The uncertainty principle: variations on a theme

TL;DR

This paper reveals that many classical uncertainty principles for the Fourier transform stem from basic properties like boundedness and unitarity, and it offers a unified framework for proofs and generalizations across various operators and settings.

Contribution

It introduces a simple, unified proof template showing that uncertainty principles follow from minimal properties of operators, enabling new proofs and broad generalizations.

Findings

Many uncertainty principles derive from weak operator properties

A unified proof approach simplifies understanding of uncertainty principles

New generalizations extend classical results to broader contexts

Abstract

We show how a number of well-known uncertainty principles for the Fourier transform, such as the Heisenberg uncertainty principle, the Donoho--Stark uncertainty principle, and Meshulam's non-abelian uncertainty principle, have little to do with the structure of the Fourier transform itself. Rather, all of these results follow from very weak properties of the Fourier transform (shared by numerous linear operators), namely that it is bounded as an operator $L^1 \to L^\infty$, and that it is unitary. Using a single, simple proof template, and only these (or weaker) properties, we obtain some new proofs and many generalizations of these basic uncertainty principles, to new operators and to new settings, in a completely unified way. Together with our general overview, this paper can also serve as a survey of the many facets of the phenomena known as uncertainty principles.