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 Hölder 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.

16 pages