Относительные средние поперечники пространств Соболевав L2(ℝd)

Yongping Liu; Weiwei Xiao

doi:10.1007/s10476-008-0107-8

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Analysis Mathematica

Volume/Issue:: Volume 34: Issue 1

Относительные средние поперечники пространств Соболевав L²(ℝ^d)

Authors: Yongping Liu¹ and Weiwei Xiao¹

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¹ Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education School of Mathematical Sciences Beijing 100 875 People’s Republic of China

DOI:: https://doi.org/10.1007/s10476-008-0107-8

Page Count:: 71–82

Publication Date:: 20 Jan 2008

Online Publication Date:: 20 Feb 2008

Article Category:: Research Article

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Abstract

In this paper, in order to consider the problems of relative width on ℝ^d, we proposed definitions of relative average width which combine the ideas of the relative width and the average width. We established the smallest number M which make the following equality

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\overline K _\sigma (U(W_2^\alpha ),M(W_2^\alpha ),L_2 ({\mathbb{R}}^d )) = \overline d _\sigma (U(W_2^\alpha ),L_2 ({\mathbb{R}}^d ))$$ \end{document}

hold, where U(W ₂ ^α) is the Riesz potential or Bessel potential of the unit ball in L ²(ℝ^k) and the notations

(·, ·,L ₂(ℝ^d)) and

(·, L ₂(ℝ^d)) denote respectively the relative average width in the sense of Kolmogorov and the average width in the sense of Kolmogorov in their given order. In 2001, Subbotin and Telyakovskii got similar results on the relative width of Kolmogorov type. We also proved that

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\overline K _\sigma (U(W_2^\alpha ) \cap B(L_2 (\mathbb{R}^d )),U(W_2^\beta ) \cap B(L_2 (\mathbb{R}^d ))L_2 (\mathbb{R}^d )) = \overline d _\sigma (U(W_2^\alpha ),L_2 (\mathbb{R}^d )),$$ \end{document}

where 0 × β × α.