On Beurling's uncertainty principle

TL;DR

This paper extends Beurling's uncertainty principle by characterizing functions with specific exponential decay conditions, showing they must be polynomials times a Gaussian, thus broadening the understanding of function localization.

Contribution

The paper generalizes Hedenmalm's result, providing a new characterization of functions satisfying a particular exponential integrability condition as polynomials times a Gaussian.

Findings

Functions with exponential decay conditions are polynomials times a Gaussian.

The result extends classical uncertainty principles to a broader class of functions.

Provides a quantitative condition linking decay rate to function form.

Abstract

We generalise a result of Hedenmalm to show that if a function $f$ on $\mathbb{R}$ is such that $\int_{\mathbb{R}^2} \bigl|f(x) \, \hat f(y)\bigr| \,e^{\lambda \left|xy\right|} \,dx\,dy = O( (1-\lambda)^{-N} )$ as $\lambda \to 1-$, then $f$ is the product of a polynomial and a gaussian.