An optimization problem and point-evaluation in Paley-Wiener spaces

TL;DR

This paper investigates the constant _p in Paley-Wiener spaces, improving bounds for certain p-values by solving an optimization problem related to point evaluation.

Contribution

It provides improved bounds for _p in the range 2 < p  5 through solving a new optimization problem.

Findings

_p < p/2 for all p > 2

Improved bounds for 2 < p  5

Method involves solving an optimization problem

Abstract

We study the constant $\mathscr{C}_p$ defined as the smallest constant $C$ such that $|f(0)|^p \leq C\|f\|_p^p$ holds for every function $f$ in the Paley-Wiener space $PW^p$. Brevig, Chirre, Ortega-Cerd\`a, and Seip have recently shown that $\mathscr{C}_p<p/2$ for all $p>2$. We improve this bound for $2<p \leq 5$ by solving an optimization problem.