This article is cited in 2 papers

On the Orders of Nonlinear Approximations for Classes of Functions of Given Form

V. N. Konovalov

Institute of Mathematics, Ukrainian National Academy of Sciences

Abstract: Suppose that $\Delta^s_+$ is the set of functions $x\colon I\to\mathbb R$ on a finite interval $I$ such that the divided differences $[x;t_0,\dots,t_s]$ of order $s\in\mathbb N$ of these functions are nonnegative for all collections from $(s+1)$ different points $t_0,\dots,t_s\in I$. For all $s\in\mathbb N$ and $1\le p\le\infty$, we establish exact orders of best approximations by splines with free nodes and rational functions in the metrics of $L_p$ for classes $\Delta^s_+B_p:=\Delta^s_+\cap B_p$, where $B_p$ is the unit ball in $L_p$. We also establish the asymptotics of pseudodimensional widths in $L_p$ of these classes of functions.

UDC: 517.5

Received: 28.06.2004

DOI: 10.4213/mzm2565

 English version: Mathematical Notes, 2005, 78:1, 88–104

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