Generalized sign Fourier uncertainty

TL;DR

This paper explores a generalized sign uncertainty principle for the Fourier transform, analyzing how functions and their transforms resonate with a given pattern outside a certain region, and identifies sharp constants in some cases.

Contribution

It extends the sign uncertainty principle to a broader setting involving a generic function P, providing new sharp constants and insights into the resonance behavior.

Findings

Identified sharp constants for the generalized sign uncertainty principle.

Extended the classical uncertainty principle to a more general framework involving a function P.

Provided conditions under which resonance can occur outside a specified region.

Abstract

We consider a generalized version of the sign uncertainty principle for the Fourier transform, first proposed by Bourgain, Clozel and Kahane in 2010 and revisited by Cohn and Gon\c{c}alves in 2019. In our setup, the signs of a function and its Fourier transform resonate with a generic given function $P$ outside of a ball. One essentially wants to know if and how soon this resonance can happen, when facing a suitable competing weighted integral condition. The original version of the problem corresponds to the case $P \equiv 1$. Surprisingly, even in such a rough setup, we are able to identify sharp constants in some cases.