The fundamental limit of what we can know about a signal is so rigid that you only need a few data points to prove it exists.
The Hardy uncertainty principle holds true even when a function is only measured at specific, discrete sets of points. This principle normally states that a wave and its frequency cannot both be squeezed into a tiny space. This research proves that you do not need a continuous, perfect signal to observe this law of physics in action. A handful of well-placed samples are enough to constrain the entire behavior of a complex system. This finding makes it much easier to detect and analyze signals in noisy environments where data is missing. It provides a new way to pull clear information out of the most chaotic backgrounds.
A discrete Hardy uncertainty principle
arXiv · 2605.03679
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