On nonlinear $n$-widths
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- by Dinh Dung and Vu Quoc Thanh
- Proc. Amer. Math. Soc. 124 (1996), 2757-2765
- DOI: https://doi.org/10.1090/S0002-9939-96-03337-0
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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
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Bibliographic 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
- 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
- © Copyright 1996 American Mathematical Society
- 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