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线性赋范空间联合最佳逼近的特征

蒋鸣和,徐康康,陈竞先,陈馥荪,张鑫宝,唐德培,施瑛   

  1. 上海市业余工业大学 ,上海市业余工业大学 ,上海市业余工业大学 ,上海市业余工业大学 ,上海市业余工业大学 ,上海市业余工业大学 ,上海市业余工业大学
  • 出版日期:1984-03-14 发布日期:1984-03-14

蒋鸣和,徐康康,陈竞先,陈馥荪,张鑫宝,唐德培,施瑛. 线性赋范空间联合最佳逼近的特征[J]. 计算数学, 1984, 6(3): 261-272.

THE CHARACTERISTICS OF BEST SIMULTANEOUS APPROXIMATION IN LINEAR NORMED SPACE

  1. Jiang Ming-he;Xu Kang-kang;Chen Jing-xian;Chen Fu-sun;Zhang Xin-bao;Tang De-pei;Shi Ying Shanghai Part-time University of Technology
  • Online:1984-03-14 Published:1984-03-14
§1.引言 线性赋范空间R中一个元对一族元的同时逼近,即联合最佳逼近问题,在实践中有广泛应用。例如,多目标规划中的有效解问题;数学物理方程中一个函数对另一个函数及其任意阶导数的同时逼近问题;最优控制问题;数理统计中Gauss-Markov线性模型最佳有偏最小方差估计的求解问题等等。本文主要研究下述形式联合最佳逼近问题。 定义1.1. 设{x_n}是线性赋范空间R中一列元,G是空间R的子集,{α_n}是一列非负实数,对给定正数p≥1,满足
In this paper, the imbedding of the normed linear space R into its weighted direct sumspace R_ω~P is applied to the discussion on the characterization theorems of functional and variat-ional forms of the best simultaneous approximation by using the elements of an arbitrary convexset G in the space R. The main result is as follows. Theorem. Let G be a convex subset in a normed linear space. Assume that a sequence{x_n} R consisting of distinct points or x is not in G. Then y_0∈G ({x_n}) if and only ifthere exists a sequence {f_n} of bounded linear functionals such thatfor p=1:(1)||f_n||=1, N=1, 2, …;(2)Re sum form n=1 to ∞α_nf_n(y-y_0)≥0, y∈G;(3)Ref_n(x_n-y_0)=||x_n-y_0||, n=1, 2, ….for p>1:(1)||f_n||=(||x_n-y_0||~(p/q))/(sum form n=1 to ∞ α_n||x_n-y_0||~p)~(1/q), N=1, 2, …;(2)Re sum from n=1 to ∞ f_n(y_0-y)≥0, y∈G; (3)Ref_n(x_n-y_0)=||f_n||·||x_n-y_0||=(||x_n-y_0||~p/(sum form n=1 to ∞ α_n||x_n-y_0||~p)~(1/q),As an example, we give the characterization of the best simultaneous L_p-approximation.
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