A note on Lototsky–Bernstein bases
Abstract In this note, we study some approximation properties on a class of special Lototsky–Bernstein bases. We focus on approximation of | x | $|x|$ on [ − 1 , 1 ] $[-1,1]$ by an approximation process generated from fixed points on Lototsky–Bernstein bases. Our first result shows that the approxim...
| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2024-02-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s13660-024-03076-7 |
| _version_ | 1797272842820124672 |
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| author | Xiao-Wei Xu Xin Yu Jia-Lin Cui Qing-Bo Cai Wen-Tao Cheng |
| author_facet | Xiao-Wei Xu Xin Yu Jia-Lin Cui Qing-Bo Cai Wen-Tao Cheng |
| author_sort | Xiao-Wei Xu |
| collection | DOAJ |
| description | Abstract In this note, we study some approximation properties on a class of special Lototsky–Bernstein bases. We focus on approximation of | x | $|x|$ on [ − 1 , 1 ] $[-1,1]$ by an approximation process generated from fixed points on Lototsky–Bernstein bases. Our first result shows that the approximation procedure p n ( x ) $p_{n}(x)$ to | x | $|x|$ preserves good shapes on [ − 1 , 1 ] $[-1,1]$ . Moreover, some convergence results and inequalities are derived. Our second main result states that the rate convergence of the approximation is O ( n − 2 ) $O(n^{-2})$ . |
| first_indexed | 2024-03-07T14:35:15Z |
| format | Article |
| id | doaj.art-9f760298326e4876985d4c7bd0872f01 |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-03-07T14:35:15Z |
| publishDate | 2024-02-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-9f760298326e4876985d4c7bd0872f012024-03-05T20:41:29ZengSpringerOpenJournal of Inequalities and Applications1029-242X2024-02-01202411610.1186/s13660-024-03076-7A note on Lototsky–Bernstein basesXiao-Wei Xu0Xin Yu1Jia-Lin Cui2Qing-Bo Cai3Wen-Tao Cheng4School of Computer and Data Engineering, NingboTech UniversitySchool of Computer and Data Engineering, NingboTech UniversitySchool of Information, NingboTech UniversitySchool of Mathematics and Computer Science, Quanzhou Normal UniversitySchool of Mathematics and Physics, Anqing Normal UniversityAbstract In this note, we study some approximation properties on a class of special Lototsky–Bernstein bases. We focus on approximation of | x | $|x|$ on [ − 1 , 1 ] $[-1,1]$ by an approximation process generated from fixed points on Lototsky–Bernstein bases. Our first result shows that the approximation procedure p n ( x ) $p_{n}(x)$ to | x | $|x|$ preserves good shapes on [ − 1 , 1 ] $[-1,1]$ . Moreover, some convergence results and inequalities are derived. Our second main result states that the rate convergence of the approximation is O ( n − 2 ) $O(n^{-2})$ .https://doi.org/10.1186/s13660-024-03076-7Lototsky–Bernstein operatorsRecursive approximationAbsolute functionsConvergence rate |
| spellingShingle | Xiao-Wei Xu Xin Yu Jia-Lin Cui Qing-Bo Cai Wen-Tao Cheng A note on Lototsky–Bernstein bases Journal of Inequalities and Applications Lototsky–Bernstein operators Recursive approximation Absolute functions Convergence rate |
| title | A note on Lototsky–Bernstein bases |
| title_full | A note on Lototsky–Bernstein bases |
| title_fullStr | A note on Lototsky–Bernstein bases |
| title_full_unstemmed | A note on Lototsky–Bernstein bases |
| title_short | A note on Lototsky–Bernstein bases |
| title_sort | note on lototsky bernstein bases |
| topic | Lototsky–Bernstein operators Recursive approximation Absolute functions Convergence rate |
| url | https://doi.org/10.1186/s13660-024-03076-7 |
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