sobolev spaces |
| title_short | A note on optimal Hermite interpolation in Sobolev spaces |
| title_sort | note on optimal hermite interpolation in sobolev spaces |
| topic | Optimal Hermite interpolation Sobolev space Worst-case setting |
| url | https://doi.org/10.1186/s13660-021-02741-5 |
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