Heisenberg's uncertainty principle in the sense of Beurling

TL;DR

This paper provides a new proof of Beurling's version of Heisenberg's uncertainty principle, relaxing assumptions and extending the results to multiple functions and higher dimensions using Fourier and Mellin transforms.

Contribution

It introduces a significantly different proof method that weakens the assumptions of Beurling's uncertainty principle and extends the results to multi-function and higher-dimensional cases.

Findings

The new proof is more general and requires weaker assumptions.

Examples demonstrate the sharpness of the results.

Extensions to multiple functions and higher dimensions are established.

Abstract

We shed new light on Heisenberg's uncertainty principle in the sense of Beurling, by offering an essentially different proof which permits us to weaken the assumptions substantially, and examples show that the result is sharp. The proof involves Fourier and Mellin transforms. We alo extend to a setting of two functions. A higher-dimensional analogue is considered as well.