Uncertainty principle via variational calculus on modulation spaces

TL;DR

This paper formulates uncertainty principles as a variational problem on modulation spaces, linking optimal constants to eigenvalues of localization operators and deriving related eigenfunction equations.

Contribution

It introduces a novel variational approach to uncertainty principles on modulation spaces and connects these to eigenvalues of localization operators.

Findings

Optimal constants are smallest eigenvalues of inverse localization operators.

Eigenfunctions satisfy derived Euler-Lagrange equations.

Generalizes Lieb's inequality to mixed-norm spaces.

Abstract

We approach uncertainty principles of Cowling-Price-Heis-\\enberg-type as a variational principle on modulation spaces. In our discussion we are naturally led to compact localization operators with symbols in modulation spaces. The optimal constant in these uncertainty principles is the smallest eigenvalue of the inverse of a compact localization operator. The Euler-Lagrange equations for the associated functional provide equations for the eigenfunctions of the smallest eigenvalue of these compact localization operators. As a by-product of our proofs we derive a generalization to mixed-norm spaces of an inequality for Wigner and Ambiguity functions due do Lieb.