Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On nonlinear $n$-widths


Authors: Dinh Dung and Vu Quoc Thanh
Journal: Proc. Amer. Math. Soc. 124 (1996), 2757-2765
MSC (1991): Primary 41A46, 41A25, 41A63, 42A10
DOI: https://doi.org/10.1090/S0002-9939-96-03337-0
MathSciNet review: 1327007
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For characterization of best nonlinear approximation, DeVore,
Howard, and Micchelli have recently suggested the nonlinear $n$-width $\delta _n(W,X)$ of a subset $W$ in a normed linear space $X$. We proved by a topological method that for $\delta _n(W,X)$ and the well-known Aleksandrov $n$-width $a_n(W,X)$ in a Banach space $X$ the following inequalities hold: $\delta _{2n+1}(W,X)\le a_n(W,X)\le \delta _n(W,X)$. Let $K_{p,\theta }^{\alpha }$ be the unit ball of Besov space $B_{p,\theta }^{\alpha },\quad \alpha >0,\quad 1\le p,\theta \le \infty $, of multivariate periodic functions. Then for approximation in $L_q,\quad 1\le q\le \infty $, with some restriction on $p,q$ and $\alpha $, we established the asymptotic degree of these $n$-widths: $a_n(K_{p,\theta }^{\alpha },L_q)\approx \delta _n(K_{p,\theta }^{\alpha }, L_q)\approx n^{-\alpha }$.


References [Enhancements On Off] (What's this?)

  • [1] P.S.Aleksandrov, Über die Urysonschen Konstanten, Fund. Math. 20 (1933), 140--150.
  • [2] P.S.Aleksandrov and V.A.Pasynkov, Introduction to dimension theory, Nauka, Moscow, 1973. MR 51:1776
  • [3] Dinh Dung, On recovery and onesided approximation of periodic functions of several variables, Dokl. Akad. Nauk USSR 313 (1990), 757--790. CMP 91:04
  • [4] R. DeVore, Degree of nonlinear approximation, Approximation theory VI:v.1, Acad. Press, 1989, 175--201. MR 92e:41021
  • [5] R.DeVore, R.Howard and C.Micchelli, Optimal nonlinear approximation, Manuscripta Mathematica 63 (1989), 469--478. MR 90c:41053
  • [6] R.DeVore and Xiang Ming Yu, Nonlinear n-width Besov space, Approximation theory VI:v.1, Acad. Press, 1989, pp. 203--206. MR 92e:41022
  • [7] R.A.DeVore, G.Kyriazis, D.Leviatan and V.M.Tikhomirov, Wavelet compression and nonlinear $n$-widths (manuscript).
  • [8] C.H.Dowker, On a theorem of Hanner, Ark. Math. 2 (1952), 307--313. MR 14:396a
  • [9] A.A.Kitbalyan, On widths of anisotropic classes of functions of a finite smoothness, Uspekhi Mat. Nauk 45 (1990), 177--178. MR 91d:46036
  • [10] P.Mathé, s-Number in information-based complexity, J. of Complexity 6 (1990), 41--66. MR 91e:41042
  • [11] S.M.Nikolskii, Approximation of functions of several variables and embedding theorems,
    Springer-Verlag, Berlin, 1975. MR 51:11073
  • [12] M.I.Stesin, Aleksandrov widths of finite dimensional set and of classes of smooth functions, Dokl. Akad. Nauk USSR 220 (1975), 1278--1281. MR 58:29729
  • [13] V.M.Tikhomirov, Some problems of approximation theory, Moscow Univ., Moscow, 1976. MR 58:6822
  • [14] A.Zygmund, Trigonometric series, II, Cambridge Univ. Press, 1959. MR 21:6498

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 41A46, 41A25, 41A63, 42A10

Retrieve articles in all journals with MSC (1991): 41A46, 41A25, 41A63, 42A10


Additional Information

Dinh Dung
Affiliation: Institute of Information Technology, Nghia Do, Tu Liem, Hanoi 10000, Vietnam
Email: ddung@math-ioit.ac.vn

Vu Quoc Thanh
Affiliation: Institute of Information Technology, Nghia Do, Tu Liem, Hanoi 10000, Vietnam

DOI: https://doi.org/10.1090/S0002-9939-96-03337-0
Keywords: Nonlinear approximation, $n$-widths, Besov space
Received by editor(s): April 8, 1992
Received by editor(s) in revised form: March 8, 1995
Additional Notes: This work was supported by Project 1.5.5 of the Vietnamese National Program for Researches in Natural Sciences.
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society