The characterization of best nonlinear Tchebycheff approximations

Abstract

JOHN R. RICE(')1.1.Let two real functions/ and F be given such that/=/(x) is a continuous function for x [0, l] and F=F(al, a2, , a", x) is a continuous function of n parameters and x(E [0, l].For simplicity of notation the point (a1, a2, , a") in Euclidean w-space, En, is denoted by a.The domain of the parameters of F is denoted by P, a subset of En.The main problem in the theory of approximation of continuous functions may be stated as follows: Determine a*EP so that the deviation of the function F(a, x) from f(x) shall be minimized.Naturally one must define the deviation of F(a, x) from/(x) and different definitions lead to different theories.In this paper the deviation of F(a, x) from f(x) is taken to be maxI(=[o,i] | F(a, x) -f(x) \.All maxima and minima are taken over x [0, l] unless otherwise stated.F(a*, x) is said to be a best approximation to/The results of this paper remain valid if [0, l] is replaced by any other closed and bounded interval.By 1920 an elegant theory of Tchebycheff approximation had been developed for F(a, x) depending linearly on the parameters [2; 3; 7].Some of the main points of this linear theory are given in the following resume.Let T be a set of n functions 4>i(x), 4>2(x), , </>"(x) continuous on [0, 1 ] and let P(a, x) = /,?_, a*'<,(x) be called a T-polynomial.P(a, x) is said to be a nontrivial T-polynomal if 2~ll=i | a*| >0.T is said to be a Tchebycheff Set if every nontrivial polynomial has at most n-1 zeros, max | F(a, x) -/(x)| is said to alternate n times if there are + l points 0^Xi<x2<Alternance is a basic concept in the theory of Tchebycheff approximations.The following theorem answers the principal questions of a general nature in the theory of linear Tchebycheff approximations.Theorem.Let T be a Tchebycheff set and let f(x) be an arbitrary function continuous on [0, l].Then A. f(x) possesses a best approximation, B. a necessary and sufficient condition that P(ao, x) be a best approximation to f(x) is that max | P(a0, x) -f(x) \ alternates at least n times, C. the best approximation to f(x) is unique.