The Theory of Best Approximation and Functional Analysis
The Theory of Best Approximation and Functional Analysis
复制标题
DOI：
10.1137/1.9781611970548
复制
发表时间：
1974
期刊：
影响因子：
--
通讯作者：
I. Singer
中科院分区：
文献类型：
--
作者：
I. Singer
在本专着中，我们提出了现代最佳逼近理论中的一些结果和问题，即以随后的方式应用泛函分析方法。这一现代理论既构成了经典最佳逼近理论（用函数论的方法处理问题）的统一基础，又是获得新结果的有力工具。在赋范线性空间的一般框架内，最佳逼近问题相当于距离的最小化，这使得我们可以使用几何直觉（但需要严格的分析证明），并且在各种特定的具体函数空间中，现象的联系比经典的最佳逼近理论更加清晰，论证也更加简单。我们希望这一点在专着[168]（这是文献中的第一本）和讲义[175]中得到了足够令人信服的证明，并将在本专着中再次得到证明（例如，参见§1，定理1.8之后的评论，或§3，对定理3.5的评论）。当然，最佳逼近理论和泛函分析之间的相互作用也适用于另一方面，例如，赋范线性空间中的最佳逼近问题导致了极值泛函极值扩展定理的发现、某些共轭空间中晶胞极值点的具体表示、集值映射半连续性的新结果等的发现。但是，我们在本专着中不考虑它们相互作用的这一方面。
In this monograph we present some results and problems in the modern theory of best approximation, i.e., in which the methods of functional analysis are applied in a consequent manner. This modern theory constitutes both a unified foundation for the classical theory of best approximation (which treats the problems with the methods of the theory of functions) and a powerful tool for obtaining new results. Within the general framework of normed linear spaces the problem of best approximation amounts to the minimization of a distance, which permits us to use geometric intuition (but rigorous analytic proofs), and the connections of the phenomena become clearer and the arguments simpler than those of the classical theory of best approximation in the various particular concrete function spaces. We hope that this has been proved convincingly enough in the monograph [168] (which was the first of this kind in the literature) and in the lecture notes [175], and will be proved again in the present monograph (see, for example, § 1, the remarks made after Theorem 1.8, or § 3, the remark to Theorem 3.5).Naturally, the interaction between the theory of best approximation and functional analysis works also in the other direction, for example, problems of best approximation in normed linear spaces have led to the discovery of the theorem on extremal extension of extremal functional, of the concrete representations for the extremal points of the unit cell in certain conjugate spaces, of new results on semi-continuity of set-valued mappings, etc. However, we shall not consider this side of their interaction in the present monograph.