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Kolmogorov, Linear and Pseudo-Dimensional Widths of Classes of s-Monotone Functions in 𝕃p, 0 < p < 1
Published online by Cambridge University Press: 20 November 2018
Abstract
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Let
${{B}_{p}}$
be the unit ball in
${{\mathbb{L}}_{p}}$
, 
$0\,<\,p\,<\,1$, and let
$\Delta _{+}^{s}$
, 
$s\,\in \,\mathbb{N}$, be the set of all 
$s$-monotone functions on a finite interval 
$I$, i.e.,
$\Delta _{+}^{s}$
consists of all functions 
$x\,:\,I\,\mapsto \,\mathbb{R}$ such that the divided differences
$[x;\,{{t}_{0}},\,...\,,\,{{t}_{s}}]$
of order $s$ are nonnegative for all choices of 
$\left( s\,+\,1 \right)$ distinct points
${{t}_{0}},\,.\,.\,.\,,{{t}_{s}}\,\in \,I.$ For the classes
$\Delta _{+}^{s}{{B}_{P}}\,:=\,\Delta _{+}^{s}\,\cap \,{{B}_{P}},$ we obtain exact orders of Kolmogorov, linear and pseudo-dimensional widths in the spaces
${{\mathbb{L}}_{q}},$
$0\,<\,q\,<\,p\,<\,1$:
$${{d}_{n}}(\Delta _{+}^{s}{{B}_{P}})_{{{\mathbb{L}}_{q}}}^{\text{psd}}\asymp {{d}_{n}}(\Delta _{+}^{s}{{B}_{P}})_{{{\mathbb{L}}_{q}}}^{\text{kol}}\asymp {{d}_{n}}(\Delta _{+}^{s}{{B}_{P}})_{{{\mathbb{L}}_{q}}}^{\text{lin}}\asymp {{n}^{-s}}.$$
- Type
- Research Article
- Information
- Copyright
- Copyright © Canadian Mathematical Society 2008
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