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On nonlinear -widths

Authors:	Dinh Dung Vu Quoc Thanh

Institution:	Institute of Information Technology, Nghia Do, Tu Liem, Hanoi 10000, Vietnam ; Institute of Information Technology, Nghia Do, Tu Liem, Hanoi 10000, Vietnam

Abstract:	For characterization of best nonlinear approximation, DeVore, Howard, and Micchelli have recently suggested the nonlinear -width $\delta _n(W,X)$ of a subset in a normed linear space . We proved by a topological method that for $\delta _n(W,X)$ and the well-known Aleksandrov -width in a Banach space 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 and $\alpha$ , we established the asymptotic degree of these -widths: $a_n(K_{p,\theta }^{\alpha },L_q)\approx \delta _n(K_{p,\theta }^{\alpha }, L_q)\approx n^{-\alpha }$ .

Keywords:	Nonlinear approximation $n$-widths Besov space

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