The uncertainty principle: Variations on a theme

Wigderson, Avi; Wigderson, Yuval

doi:10.1090/bull/1715

The uncertainty principle: Variations on a theme
HTML articles powered by AMS MathViewer

by Avi Wigderson and Yuval Wigderson HTML | PDF

Bull. Amer. Math. Soc. 58 (2021), 225-261 Request permission

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 nonabelian 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.

References

W. O. Amrein and A. M. Berthier, On support properties of $L^{p}$-functions and their Fourier transforms, J. Functional Analysis 24 (1977), no. 3, 258–267. MR 0461025, DOI 10.1016/0022-1236(77)90056-8
T. Banica, Complex Hadamard matrices and applications, 2019. https://banica.u-cergy.fr/pdf/ham.pdf.
Martin J. Bastiaans and Tatiana Alieva, The linear canonical transformation: definition and properties, Linear canonical transforms, Springer Ser. Optical Sci., vol. 198, Springer, New York, 2016, pp. 29–80. MR 3467302, DOI 10.1007/978-1-4939-3028-9_{2}
William Beckner, Inequalities in Fourier analysis, Ann. of Math. (2) 102 (1975), no. 1, 159–182. MR 385456, DOI 10.2307/1970980
William Beckner, Pitt’s inequality and the uncertainty principle, Proc. Amer. Math. Soc. 123 (1995), no. 6, 1897–1905. MR 1254832, DOI 10.1090/S0002-9939-1995-1254832-9
Michael Benedicks, On Fourier transforms of functions supported on sets of finite Lebesgue measure, J. Math. Anal. Appl. 106 (1985), no. 1, 180–183. MR 780328, DOI 10.1016/0022-247X(85)90140-4
Iwo Białynicki-Birula and Jerzy Mycielski, Uncertainty relations for information entropy in wave mechanics, Comm. Math. Phys. 44 (1975), no. 2, 129–132. MR 386550
T. Carroll, X. Massaneda, and J. Ortega-Cerda, An enhanced uncertainty principle for the Vaserstein distance, 2020. Preprint available at arXiv:2003.03165.
K. Conrad, Characters of finite abelian groups, accessed 2020. https://kconrad.math.uconn.edu/blurbs/grouptheory/charthy.pdf.
Michael G. Cowling and John F. Price, Bandwidth versus time concentration: the Heisenberg-Pauli-Weyl inequality, SIAM J. Math. Anal. 15 (1984), no. 1, 151–165. MR 728691, DOI 10.1137/0515012
R. Craigen and H. Kharaghani, Orthogonal designs, in C. J. Colbourn and J. H. Dinitz (eds.), Handbook of combinatorial designs, Discrete Mathematics and its Applications (Boca Raton), second ed., Chapman & Hall/CRC, Boca Raton, FL, 2007, pp. 280–295.
Amir Dembo, Thomas M. Cover, and Joy A. Thomas, Information-theoretic inequalities, IEEE Trans. Inform. Theory 37 (1991), no. 6, 1501–1518. MR 1134291, DOI 10.1109/18.104312
David L. Donoho and Xiaoming Huo, Uncertainty principles and ideal atomic decomposition, IEEE Trans. Inform. Theory 47 (2001), no. 7, 2845–2862. MR 1872845, DOI 10.1109/18.959265
David L. Donoho and Philip B. Stark, Uncertainty principles and signal recovery, SIAM J. Appl. Math. 49 (1989), no. 3, 906–931. MR 997928, DOI 10.1137/0149053
W. Erb, Shapes of uncertainty in spectral graph theory, 2019. Preprint available at arXiv:1909.10865.
Gerald B. Folland and Alladi Sitaram, The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl. 3 (1997), no. 3, 207–238. MR 1448337, DOI 10.1007/BF02649110
F. Gonçalves, D. Oliveira e Silva, and J. P. G. Ramos, New sign uncertainty principles, 2020. Preprint available at arXiv:2003.10771.
G. H. Hardy, A Theorem Concerning Fourier Transforms, J. London Math. Soc. 8 (1933), no. 3, 227–231. MR 1574130, DOI 10.1112/jlms/s1-8.3.227
J. J. Healy, M. A. Kutay, H. M. Ozaktas, and J. T. Sheridan (eds.), Linear canonical transforms: Theory and applications, Springer Series in Optical Sciences, vol. 198, Springer, New York, 2016.
W. Heisenberg, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Z. Physik 43 (1927), 172–198.
I. I. Hirschman Jr., A note on entropy, Amer. J. Math. 79 (1957), 152–156. MR 89127, DOI 10.2307/2372390
A. Jaffe, C. Jiang, Z. Liu, Y. Ren, and J. Wu, Quantum fourier analysis, Proceedings of the National Academy of Sciences 117 (2020), 10715–10720.
Tiefeng Jiang, Maxima of entries of Haar distributed matrices, Probab. Theory Related Fields 131 (2005), no. 1, 121–144. MR 2105046, DOI 10.1007/s00440-004-0376-5
E. H. Kennard, Zur Quantenmechanik einfacher bewegungstypen, Z. Physik 44 (1927), 326–352.
Tamás Matolcsi and József Szűcs, Intersection des mesures spectrales conjuguées, C. R. Acad. Sci. Paris Sér. A-B 277 (1973), A841–A843 (French). MR 326460
Roy Meshulam, An uncertainty inequality for groups of order $pq$, European J. Combin. 13 (1992), no. 5, 401–407. MR 1181081, DOI 10.1016/S0195-6698(05)80019-8
M. Northington V, Uncertainty principles for Fourier multipliers, 2019. Preprint available at arXiv:1907.08812.
R. E. A. C. Paley, On orthogonal matrices, J. Math. and Phys. 12 (1933), 311–320.
A. Poria, Uncertainty principles for the Fourier and the short-time Fourier transforms, 2020. Preprint available at arXiv:2004.04184.
S. Quader, A. Russell, and R. Sundaram, Small-support uncertainty principles on $\mathbb {Z}/p$ over finite fields, 2019. Preprint available at arXiv:1906.05179.
M. Ram Murty, Some remarks on the discrete uncertainty principle, Highly composite: papers in number theory, Ramanujan Math. Soc. Lect. Notes Ser., vol. 23, Ramanujan Math. Soc., Mysore, 2016, pp. 77–85. MR 3692726
Michael Reed and Barry Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0493420
Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 0450380
Adrian Stern, Uncertainty principles in linear canonical transform domains and some of their implications in optics, J. Opt. Soc. Amer. A 25 (2008), no. 3, 647–652. MR 2449194, DOI 10.1364/JOSAA.25.000647
Ran Tao and Juan Zhao, Uncertainty principles and the linear canonical transform, Linear canonical transforms, Springer Ser. Optical Sci., vol. 198, Springer, New York, 2016, pp. 97–111. MR 3467304, DOI 10.1007/978-1-4939-3028-9_{4}
Terence Tao, An uncertainty principle for cyclic groups of prime order, Math. Res. Lett. 12 (2005), no. 1, 121–127. MR 2122735, DOI 10.4310/MRL.2005.v12.n1.a11
Hermann Weyl, Gruppentheorie und Quantenmechanik, 2nd ed., Wissenschaftliche Buchgesellschaft, Darmstadt, 1977 (German). MR 0450450
David N. Williams, New mathematical proof of the uncertainty relation, Amer. J. Phys. 47 (1979), no. 7, 606–607. MR 534914, DOI 10.1119/1.11763
Kurt Bernardo Wolf, Integral transforms in science and engineering, Mathematical Concepts and Methods in Science and Engineering, vol. 11, Plenum Press, New York-London, 1979. MR 529766