MSC: 41A05, 41A25, 41A26
DOI: 10.21538/0134-4889-2024-30-1-270-279
In this paper, we study the optimal interpolation problem in the Sobolev class $W^{n}_{1}([-1,1])$, $n\in\mathbb N$, with Hermite information. By some properties of spline functions, we proved that the Lagrange interpolation based on the extreme points of Chebyshev polynomials is optimal for $W^{n}_{1}([-1,1])$, and we obtained the approximation error for the optimal interpolation problem.
Keywords: Hermite interpolation, spline function, optimal interpolation, Sobolev class
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Received August 17, 2021
Revised December 29, 2023
Accepted January 10, 2024
Funding Agency: This research was supported by the National Natural Science Foundation of China (11871006).
Conflicts of interest: The authors declare no conflict of interest.
Authors’ contributions: Dandan Guo sorted out the contents of the paper, Yongping Liu and Guiqiao Xu provided the ideas and methods of this paper. All authors have read and approved the final manuscript.
Dandan Guo, student, College of Mathematical Science, Beijing Normal University, Beijing 100875, China, e-mail: guodandan0311@163.com
Yongping Liu, Ph. D., Prof., College of Mathematical Science, Beijing Normal University, Beijing 100875, China, e-mail: ypliu@bnu.edu.cn
Guiqiao Xu, Ph. D., Prof., College of Mathematical Science, Tianjin Normal University, Tianjin 300387, China, e-mail: xuguiqiao@aliyun.com
Cite this article as: Dandan Guo, Yongping Liu, and Guiqiao Xu. An optimal interpolation problem with Hermite information in the Sobolev class W1n([-1,1]). Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 1, pp. 270–279.
Русский
Даньдань Го, Йонг Пинг Ли, Гуйцяо Сюй. Оптимальная интерполяционная задача с эрмитовой информацией в классе Соболева $W^{n}_{1}([-1,1])$
В данной работе изучается задача оптимальной интерполяции в классе Соболева W1n([-1, 1]), n ∈ ℕ с информацией Эрмита. С помощью некоторых свойств сплайн-функций мы доказали, что интерполяция Лагранжа по экстремальным точкам полиномов Чебышева является оптимальной для W1n([-1, 1]), и получили ошибку аппроксимации для задачи оптимальной интерполяции.
Ключевые слова: эрмитова интерполяция, сплайн-функция, оптимальная интерполяция, класс Соболева
