Study On Problems Of Some Classes Of Simultaneous Approximations In Complex Normed Spaces

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.