A discrete Hardy uncertainty principle

TL;DR

This paper establishes a discrete version of Hardy's uncertainty principle, showing that decay conditions on a function and its Fourier transform on discrete sets imply global decay, with applications to Gaussian characterization.

Contribution

It generalizes Morgan's uncertainty principle to discrete sets and provides a new discrete Hardy's uncertainty principle, answering open questions in the field.

Findings

Decay transfer from discrete sets implies global decay of functions.

Functions with specific decay on discrete sets are Gaussian when parameters are optimal.

The results apply to supercritical pairs, extending classical uncertainty principles to discrete settings.

Abstract

We show that knowing the decay of a function $f$ on a discrete set $\Lambda\subset\mathbb{R}$ and the decay of its Fourier transform $\hat{f}$ on a discrete set $M\subset\mathbb{R}$ is enough to determine the global decay of $f$ and $\hat{f}$, provided that $(\Lambda,M)$ is a supercritical pair in the sense of Kulikov, Nazarov, and Sodin. This decay transfer result leads to a discrete generalization of Morgan's uncertainty principle: it is enough to require $|f(\lambda)|\lesssim e^{-\frac{2}{p}A\pi|\lambda|^p}$ for all $\lambda\in\Lambda$ and $|\hat{f}(\mu)|\lesssim e^{-\frac{2}{q}A\pi|\mu|^q}$ for all $\mu\in M$, where $(p,q)$ are H\"{o}lder conjugates, $A>|\cos(\frac{r\pi}{2})|^\frac{1}{r}$, and $r:=\min\{p,q\}$. For $A=1$ and $p,q=2$, we also show that any such function must be a scaled Gaussian. This yields a discrete version of Hardy's uncertainty principle and resolves two questions posed by Ramos and Sousa.