Tchebycheff approximation in several variables

Abstract

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