Extreme points in spaces of polynomials

Abstract

We determine the extreme points of the unit ball in spaces of complex polynomials (of a fixed degree), living either on the unit circle or on a subset of the real line and endowed with the supremum norm. Introduction Let Pn stand for the space of all polynomials with complex coefficients of degree not exceeding n. Given a compact set E ⊂ C, one may treat Pn as a subspace of C(E), the space of continuous functions on E, and equip it with the maximum norm ‖P‖∞ = ‖P‖∞,E := max z∈E |P (z)| (P ∈ Pn). The resulting space will be denoted by Pn(E). We write ball(Pn(E)) := {P ∈ Pn : ‖P‖∞,E ≤ 1} for the unit ball of Pn(E), and we shall be concerned with the extreme points of this ball. (As usual, an element of a convex set S is said to be its extreme point if it is not the midpoint of any nondegenerate segment contained in S.) In this paper, we explicitly characterize the extreme points of ball(Pn(E)) in the case where E is either the circle T := {z ∈ C : |z| = 1} or a perfect compact subset of the real line R. The description obtained is, perhaps, a bit more complicated than one could at first expect; however, the complexity seems to be in the nature of things. Let us begin by recalling that the extreme points of the unit ball in L∞(T) – or in C(T) – are precisely the functions of modulus 1. (The same applies to other sets in place of T.) Further, in the space H∞ of bounded analytic functions on {|z| < 1}, as well as in the disk algebra H∞ ∩ C(T), the extreme points are known to be the unit-norm functions f with ∫ T log(1 − |f(z)|2)|dz| = −∞; see [H, Chap. 9]. Received September 30, 2002. 2000 Mathematics Subject Classification. 46B20, 46E30 Supported in part by Grant 02-01-00267 from the Russian Foundation for Fundamental Research, by a PIV fellowship from Generalitat de Catalunya, and by the Ramon y Cajal program (Spain).