Unisolvent functions

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 the properties of unisolvence and the 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 the 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 the 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 the characterization of best Tchebycheff approximations is given. Also given is an interesting theorem on a topological property of the parameter space. The final section contains some examples which illustrate the ideas of the paper. 2. Euclidean n-dimensional space is denoted by E.; points in E. are denoted by a, b, etc. and the 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 the 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