A discrete Hardy uncertainty principle

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.