Saddle-point theorems for rational approximation

Abstract

Rational approximation is considered where the ( n + m) functions involved are supposed to be continuous on a general compact set. With the aid of Helly-type theorems, the approximation viewed as a mathematical program is reduced to one that is discrete, without any assumption regarding the existence of solutions. This discrete problem, which is the rational approximation considered on an at most ( n + m) element subset of our compact set, has the same value as the original problem, while its solution set includes that of the original problem. Moreover, all the above sets of cardinality at most ( n + m) are found by max-inf statements, where the maximum interchange with the infimum and a finite number of variables are involved. If the original approximation problem has a solution, then all of its solutions, as well as all the above-mentioned finite subsets, are expressed by the saddle points of our minimax statements.