On geometry of the unit ball of Paley-Wiener space over two symmetric intervals

TL;DR

This paper characterizes the geometric structure of the unit ball in Paley-Wiener spaces over two symmetric intervals, focusing on extreme and exposed points, with results depending on the relationship between the interval parameters.

Contribution

It provides a complete description of extreme and exposed points of the unit ball in Paley-Wiener spaces for certain interval configurations, advancing geometric understanding.

Findings

Complete description for $ ho>\sigma/2$

Complex structure when $ ho<\sigma/2$

Insights into the geometry of Paley-Wiener space unit balls

Abstract

Let $PW_S^1$ be the space of integrable functions on $\mathbb{R}$ whose Fourier transform vanishes outside $S$, where $S = [-\sigma,-\rho]\cup[\rho,\sigma]$, $0<\rho<\sigma$. In the case $\rho>\sigma/2$, we present a complete description of the set of extreme and the set of exposed points of the unit ball of $PW^1_S$. The structure of these sets becomes more complicated when $\rho<\sigma/2$.