Kolmogorov, Linear and Pseudo-Dimensional Widths of Classes of s-Monotone Functions in Lp , 0 < p < 1

Konovalov, Victor N. ; Kopotun, Kirill A.

Canadian mathematical bulletin, Tome 51 (2008) no. 2, pp. 236-248

Résumé

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 ${{\mathbb{L}}_{q}},$ $0\,<\,q\,<\,p\,<\,1$ : $${{d}_{n}}(\Delta _{+}^{s}{{B}_{P}})_{{{\mathbb{L}}_{q}}}^{\text{psd}}\asymp {{d}_{n}}(\Delta _{+}^{s}{{B}_{P}})_{{{\mathbb{L}}_{q}}}^{\text{kol}}\asymp {{d}_{n}}(\Delta _{+}^{s}{{B}_{P}})_{{{\mathbb{L}}_{q}}}^{\text{lin}}\asymp {{n}^{-s}}.$$

Détail

DOI : 10.4153/CMB-2008-025-6

Mots-clés : 41A46, 46E35, 41A29, Kolmogorov width, linear width, pseudo-dimensional widths, s-monotone functions