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A theorem concerning Fourier transforms: A survey

Aingeru Fernández-Bertolin, Luis Vega

TL;DR

Hardy’s uncertainty principle links simultaneous Gaussian-type decay of a function and its Fourier transform to rigid structural constraints, highlighting fundamental limits of time–frequency localization. The paper surveys Hardy’s original theorem, an abstract commutator-based framework, and a broad spectrum of generalizations—from Morgan and Beurling–Hörmander-type results to multidimensional and endpoint forms—together with real-variable proofs and dynamical interpretations in PDEs. It also surveys extensions to non-Euclidean settings, including Heisenberg groups and discrete structures, and connects these results to time–frequency analysis and metaplectic/Wigner formalisms. Collectively, the work illuminates deep links between harmonic analysis, partial differential equations, and signal processing, with wide theoretical and applied implications.

Abstract

In this note, we highlight the impact of the paper G. H. Hardy, A theorem concerning Fourier transforms, J. Lond. Math. Soc. (1) 8 (1933), 227--231 in the community of harmonic analysis in the last 90 years, reviewing, on the one hand, the direct generalizations of the main results and, on the other, the different connections to related areas and new perspectives.

A theorem concerning Fourier transforms: A survey

TL;DR

Hardy’s uncertainty principle links simultaneous Gaussian-type decay of a function and its Fourier transform to rigid structural constraints, highlighting fundamental limits of time–frequency localization. The paper surveys Hardy’s original theorem, an abstract commutator-based framework, and a broad spectrum of generalizations—from Morgan and Beurling–Hörmander-type results to multidimensional and endpoint forms—together with real-variable proofs and dynamical interpretations in PDEs. It also surveys extensions to non-Euclidean settings, including Heisenberg groups and discrete structures, and connects these results to time–frequency analysis and metaplectic/Wigner formalisms. Collectively, the work illuminates deep links between harmonic analysis, partial differential equations, and signal processing, with wide theoretical and applied implications.

Abstract

In this note, we highlight the impact of the paper G. H. Hardy, A theorem concerning Fourier transforms, J. Lond. Math. Soc. (1) 8 (1933), 227--231 in the community of harmonic analysis in the last 90 years, reviewing, on the one hand, the direct generalizations of the main results and, on the other, the different connections to related areas and new perspectives.

Paper Structure

This paper contains 5 sections, 6 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. An abstract Uncertainty Principle
  3. Generalizations of Hardy's uncertainty principle
  4. Hardy's uncertainty principle and real variable arguments
  5. Hardy's uncertainty principle in different settings

Key Result

Theorem 1

If $f$ and $\widehat{f}$ are both $O(|x|^m e^{-x^2/2})$ for large $x$ and some $m$, then each is a finite liner combination of Hermite functions.

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Theorem 4
  • proof : Sketch of the proof
  • Theorem 5
  • Theorem 6