LINEAR PROBLEMS IN THE SPACE OF POLYNOMIALS OF DEGREE AT MOST 3

Abstract

enote by P n, n∈N the linear space of real polynomials p of degree at most n. There are various ways in which we can introduce norm in P n, here the problem is investigated when |p|=max{|p(x)|:x∈[-1;1]}. Let B n={p∈ P_n:|p|≤1} be the unit ball and let EB n be the set of the extreme points of B n, i.e. such points p∈ B n that B n\{p} is convex. The sets EB 0, EB 1 and EB 2 are known and it turns out that also EB 3 has a particularly simple form. In this paper we determine EB 3 and give some conclusions and applications of the main results. Moreover, several examples are included. The coefficient region for the polynomials of degree exceeding 3 seems very complicated.