Best Uniform Polynomial Approximation

Abstract

This chapter is a continuation of Chapter 4, in that the best uniform polynomial approximation $P^{*} \in \pi_{n} \,{\text{of}}\,f \in C[a,\,b],$ with existence of $P^{*}$ guaranteed by Theorem 4.1.2, will be characterized in terms of the alternation properties of the error function $f - P^{*} .$ As an application, the uniqueness of $P^{*} \in \pi_{n}$ as the only best uniform polynomial approximant of $f \in C[a,b]$ is assured