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

MSC: 41A63, 65Y20, 68Q25

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

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

REFERENCES

1.   N. I. Achieser. Theory of approximation. N Y: Dover Publications, INC., 1992, 307 p. ISBN: 0486671291 .

2.   V. F. Babenko. Approximations in the mean with constraints on the derivatives of approximating functions. In : Questions in Analysis and Approximations. Kiev: Akad. Nauk Ukrain. SSR, Inst. Mat., 1989, pp. 9–18 (in Russian).

3.   Dirong Chen. Average n-widths and optimal recovery of Sobolev classes in $L_p(\mathbb R).$ Chinese Ann. Math. Ser. B, 1992, vol. 13, no. 4, pp. 396–405.

4.   R. A. DeVore, G. G. Lorentz. Constructive approximation. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 303, Berlin: Springer-Verlag, 1993, 449 p. ISBN: 3-540-50627-6 .

5.   R. S. Ismagilov On n-dimensional diameters of compacts in a Hilbert space. Funct. Anal. Its. Appl., 1968, vol. 2, no. 2, pp. 125–132. doi: 10.1007/BF01075946

6.   Yanjie Jiang. Widths and optimal recovery of smooth function classes. PhD thesis, Beijing Normal University, 1998.

7.   V. N. Konovalov. Estimates of Kolmogorov-type widths for classes of differentiable periodic functions. Math. Notes, 1984, vol. 35, no. 3, pp. 193–199. doi: 10.1007/BF01139916

8.   V. N. Konovalov. Approximation of Sobolev classes by their finite-dimensional sections. Math. Notes, 2002, vol. 72, no. 3, pp. 337–349. doi: 10.1023/A:1020547320561

9.   V. N. Konovalov. Approximation of Sobolev classes by their sections of finite dimension. Ukraine Math. J., 2002, vol. 54, no. 5, pp. 795–805. doi: 10.1023/A:1021635530578

10.   N. P. Korneichuk. Ekstremal’nye zadachi teorii priblizheniya [Extremal problems of approximation theory]. Moscow: Nauka Publ., 1976, 320 p.

11.   N. P. Korneichuk. Exact constants in approximation theory. Cambridge: Cambridge University Press, 1991, Encyclopedia Math. Appl., vol. 38, 466 p. ISBN: 9781107094277 .

12.   Bo Ling, Yongping Liu. Best restriction approximation of Sobolev classes by entire functions of exponential type. Acta Mathematica Sinica, Chinese Series, 2017, vol. 60, no. 3, pp. 389–400.

13.   Yongping Liu. Infinite dimensional widths and optimal recovery on the S - W spaces. PhD thesis, Beijing Normal University, 1993.

14.   Yongping Liu, Weiwei Xiao. Relative average widths of Sobolev spaces in $L_2(\mathbb R^d)$. Anal. Math., 2008, vol. 34, no. 1, pp. 71–82. doi: 10.1007/s10476-008-0107-8

15.   G. G. Lorentz, M. V. Golitschek, Y. Makovoz. Constructive approximation. Advanced problems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 304. Berlin: Springer-Verlag, 1996, 649 p. ISBN: 3-540-57028-4 .

16.   G. G. Magaril-Il’yaev. $\varphi$-mean diameters of classes of functions on the line. Russian Math. Surv., 1990, vol. 45, no. 2, pp. 218–219. doi: 10.1070/RM1990v045n02ABEH002340

17.   G. G. Magaril-Il’yaev. Mean dimension, widths, and optimal recovery of Sobolev classes of functions on the line. Math. USSR-Sb., 1993, vol. 74, no. 2, pp. 381–403. doi: 10.1070/SM1993v074n02ABEH003352

18.   G. G. Magaril-Il’yaev, V. M. Tikhomirov. Average dimension and ν-widths of classes of functions on the whole line. J. Complexity, 1992, vol. 8, no. 1, pp. 64–71. doi: 10.1016/0885-064X(92)90034-9

19.   S. M. Nikol’skii. Approximation of functions of several variables and embedding theorems. Berlin; N Y: Springer-Verlag, 1975, 420 p. ISBN: 0387064427 .

20.   Heping Wang. Approximation and quadrature formula on function classes with mixed smoothness. PhD thesis, Beijing Normal University, 1996.

21.   Yongsheng Sun, Yongping Liu, Dirong Chen. Extremal problems in approximation theory for some classes of smooth functions defined on $\mathbb R^d.$ J. Beijing Normal University (Natural Sciences), 1999, vol. 35(supp.), pp. 79–144.

22.   V. M. Tikhomirov. Some remarks on relative diameters. In: Approximation and function spaces, Proc. 27th Semest., Warsaw/Pol. 1986, Banach Cent. Publ., 1989, vol. 22, pp. 471–474.

23.   V. M. Tikhomirov. On approximation properties of smooth functions. Proc. Conf. on Differential Equations and Numerical Mathematics, (Novosibirsk: Nauka Publ., 1980), pp. 183–188 (in Russian).

24.   Guiqiao Xu. Widths and optimal recovery of multivariate smooth functions. PhD thesis, Beijing Normal University, 2001.

25.   Wei Yang. Relative widths of differentiable function classes and convolution classes with $2\pi$  periodic in one variable case and hexagonal periodic in 2-dimensional case. PhD thesis, Beijing Normal University, 2009.

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