The Multi-Dimensional Hardy Uncertainty Principle and its Interpretation in Terms of the Wigner Distribution; Relation With the Notion of Symplectic Capacity

TL;DR

This paper generalizes Hardy's uncertainty principle to multiple dimensions using symplectic geometry, linking it to the Wigner distribution and symplectic capacity, and extends it to Lagrangian frames and convex exponents.

Contribution

It introduces a multidimensional extension of Hardy's uncertainty principle via symplectic diagonalization and relates it to geometric and phase space concepts.

Findings

Hardy's uncertainty principle extended to multidimensional case.

Equivalence between Hardy's principle and Wigner distribution properties.

Geometric interpretation using symplectic capacity.

Abstract

We extend Hardy's uncertainty principle for a square integrable function and its Fourier transform to the multidimensional case using a symplectic diagonalization. We use this extension to show that Hardy's uncertainty principle is equivalent to a statement on the Wigner distribution of the function. We give a geometric interpretation of our results in terms of the notion of symplectic capacity of an ellipsoid. Furthermore, we show that Hardy's uncertainty principle is valid for a general Lagrangian frame of the phase space. Finally, we discuss an extension of Hardy's theorem for the Wigner distribution for exponentials with convex exponents.