The uncertainty principle: variations on a theme

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.

[1]  E. H. Kennard Zur Quantenmechanik einfacher Bewegungstypen , 1927 .

[2]  W. Heisenberg Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik , 1927 .

[3]  Berger Gruppentheorie und Quantenmechanik , 1931 .

[4]  G. Hardy A Theorem Concerning Fourier Transforms , 1933 .

[5]  I. Bialynicki-Birula,et al.  Uncertainty relations for information entropy in wave mechanics , 1975 .

[6]  W. Beckner Inequalities in Fourier analysis , 1975 .

[7]  M. Reed,et al.  Methods of Modern Mathematical Physics. 2. Fourier Analysis, Self-adjointness , 1975 .

[8]  A. Berthier,et al.  On support properties of Lp-functions and their Fourier transforms , 1977 .

[9]  Kurt Bernardo Wolf,et al.  Integral transforms in science and engineering , 1979 .

[10]  David N. Williams New mathematical proof of the uncertainty relation , 1979 .

[11]  M. Cowling,et al.  Bandwidth Versus Time Concentration: The Heisenberg–Pauli–Weyl Inequality , 1984 .

[12]  M. Benedicks On Fourier transforms of functions supported on sets of finite Lebesgue measure , 1985 .

[13]  D. Donoho,et al.  Uncertainty principles and signal recovery , 1989 .

[14]  Amir Dembo,et al.  Information theoretic inequalities , 1991, IEEE Trans. Inf. Theory.

[15]  Roy Meshulam,et al.  An uncertainty inequality for groups of order pq , 1992, Eur. J. Comb..

[16]  William Beckner,et al.  Pitt’s inequality and the uncertainty principle , 1995 .

[17]  G. Folland,et al.  The uncertainty principle: A mathematical survey , 1997 .

[18]  Xiaoming Huo,et al.  Uncertainty principles and ideal atomic decomposition , 2001, IEEE Trans. Inf. Theory.

[19]  T. Tao An uncertainty principle for cyclic groups of prime order , 2003, math/0308286.

[20]  M. Behbahani On orthogonal matrices , 2004 .

[21]  Tiefeng Jiang,et al.  Maxima of entries of Haar distributed matrices , 2005 .

[22]  Eliahu Levy,et al.  A note about entropy , 2006 .

[23]  CHARACTERS OF FINITE ABELIAN GROUPS , 2007 .

[24]  Randall R. Holmes Linear Representations of Finite Groups , 2008 .

[25]  Adrian Stern,et al.  Uncertainty principles in linear canonical transform domains and some of their implications in optics. , 2008, Journal of the Optical Society of America. A, Optics, image science, and vision.

[26]  W. Marsden I and J , 2012 .

[27]  Ran Tao,et al.  Uncertainty Principles and the Linear Canonical Transform , 2016 .

[28]  Tatiana Alieva,et al.  The Linear Canonical Transformation: Definition and Properties , 2016 .

[29]  M. Ram Murty Some Remarks on the Discrete Uncertainty Principle , 2016 .

[30]  Kristian Kirsch,et al.  Methods Of Modern Mathematical Physics , 2016 .

[31]  T. Banica Complex Hadamard Matrices and Applications , 2019 .

[32]  V MichaelNorthington,et al.  Uncertainty Principles for Fourier Multipliers , 2019, 1907.08812.

[33]  Alexander Russell,et al.  Small-Support Uncertainty Principles on $\mathbb{Z}/p$ over Finite Fields. , 2019, 1906.05179.

[34]  New Sign Uncertainty Principles , 2020, 2003.10771.

[35]  J. Ortega-Cerdà,et al.  An enhanced uncertainty principle for the Vaserstein distance , 2020, Bulletin of the London Mathematical Society.

[36]  Chunlan Jiang,et al.  Quantum Fourier analysis , 2020, Proceedings of the National Academy of Sciences.

[37]  Danna Zhou,et al.  d. , 1840, Microbial pathogenesis.

[38]  Wolfgang Erb,et al.  Shapes of Uncertainty in Spectral Graph Theory , 2019, IEEE Transactions on Information Theory.

[39]  Anirudha Poria Uncertainty principles for the Fourier and the short-time Fourier transforms , 2021, Journal of Mathematical Physics.