Majorants and Extreme Points of Unit Balls in Bernstein Spaces

Abstract

The Bernstein space Bp (σ) (1\( \leqslant p \leqslant \infty ,\sigma >\)0) is the set of functions from Lp(\(\mathbb{R}\)) having Fourier transforms (in the sense of generalized functions) with supports in the compact segment [-σ , σ ]. Every function f\( \in B^p (\sigma )\) has an analytic continuation onto the complex plane, which is an entire function of exponential type ≤ σ. The spaces Bp (σ)\, are conjugate Banach spaces. Therefore, the closed unit ball \(\mathcal{D}(B^p (\sigma ))\) in Bp (σ) has a rich set of extreme (boundary) points: \(\mathcal{D}(B^p (\sigma ))\) coincides with the weakly * closed convex hull of its extreme points. Since, for 1< p< ∞, Bp (σ) is a uniformly convex space, only the balls \(\mathcal{D}(B^1 (\sigma ))\) and \(\mathcal{D}(B^\infty (\sigma ))\) have nontrivially arranged sets of extreme points. In this paper, in terms of zeros of entire functions, we obtain necessary and sufficient conditions of extremeness for functions from \(\mathcal{D}(B^1 (\sigma ))\).