On exposed functions in Bernstein spaces of functions of exponential type

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 $D(B^1_{\sigma})$.