Sign uncertainty and de Branges spaces

TL;DR

This paper explores sign uncertainty for bandlimited functions using de Branges spaces, providing sharp constants, classifying extremizers, and applying results to bounds on zeros of L-functions in number theory.

Contribution

It introduces a framework based on de Branges spaces to analyze sign uncertainty, determining sharp constants and extremizers in various contexts.

Findings

Established a general framework for sign uncertainty using de Branges spaces.

Derived sharp constants and classified extremizers for the problem.

Applied the theory to obtain bounds on zeros of L-functions.

Abstract

We investigate here the sign uncertainty phenomenon for bandlimited functions, with a competing condition given by integration with respect to a general measure. Our main result provides a framework related to the theory of de Branges spaces of entire functions, that allows one to find the sharp constants and classify the extremizers in a broad range of situations. We discuss an application in number theory, in connection to bounds for zeros of $L$-functions.