Tchebycheff approximation in several variables
Tchebycheff approximation in several variables
复制标题
多个变量的切比雪夫近似
DOI：
10.1090/s0002-9947-1963-0157165-0
复制
发表时间：
1963
影响因子：
1.3
通讯作者：
J. Rice
中科院分区：
文献类型：
--
作者：
J. Rice
1.导论.理论切比雪夫逼近的职能，一个真实的变量已了解了一段时间，是相当优雅。大约50年来，人们一直试图将这一理论推广到多元函数。这些尝试都失败了，因为缺乏唯一性的最佳近似函数的一个以上的变量。事实上，Mairhuber [6]已经证明，除非我们的函数定义在同胚于单位圆子集的空间上，否则最佳逼近不可能是唯一的。Rivlin和Shapiro [11]得到了进一步的否定结果，结论是在最佳逼近唯一的多个变量中不存在有趣的切比雪夫逼近问题。勋伯格[12]给出了一个多变量理论的说明，但他的假设基本上限制了对一个变量函数的适用性。本文发展了多元函数切比雪夫逼近的两个理论。不能说它们和单变量理论一样令人满意，但它们是可行的理论。对于第一个理论，给出了临界点集的概念（定义与单变量的定义有些不同），并证明了最佳逼近的临界点集的集合是唯一的。给出了最佳逼近的临界点集的特征定理，并证明了切比雪夫逼近理论中的其它定理。Lawson [16]、Rivlin和Shapiro [17]最近也研究了类似的观点。第二个理论定义了定义在有限点集上的函数的严格逼近。除了从第一理论得到的结果，严格近似是唯一的。它也是通常意义上的最佳近似。单变量理论有两个不容易推广的基本特征。第一个是切比雪夫集的概念。在本文中不存在这些集合的对应物。一个启发式的参数表明，有没有可能定义Tchebycheff集在几个变量的重要属性存在于一个变量的情况下。第二个特点是，
1. Introduction. The theory of Tchebycheff approximation for functions of one real variable has been understood for some time and is quite elegant. For about fifty years attempts have been made to generalize this theory to functions of several variables. These attempts have failed because of the lack of uniqueness of best approximations to functions of more than one variable. Indeed, Mairhuber [6] has shown that best approximations cannot be unique unless our functions are defined on a space homeomorphic to a subset of the unit circle. Further negative results have been obtained by Rivlin and Shapiro [11] and the conclusion is that there is no interesting Tchebycheff approximation problem in several variables for which best approximations are unique. Schoenberg [12] gives an account of a theory in several variables, but his assumptions essentially limit the applicability to functions of one variable. In this paper two theories of Tchebycheff approximation to functions of several variables are developed. It cannot be said that they are as satisfying as the theory for one variable, but they are workable theories. For the first theory the concept of a critical point set is given (the definition is somewhat different from that in one variable) and it is shown that the set of critical point sets of a best approximation is unique. A characterization theorem for best approximations is given in terms of critical point sets and other theorems normally found in the theory of Tchebycheff approximation are valid. Similar viewpoints have recently been investigated by Lawson [16] and Rivlin and Shapiro [17]. The second theory defines the strict approximation for functions defined on a finite point set. In addition to the results obtained from the first theory, the strict approximation is unique. It is also a best approximation in the usual sense. There are two basic features of the one variable theory which do not generalize readily. The first of these is the idea of a Tchebycheff set. No counterpart of these sets exists in this paper. A heuristic argument is given that shows that there is no possibility of defining Tchebycheff sets in several variables with the important properties present in the case of one variable. The second feature which does