由分数次导数所确定的L2(T)中的一元周期函数类在Lq(T)中的相对宽度

数学物理学报 ›› 2009, Vol. 29 ›› Issue (4): 833-842.

• 论文 • 下一篇

由分数次导数所确定的L₂(T)中的一元周期函数类在L_q(T)中的相对宽度

(1. 北京师范大学数学科学学院, 数学与复杂系统教育部重点实验室, 北京 100875, 2. 北方工业大学理学院, 北京 100144)

收稿日期:2007-10-29 修回日期:2009-03-25 出版日期:2009-08-25 发布日期:2009-08-25
基金资助:
国家自然科学基金(10771016)、北京师范大学``985项目"和北京自然科学基金(1062004)资助

Relative Widths of Function Classes of L₂(T) Determined by Fractional Order Derivatives in L_q(T)

(1. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875 2. College of Sciences, North China University of Technology, |Beijing 100144)

Received:2007-10-29 Revised:2009-03-25 Online:2009-08-25 Published:2009-08-25
Supported by:
国家自然科学基金(10771016)、北京师范大学``985项目"和北京自然科学基金(1062004)资助

摘要/Abstract

摘要：

作者研究了相对宽度K_n(W₂^α(T), MW₂^β(T), L₂(T)), T=[0,2 π], 确定了使等式K_n(W₂^α(T), MW₂^β(T), L₂(T))=d_n(W₂^α(T), L₂(T))成立的最小M值, 得到了相对宽度K_n(W₂^α(T), W₂^α(T), L_q(T))的渐近阶, 其中α≥β>0, 1≤q ≤ ∞ , K_n (., ., L_q(T)) 和 d_n(., L_q(T))分别表示Kolmogorov意义下L_q(T)尺度下的相对宽度和宽度, MW_p^α(T), 1≤ p ≤ ∞ , 表示有如下卷积表达式的2 π 周期函数类, f(t)=c+(B_α* g)(t), c∈ R, B_α* g 表示 B_α 和g 的卷积, g∈ L_p(T) 满足∫₀^2πg(τ)dτ=0 和||g||_p≤ M, B_α∈ L₁(T) 有如下Fourier展开: B_α(t)=1/2π ∑' _k∈ _Z(ik)^-αe^ikt, ∑^'表示去掉 k=0的项.

关键词: 相对宽度, n-K宽度, 分数次导数

Abstract:

The relative widths K_n(W₂^α(T), MW₂^β(T), L₂(T)), T=[0, 2π], is studied and the smallest number M which makes the equality K_n(W₂^α(T), MW₂^β(T), L₂(T))=d_n(W₂^α(T), L₂(T)) valid is obtained, and the asymptotic order of relative widths K_n(W₂^α(T), W₂^α(T), L_q(T) ) is obtained, where α≥β>0, 1≤q ≤ ∞ , K_n (., ., L_q(T)) and d_n(., L_q(T)) denote respectively the relative widths and the widths in the sense of Kolmogorov in L_q(T), and MW_p^α(T), 1≤ p ≤ ∞ , denotes the collection of 2π-periodic and continuous functions f representable as a convolution f(t)=c+(B_α* g)(t), where B_α* g denotes the convolution of B_αand g, for g ∈ L_p(T) satisfying ∫₀^2πg(τ)dτ=0 and ||g||_p≤ M. Here B_αis in L₁(T) with the Fourier expansion
B_α(t) )=1/2π ∑' _k∈ _Z(ik)^-αe^ikt, where ∑^'means that the term is omitted when k=0.

Key words: Relative widths, n-K widths, Derivatives of fractional order

中图分类号:

41A46

肖维维, 刘永平. 由分数次导数所确定的L₂(T)中的一元周期函数类在L_q(T)中的相对宽度[J]. 数学物理学报, 2009, 29(4): 833-842.

XIAO Wei-Wei, LIU Yong-Ping. Relative Widths of Function Classes of L₂(T) Determined by Fractional Order Derivatives in L_q(T)[J]. Acta mathematica scientia,Series A, 2009, 29(4): 833-842.

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由分数次导数所确定的L₂(T)中的一元周期函数类在L_q(T)中的相对宽度

Relative Widths of Function Classes of L₂(T) Determined by Fractional Order Derivatives in L_q(T)

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