Yongping Liu, Quiqiao Xu, Jie Zhang. Best restricted approximation of smooth function classes ... P. 283-294

Vol. 24, no. 4, 2018

MSC: 41A63, 65Y20, 68Q25

DOI: 10.21538/0134-4889-2018-24-4-283-294

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This work was supported by the National Natural Science Foundation of China (Grant No. 11471043, 11671271) and by the Beijing Natural Science Foundation (Grant No. 1172004). Jie Zhang is the corresponding author.

We first discuss the relative Kolmogorov $n$-widths of classes of smooth $2\pi$-periodic functions for which the modulus of continuity of their $r$-th derivatives does not exceed a given modulus of continuity, and then discuss the best restricted approximation of classes of smooth bounded functions defined on the real axis $\mathbb R$ such that the modulus of continuity of their $r$-th derivatives does not exceed a given modulus of continuity by taking the classes of the entire functions of exponential type as approximation tools. Asymptotic results are obtained for these two problems.

Keywords: modulus of continuity, best restricted approximation, average width

Юнпин Лю, Гуйцяо Сюй, Цзе Чжан. Наилучшая аппроксимация с ограничениями для классов гладких функций

Обсуждаются относительные $n$-поперечники по Колмогорову для классов гладких $2\pi$-периодических функций, определяемых модулем непрерывности, а также наилучшая аппроксимация с ограничениями целыми функциями экспоненциального типа для классов гладких ограниченных функций, определенных на числовой оси $\mathbb R$ и таких, что модуль непрерывности их $r$-й производной не превосходит заданного модуля непрерывности. Для этих двух задач получены асимптотические результаты.

Ключевые слова: модуль непрерывности, наилучшая аппроксимация с ограничениями, средний поперечник

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Received August 31, 2018

Revised October 25, 2018

Accepted October 29, 2018

Yongping Liu, Prof., School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China, e-mail: ypliu@bnu.edu.cn

Guiqiao Xu, Prof., School of Mathematical Sciences, Tianjin Normal University, Tianjin, 300387, China, e-mail: xuguiqiao@tjnu.edu.cn

Jie Zhang, Dr., School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China, e-mail: zhangjie91528@163.com