Tchebycheff approximations by functions unisolvent of variable degree

Abstract

1. The first steps toward a theory of nonlinear Tchebycheff approximations were made by T. S. Motzkin [2] and L. Tornheim [4]. They introduced properties of unisolvence and equivalent concept of n-parameter families. For approximating functions which have these properties an elegant theory of Tchebycheff approximations may be developed. Unfortunately, this class of functions does not contain any well-known functions, except for degenerate case of linear approximating functions and transformations thereof. Motzkin [3] has shown, however, that essentially nonlinear unisolvent functions do exist. In this paper a property weaker than unisolvence will be introduced which also allows development of an elegant theory of Tchebycheff approximations. The class of functions possessing this property contains many elementary nonlinear approximating functions. The main definitions are given in ?2. Existence and uniqueness are discussed in ?3 and a theorem on characterization of best Tchebycheff approximations is given. Also given is an interesting theorem on a topological property of parameter space. The final section contains some examples which illustrate ideas of paper. 2. Euclidean n-dimensional space is denoted by E.; points in E. are denoted by a, b, etc. and coordinates of a are (a', a2, * * , an). Curly brackets, { I, denote a set and {xl .. *} is read as the set of x such that .. All maxima and minima are taken over xe [0, 1] unless otherwise stated. The real function F= F(a, x) is defined for xE [0, 1] and aGP where P is a nonvoid subset of En. F is continuous in sense that given aoCP, xoG[0, 1] and e>O there is a 8>0 such that aEP, xe[0, 1], lao-al + xo x <5 implies that I F(ao, xo) F(a, x) I <e. f(x) will denote a continuous function on [0, 1]. The Tchebycheff Approximation Problem for f(x) may be stated as: Determine a*EP such that max |F(a*, x) -f(x) I _ max |F(a, x) -f(x) for all aSP. A solution, F(a*, x), to approximation problem is called a best approximation to f(x).