【作者】 罗先发；

【导师】 李冲；

【作者基本信息】 东南大学 ， 应用数学， 2005， 博士

【摘要】 由Garkavi A L提出的赋范线性空间中集合的限制Chebyshev中心(或最佳同时逼近)问题的研究已有四十年的历史。由于它同连续复杂性问题，集值映射问题和经济决策问题的研究有密切联系，因而自此课题提出以后，就得到了大量研究，取得了许多成果。下面简单介绍一下本文在此领域所作的相关研究。 一．赋范空间中集的限制Chebyshev中心在局部凸空间中的推广—限制p-中心—问题的研究只有十余年的历史。本文首先在复局部凸空间中引入了几种紧性及几种太阳集的概念，它们分别是已有紧性和凸集概念的推广；然后通过建立局部凸空间中集的限制p-中心和关于ker(p)的商空间中对应集的限制Chebyshev中心的关系，来研究局部凸空间中集的限制p-中心问题，本文即使是在实赋范空间和在实局部凸空间中所得的结果都分别是有关已知结果的深化和推广。 二．一般赋范空间中对有限或无限序列的最佳同时逼近问题首先由李冲提出，该问题包括逼近的特征，唯一性和强唯一性等。本文在复赋范空间中研究了同时太阳集对无限序列的加权同时逼近问题，在权满足一定条件时，通过把所研究问题转化为连续向量值函数空间中相应集对一个上半连续函数的最佳逼近问题，得到了逼近的特征和RS集逼近的唯一性定理，有例表明它们是已有结果的本质推广；在权不受限制时，得到了凸集对全有界序列逼近的特征。 三．实值连续函数空间中最佳限制值域逼近问题的研究有较长历史，而复值连续函数空间中相应问题的研究却是近几年来的事情。史应光以“交错”概念为基础，建立了前一问题的Chebyshev极限理论。本文在后一问题的研究中，成功地找到了“交错”的替代概念—二元极支柱，由此得到了后一种逼近的新的特征定理；进而引进了用以刻画逼近集而使得逼近的特征或唯一性结果成立的几种性质，得到了逼近集分别有这些性质的充要条件，建立了本文所考虑的问题的Chebyshev极限理论。更多还原

【Abstract】 It has a history of 40 years on the studies of the theme of restricted Chebyshev centers (best simultaneous approximations) of a set in normed linear spaces proposed by Garkavi. Since it has closed connection with the studies on the problems of continuation complexity, set-valued mappings and economic decisions, it has received very much interests, and a great deal of results have been obtained. The following are the brief descriptions of our studies on the problem in this dissertation.The generalization in locally convex space, restricted p-centers, of restricted Chebyshev centers of a set in normed linear spaces has been studied for only about a little bit more than ten years. In this paper, a varieties of concepts of compactness of sets and sunsets are introduced in a complex locally convex space, which are respectively the generalizations of the known sunsets and convex sets. By establishing the relationship between restricted p-centers of a set in a complex locally convex space and restricted Chebyshev centers of the corresponding set in the quotient space with respect to ker(p), characterizations and uniqueness of restricted p-centers of sets in a complex locally convex space are obtained. These results are better than any other known ones when we restrict them in real normed linear spaces and real locally convex space.The problem of the best simultaneous approximation to a finite or infinite sequence in general normed linear spaces was first introduced by Li, who studied the characterization, uniqueness and strong uniqueness of the approximations. In this dissertation, the best simultaneous approximation with weight to infinite sequence from simultaneous-suns in complex normed linear space is considered, and in the case when the weight satisfies a certain condition, the characterizations of approximation from a simultaneous-sun and uniqueness of approximation from a RS-set are obtained; moreover, in the case when the weight is free, the characterizations of approximation to a totally bounded sequence from a convex set are established.It has a long history to study on the problem of best restricted range approximationin a space of real-valued continuous functions on a closed interval, while the corresponding problem in a space of complex-valued continuous functions on a compact metric space was just introduced in recent years. The Chebyshev limit theory of the former problem was established by Shi by using the notion of "alternation". While on the study of the latter problem in this dissertation, a substitution of the notion of "alternation" is successfully obtained, so that the characterizations of the approximation are given. Furthermore, some properties, which will be used to characterize a proximal set with the characterization and uniqueness, are introduced; sufficient and necessary conditions for a proximal set with these properties are obtained, and consequently Chebyshev limit theory is established.更多还原