On exposed functions in Bernstein spaces of functions of exponential type

TL;DR

This paper studies the geometric structure of the unit ball in Bernstein spaces of exponential type functions, focusing on exposed points and their relationships, revealing the convex hull properties and constructing examples.

Contribution

It characterizes the exposed and strongly exposed points of the unit ball in Bernstein spaces, showing their convex hull relations and providing explicit examples.

Findings

The unit ball's convex hull is generated by its strongly exposed points.

Explicit examples of exposed points are constructed.

Relations between various types of exposed points are established.

Abstract

For $\sigma>0$, the Bernstein space \ $B^1_{\sigma}$ consists of those $L^1(R)$\ functions whose Fourier transforms are supported by $[-\sigma,\sigma]$. Since $B^1_{\sigma}$ is separable and dual to some Banach space, the closed unit ball $D(B^1_{\sigma})$ of $B^1_{\sigma}$\ has sufficiently large sets of both exposed and strongly exposed points. Moreover, $D(B^1_{\sigma})$ coincides with the closed convex hull of its strongly exposed points. We investigate some properties of exposed points, construct several examples and obtain as corollaries the relations between the sets of exposed, strongly exposed, weak$^{\ast}$ exposed, and weak$^{\ast}$ strongly exposed points of $D(B^1_{\sigma})$.