Skip to main content
SpringerLink
Log in

The uncertainty principle: A mathematical survey

  • Research Tutorial
  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript

Abstract

We survey various mathematical aspects of the uncertainty principle, including Heisenberg’s inequality and its variants, local uncertainty inequalities, logarithmic uncertainty inequalities, results relating to Wigner distributions, qualitative uncertainty principles, theorems on approximate concentration, and decompositions of phase space.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ali, S. T. and Prugovecki, E. (1977). Systems of imprimitivity and representations of quantum mechanics in fuzzy phase space.J. Math. Phys. 18, 219–228. MR 58 #8964

    Article  MATH  MathSciNet  Google Scholar 

  2. Amrein, W. O. and Berthier, A.M. (1977). On support properties ofL pfunctions and their Fourier transforms.J. Funct. Anal. 24, 258–267. MR 57 #1013

    Article  MATH  MathSciNet  Google Scholar 

  3. Auscher, P. (1994). Remarks on the local Fourier bases.Wavelets: Mathematics and Applications (J. J. Benedetto and M. W. Frazier, eds.). CRC Press, Boca Raton, FL, 203–218. MR 94i:42039

    Google Scholar 

  4. Auscher, P., Weiss, G., and Wickerhauser, M. V. (1992). Local sine and cosine bases of Coifman and Meyer and the construction of smooth wavelets.Wavelets: A Tutorial in Theory and Applications (C. K. Chui, ed.). Academic Press, Boston, 237–256. MR 93e:42042

    Google Scholar 

  5. Barceló, J. A. and Córdoba, A. J. (1989). Band-limited functions:L pconvergence.Trans. Amer. Math. Soc. 313, 655–669. MR 90g:42017

    Article  MATH  MathSciNet  Google Scholar 

  6. Beckner, W. (1975). Inequalities in Fourier analysis.Ann. of Math. 102, 159–182. MR 52 #6317

    Article  MathSciNet  Google Scholar 

  7. —————. (1995). Pitt’s inequality and the uncertainty principle.Proc. Amer. Math. Soc. 123, 1897–1905. MR 95g:42021

    Article  MATH  MathSciNet  Google Scholar 

  8. Benedetto, J.J. (1985). An inequality associated to the uncertainty principle.Rend. Circ. Mat. Palermo (2)34, 407–421. MR 87g:26024

    MATH  MathSciNet  Google Scholar 

  9. —————. (1985). Fourier uniqueness criteria and spectrum estimation theorems.Fourier Techniques and Applications (J. F. Price, ed.). Plenum, New York, 149–170.

    Google Scholar 

  10. —————. (1990). Uncertainty principle inequalities and spectrum estimation.Recent Advances in Fourier Analysis and Its Applications (J. S. Byrnes and J. L. Byrnes, eds.). Kluwer Acad. Publ., Dordrecht. MR 91i:94010

    Google Scholar 

  11. —————. (1994). Frame decompositions, sampling, and uncertainty principle inequalities.Wavelets: Mathematics and Applications (J. J. Benedetto and M. W. Frazier, eds.). CRC Press, Boca Raton, FL, 247–304. MR 94i:94005

    Google Scholar 

  12. Benedetto, J. J. and Frazier, M. W. (eds.). (1994).Wavelets: Mathematics and Applications. CRC Press, Boca Raton, FL.

    MATH  Google Scholar 

  13. Benedetto, J. J., Heil, C., and Walnut, D. (1992). Uncertainty principles for time-frequency operators.Continuous and Discrete Fourier Transforms, Extension Problems, and Wiener-Hopf Equations (I. Gohberg, ed.). Birkhäuser, Basel, 1–25. MR 94a:42041

    Google Scholar 

  14. Benedetto, J. J. and Walnut, D. F. (1994). Gabor frames forL 2 and related spaces.Wavelets: Mathematics and Applications (J. J. Benedetto and M. W. Frazier, eds.). CRC Press, Boca Raton, FL, 97–162. MR 94i:42040

    Google Scholar 

  15. Benedicks, M. (1984). The support of functions and distributions with a spectral gap.Math. Scand. 55, 255–309. MR 86f:43006

    MathSciNet  Google Scholar 

  16. —————. (1985). On Fourier transforms of functions supported on sets of finite Lebesgue measure.J. Math. Anal. Appl. 106, 180–183. MR 86j:42017

    Article  MATH  MathSciNet  Google Scholar 

  17. Bialynicki-Birula, I. (1985). Entropic uncertainty relations in quantum mechanics.Quantum Probability and Applications III (L. Accardi and W. von Waldenfels, eds.).Lecture Notes in Math. 1136. Springer, Berlin, 90–103. MR 87b:81007

    Chapter  Google Scholar 

  18. Bialynicki-Birula, I. and Mycielski, J. (1975). Uncertainty relations for information entropy in wave mechanics.Comm. Math. Phys. 44, 129–132. MR 52 #7403

    Article  MathSciNet  Google Scholar 

  19. Borchi, E. and Marsaglia, S. (1985). A note on the uncertainty product of bandlimited functions.Signal Process. 9, 277–282. MR 87a:94007

    Article  MathSciNet  Google Scholar 

  20. Bourgain, J. (1988). A remark on the uncertainty principle for Hilbertian bases.J. Funct. Anal. 79, 136–143. MR 89f:81025

    Article  MATH  MathSciNet  Google Scholar 

  21. Bowie, P. C. (1971). Uncertainty inequalities for Hankel transforms.SIAM J. Math. Anal. 2, 601–606. MR 46 #4113

    Article  MATH  MathSciNet  Google Scholar 

  22. Busch, P. (1984). On joint lower bounds of position and momentum observables in quantum mechanics.J. Math. Phys. 25, 1794–1797. MR 85h:81004

    Article  MathSciNet  Google Scholar 

  23. Byrnes, J. S. (1994). Quadrature mirror filters, low crest factor arrays, functions achieving optimal uncertainty principle bounds, and complete orthonormal sequences—a unified approach.Appl. Comput. Harmonic Anal. 1, 261–266. MR 95i:94004

    Article  MATH  MathSciNet  Google Scholar 

  24. Chistyakov, A. L. (1976). On uncertainty relations for vector-valued operators.Teoret. Mat. Fiz. 27, 130–134; English transl.,Theor. Math. Phys. 27, 380–382. MR 56 #7615

    MATH  MathSciNet  Google Scholar 

  25. Cowling, M. G. and Price, J. F. (1983). Generalisations of Heisenberg’s inequality.Harmonic Analysis (G. Mauceri, F. Ricci, and G. Weiss, eds.).Lecture Notes in Math. 992. Springer, Berlin, 443–449. MR 86g:42002b

    Chapter  Google Scholar 

  26. —————. (1984). Bandwidth versus time concentration: the Heisenberg-Pauli-Weyl inequality.SIAM J. Math. Anal. 15, 151–165. MR 86g:42002a

    Article  MathSciNet  Google Scholar 

  27. Cowling, M. G., Price, J. F., and Sitaram, A. (1988). A qualitative uncertainty principle for semisimple Lie groups.J. Austral. Math. Soc. Ser. A 45, 127–132. MR 89d:43005

    MATH  MathSciNet  Google Scholar 

  28. Daubechies, I. (1992).Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics, Philadelphia, PA. MR 93e:42045

    MATH  Google Scholar 

  29. de Bruijn, N. G. (1967). Uncertainty principles in Fourier analysis.Inequalities (O. Shisha, ed.). Academic Press, New York, 55–71. MR 36 #3047

    Google Scholar 

  30. de Jeu, M. F. E. (1994). An uncertainty principle for integral operators.J. Funct. Anal. 122, 247–253. MR 95h:43009

    Article  MATH  MathSciNet  Google Scholar 

  31. Donoho, D. L. and Stark, P. B. (1989). Uncertainty principles and signal recovery.SIAM J. Appl. Math. 49, 906–931. MR 90c:42003

    Article  MATH  MathSciNet  Google Scholar 

  32. Dym, H. and McKean, H. P. (1972).Fourier Series and Integrals. Academic Press, New York. MR 56 #945

    MATH  Google Scholar 

  33. Echterhoff, S., Kaniuth, E., and Kumar, A. (1991). A qualitative uncertainty principle for certain locally compact groups.Forum Math. 3, 355–369. MR 93a:43005

    Article  MATH  MathSciNet  Google Scholar 

  34. Faris, W. G. (1978). Inequalities and uncertainty principles.J. Math. Phys. 19, 461–466. MR 58 #4066

    Article  MathSciNet  Google Scholar 

  35. Fefferman, C. (1983). The uncertainty principle.Bull. Amer. Math. Soc. (N.S.) 9, 129–206. MR 85f:35001

    MATH  MathSciNet  Google Scholar 

  36. Fefferman, C. and Phong, D. H. (1981). The uncertainty principle and sharp Garding inequalities.Comm. Pure Appl. Math. 34, 285–331. MR 82j:35140

    Article  MATH  MathSciNet  Google Scholar 

  37. Folland, G. B. (1989).Harmonic Analysis in Phase Space. Princeton University Press, Princeton, NJ. MR 92k:22017

    MATH  Google Scholar 

  38. —————. (1995).A Course in Abstract Harmonic Analysis. CRC Press, Boca Raton, FL.

    MATH  Google Scholar 

  39. Fuchs, W. H. J. (1954). On the magnitude of Fourier transforms.Proc. Intern. Congress Math. 1954, vol. II. North-Holland, Amsterdam.

    Google Scholar 

  40. Gabor, D. (1946). Theory of communication.J. Inst. Elec. Engr. 93, 429–457.

    Google Scholar 

  41. Garofalo, N. and Lanconelli, E. (1990). Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation.Ann. Inst. Fourier (Grenoble)40, 313–356. MR 91i:22014

    MATH  MathSciNet  Google Scholar 

  42. Gesztesy, F. and Pittner, L. (1978). Uncertainty relations and quadratic forms.J. Phys. A 11, 1765–1770. MR 81a:81028

    Article  MathSciNet  Google Scholar 

  43. Grabowski, M. (1984). Wehrl-Lieb’s inequality for entropy and the uncertainty relation.Rep. Math. Phys. 20, 153–155. MR 86f:82008

    Article  MATH  MathSciNet  Google Scholar 

  44. Gross, L. (1975). Logarithmic Sobolev inequalities.Amer. J. Math. 97, 1061–1083. MR 54 #8263

    Article  MathSciNet  Google Scholar 

  45. —————. (1978). Logarithmic Sobolev inequalities—a survey.Vector Space Measures and Applications I (R. M. Aron and S. Dineen, eds.).Lecture Notes in Math. 644. Springer, Berlin, 196–203. MR 80f:46035

    Google Scholar 

  46. Grünbaum, F. A. (1981). Eigenvectors of a Toeplitz matrix: discrete version of the prolate spheroidal wave functions.SIAM J. Alg. Disc. Methods 2, 136–141. MR 82m:15013

    MATH  Google Scholar 

  47. Grünbaum, F. A., Longhi, L., and Perlstadt, M. (1982). Differential operators commuting with finite convolution integral operators: some nonabelian examples.SIAM J. Appl. Math. 42, 941–955. MR 84f:43013

    Article  MATH  MathSciNet  Google Scholar 

  48. Havin, V. and Jöricke, B. (1994).The Uncertainty Principle in Harmonic Analysis. Springer, Berlin. MR 96c:42001

    MATH  Google Scholar 

  49. Hardy, G. H. (1933). A theorem concerning Fourier transforms.J. London Math. Soc. 8, 227–231.

    Article  MATH  Google Scholar 

  50. Heinig, H. P. and Smith, M. (1986). Extensions of the Heisenberg-Weyl inequality.Internat. J. Math. Math. Sci. 9, 185–192. MR 87i:26013

    Article  MATH  MathSciNet  Google Scholar 

  51. Heisenberg, W. (1927). Über den anschaulichen Inhalt der quantentheoretischen Kinematic und Mechanik.Zeit. Physik 43, 172–198.

    Article  Google Scholar 

  52. Helffer, B. and Nourrigat, J. (1988). Remarques sur le principe d’incertitude.J. Funct. Anal. 80, 33–46. MR 90b:35254

    Article  MATH  MathSciNet  Google Scholar 

  53. Hilberg, W. and Rothe, F. G. (1971). The general uncertainty relation for real signals in communication theory.Inform. and Control 18, 103–125. MR 43 #4540

    Article  MATH  MathSciNet  Google Scholar 

  54. Hirschman, I. I. (1957). A note on entropy.Amer. J. Math. 79, 152–156. MR 19, 622i

    Article  MATH  MathSciNet  Google Scholar 

  55. Hogan, J. A. (1988). A qualitative uncertainty principle for locally compact Abelian groups.Miniconferences on Harmonic Analysis and Operator Algebras (Proc. Centre Math. Anal. 16; M. Cowling, C. Meaney, and W. Moran, eds.). Australian National University, Canberra, 133–142. MR 89k:43006

    Google Scholar 

  56. -----. (1989).Fourier Uncertainty on Groups. Ph.D. Dissertation, University of New South Wales.

  57. Hogan, J. A. (1993). A qualitative uncertainty principle for unimodular groups of type I.Trans. Amer. Math. Soc. 340, 587–594. MR 94b:43003

    Article  MATH  MathSciNet  Google Scholar 

  58. Hörmander, L. (1991). A uniqueness theorem of Beurling for Fourier transform pairs,Ark. Mat. 29, 237–240. MR 93b:42016

    Article  MATH  MathSciNet  Google Scholar 

  59. Ishigaki, T. (1991). Four mathematical expressions of the uncertainty relation.Found. Phys. 21, 1089–1105. MR 92k:81012

    Article  MathSciNet  Google Scholar 

  60. Kacnel’son, V. E. (1973). Equivalent norms in spaces of entire functions.Mat. Sb. 92, 34–54; English transl.,Math. USSR Sb.21, 33–55. MR 49 #3171

    MathSciNet  Google Scholar 

  61. Kahane, J.-P., Lévy-Leblond, J.-M., and Sjöstrand, J. (1993). Une inegalité de Heisenberg inverse.J. Anal. Math. 60, 135–155. MR 95a:34124

    MATH  MathSciNet  Google Scholar 

  62. Kargaev, P. P. (1982/1983). The Fourier transform of the characteristic function of a set, vanishing on an interval.Mat. Sb. 117, 397–411; English transl.Math. USSR Sb.45, 397–410. MR 83f:42010

    MathSciNet  Google Scholar 

  63. Kargaev, P. P and Volberg, A. L. (1992). Three results concerning the support of functions and their Fourier transforms.Indiana Univ. Math. J. 41, 1143–1164. MR 94d:42018

    Article  MATH  MathSciNet  Google Scholar 

  64. Kay, I. and Silverman, R. A. (1959). On the uncertainty relation for real signals: postscript.Inform. and Control 2, 396–397. MR 23 #B2091

    MathSciNet  Google Scholar 

  65. Kempf, A. (1994). Uncertainty relation in quantum mechanics with quantum group symmetry.J. Math. Phys. 35, 4483–4496. MR 95k:81063

    Article  MATH  MathSciNet  Google Scholar 

  66. Kennard, E. H. (1927). Zur Quantenmechanik einfacher Bewegungstypen.Zeit. Physik 44, 326–352.

    Article  Google Scholar 

  67. Knapp, A. W. (1986).Representation Theory of Semisimple Groups. Princeton University Press, Princeton, NJ. MR 87j:22022

    MATH  Google Scholar 

  68. Koppinen, M. (1994). Uncertainty inequalities for Hopf algebras.Comm. Algebra 22, 1083–1101. MR 95a:16051

    MATH  MathSciNet  Google Scholar 

  69. Kraus, K. (1967). A further remark on uncertainty relations.Zeit. Physik 201, 134–141. MR 35 #2570

    Article  MathSciNet  Google Scholar 

  70. Kumar, A. (1992). A qualitative uncertainty principle for hypergroups.Functional Analysis and Operator Theory (B. S. Yadav and D. Singh, eds.).Lecture Notes in Math. 1511. Springer, Berlin, 1–9. MR 94a:43012

    Chapter  Google Scholar 

  71. Lahti, P. J. and Maczynski, M. J. (1987). Heisenberg inequality and the complex field in quantum mechanics.J. Math. Phys. 28, 1764–1769. MR 88g:81011

    Article  MathSciNet  Google Scholar 

  72. Landau, H. J. (1985). An overview of time and frequency limiting.Fourier Techniques and Applications (J. F. Price, ed.). Plenum, New York, 201–220.

    Google Scholar 

  73. Landau, H. J. and Pollak, H. O. (1961). Prolate spheroidal wave functions, Fourier analysis and uncertainty II.Bell Syst. Tech. J. 40, 65–84. MR 25 #4147

    MathSciNet  MATH  Google Scholar 

  74. Landau, H. J. and Pollak, H. O. (1962). Prolate spheroidal wave functions, Fourier analysis and uncertainty III: the dimension of the space of essentially time- and band-limited signals.Bell Syst. Tech. J. 41, 1295–1336. MR 26 #5200

    MathSciNet  MATH  Google Scholar 

  75. Landau, H. J. and Widom, H. (1980). Eigenvalue distribution of time and frequency limiting.J. Math. Anal. Appl. 77, 469–481. MR 81m:45023

    Article  MATH  MathSciNet  Google Scholar 

  76. Leipnik, R. (1959). Entropy and the uncertainty principle.Inform. and Control 2, 64–79.

    Article  MathSciNet  MATH  Google Scholar 

  77. Lieb, E. H. (1978). Proof of an entropy conjecture of Wehrl.Comm. Math. Phys. 62, 35–41. MR 80d:82032

    Article  MATH  MathSciNet  Google Scholar 

  78. —————. (1990). Integral bounds for radar ambiguity functions and Wigner distributions.J. Math. Phys. 31, 591–599. MR 91f:81076

    Article  MathSciNet  Google Scholar 

  79. Logvinenko, V. N. and Sereda, Ju. F. (1974). Equivalent norms in spaces of entire functions of exponential type.Teor. Funksii Funktsional. Anal. i Prilozhen. 20, 102–111. MR 57 #17232

    MATH  MathSciNet  Google Scholar 

  80. Matolcsi, T. and Szücs, J. (1973). Intersection des mesures spectrales conjuguées.C. R. Acad. Sci. Paris 277, A841-A843. MR 48 #4804

    Google Scholar 

  81. Melenk, J. M. and Zimmerman, G. (1996). Functions with time and frequency gaps.J. Fourier Anal. Appl. 2, 611–614.

    Article  MATH  MathSciNet  Google Scholar 

  82. Meshulam, R. (1992). An uncertainty principle for groups of orderpq.European J. Combin. 13, 401–407. MR 93e:20036

    Article  MATH  MathSciNet  Google Scholar 

  83. Mustard, D. (1991). Uncertainty principles invariant under the fractional Fourier transform.J. Austral. Math. Soc. Ser. B 33, 180–191. MR 93c:42007

    MATH  MathSciNet  Google Scholar 

  84. Nahmod, A. R. (1994). Generalized uncertainty principles on spaces of homogeneous type.J. Funct. Anal. 119, 171–209. MR 95g:42027

    Article  MATH  MathSciNet  Google Scholar 

  85. Narcowich, F. J. (1990). Geometry and uncertainty.J. Math. Phys. 31, 354–364. MR 91a:81088

    Article  MATH  MathSciNet  Google Scholar 

  86. Nazarov, F. L. (1993). Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type.Algebra i Analiz 5, 3–66. MR 94k:42004

    Google Scholar 

  87. Nelson, E. (1959). Anlytic vectors.Ann. of Math. (2)70, 572–615. MR 21 #5901

    Article  MathSciNet  Google Scholar 

  88. Pati, V., Sitaram, A., Sundari, M., and Thangavelu, S. (1996). An uncertainty principle for eigenfunction expansions.J. Fourier Anal. Appl. 2, 427–433.

    Article  MATH  MathSciNet  Google Scholar 

  89. Pearl, J. (1973). Time, frequency, sequency, and their uncertainty relations.IEEE Trans. Inform. Theory IT-19, 225–229. MR 54 #12344

    Article  MathSciNet  Google Scholar 

  90. Perlstadt, M. (1986). Polynomial analogues of prolate spheroidal wave functions and uncertainty.SIAM J. Math. Anal. 17, 340–248. MR 87f:42026

    Article  MathSciNet  Google Scholar 

  91. Price, J. F. (1983). Inequalities and local uncertainty principles.J. Math. Phys. 24, 1711–1714. MR 85e:81056

    Article  MATH  MathSciNet  Google Scholar 

  92. Price, J. F., (ed.). (1985).Fourier Techniques and Applications. Plenum, New York.

    MATH  Google Scholar 

  93. Price, J. F. (1987). Sharp local uncertainty inequalities.Studia Math. 85, 37–45. MR 88f:42029

    MathSciNet  Google Scholar 

  94. Price, J. F. and Racki, P. C. (1985). Local uncertainty inequalities for Fourier series.Proc. Amer. Math. Soc. 93, 245–251. MR 86e:42018

    Article  MATH  MathSciNet  Google Scholar 

  95. Price, J. F. and Sitaram, A. (1988). Functions and their Fourier transforms with supports of finite measure for certain locally compact groups.J. Funct. Anal. 79, 166–182. MR 90e:43003

    Article  MATH  MathSciNet  Google Scholar 

  96. ————— (1988). Local uncertainty inequalities for compact groups.Proc. Amer. Math. Soc. 103, 441–447. MR 89i:43004

    Article  MATH  MathSciNet  Google Scholar 

  97. ————— (1988). Local uncertainty inequalities for locally compact groups.Trans. Amer. Math. Soc. 308, 105–114. MR 89h:22017

    Article  MATH  MathSciNet  Google Scholar 

  98. Shahshahani, M. (1989). Poincaré inequality, uncertainty principle, and scattering theory on symmetric spaces.Amer. J. Math. 111, 197–224. MR 90g:22017

    Article  MATH  MathSciNet  Google Scholar 

  99. Shannon, C. (1948)/(1949). A mathematical theory of communication.Bell System Tech. J. 27, 379–423 and 623–656. Reprinted: Shannon, C. and Weaver, W.The Mathematical Theory of Communication. Univ. of Illinois Press, Urbana. MR 10, 133e

    MathSciNet  MATH  Google Scholar 

  100. Sitaram, A. and Sundari, M. An analogue of Hardy’s theorem for very rapidly decreasing functions on semi-simple Lie groups.Pacific J. Math. (to appear).

  101. Sitaram, A., Sundari, M., and Thangavelu, S. (1995). Uncertainty principles on certain Lie groups.Proc. Indian Acad. Sci. Math. Sci. 105, 135–151.

    MATH  MathSciNet  Google Scholar 

  102. Skoog, R. A. (1970). An uncertainty principle for functions vanishing on a half-line.IEEE Trans. Circuit Theory CT-17, 241–243. MR 41 #8926

    MathSciNet  Google Scholar 

  103. Slepian, D. (1964). Prolate spheroidal wave functions, Fourier analysis and uncertainty IV: extensions to many dimensions; generalized prolate spheroidal wave functions.Bell Syst. Tech. J. 43, 3009–3057. MR 31 #5993

    MathSciNet  MATH  Google Scholar 

  104. ————— (1976). On bandwidth.Proc. IEEE 64, 292–300. MR 57 #2738

    Article  MathSciNet  Google Scholar 

  105. ————— (1978). Prolate spheroidal wave functions, Fourier analysis and uncertainty V: the discrete case.Bell Syst. Tech. J. 57, 1371–1430.

    Google Scholar 

  106. ————— (1983). Some comments on Fourier analysis, uncertainty and modeling.SIAM Rev. 25, 379–393. MR 84i:94016

    Article  MATH  MathSciNet  Google Scholar 

  107. Slepian, D. and Pollak, H. O. (1961). Prolate spheroidal wave functions, Fourier analysis and uncertainty I.Bell Syst. Tech. J. 40, 43–63. MR 25 #4146

    MathSciNet  MATH  Google Scholar 

  108. Smith, K. T. (1990). The uncertainty principle on groups.SIAM J. Appl. Math. 50, 876–882. MR 91i:94008

    Article  MathSciNet  Google Scholar 

  109. Spera, M. (1993). On a generalized uncertainty principle, coherent states, and the moment map.J. Geom. Phys. 12, 165–182. MR 94m:58097

    Article  MATH  MathSciNet  Google Scholar 

  110. Strichartz, R. S. (1989). Uncertainty principles in harmonic analysis.J. Funct. Anal. 84, 97–114. MR 91a:42017

    Article  MATH  MathSciNet  Google Scholar 

  111. -----. (1994). Construction of orthonormal wavelets.Wavelets: Mathematics and Applications (J. J. Benedetto and M. W. Frazier, eds.). 23–50. MR 94i:42047

  112. Sun, L. (1994). An uncertainty principle on hyperbolic space.Proc. Amer. Math. Soc. 121, 471–479. MR 94h:43005

    Article  MATH  MathSciNet  Google Scholar 

  113. Sundari, M. Hardy’s theorem for then-dimensional Euclidean motion group.Proc. Amer. Math. Soc. (to appear).

  114. Thangavelu, S. (1990). Some uncertainty inequalities.Proc. Indian Acad. Sci. Math. Sci. 100, 137–145. MR 91h:43004

    Article  MATH  MathSciNet  Google Scholar 

  115. Uffink, J. B. M. and Hilgevoord, J. (1984). New bounds for the uncertainty principle.Phys. Lett. 105A, 176–178.

    Google Scholar 

  116. Uffink, J. B. M. and Hilgevoord, J. (1985). Uncertainty principle and uncertainty relations.Found. Phys. 15, 925–944. MR 87f:81012

    Article  MathSciNet  Google Scholar 

  117. Voit, M. (1993). An uncertainty principle for commutative hypergroups and Gelfand pairs.Math. Nachr. 164, 187–195. MR 95d:43005

    Article  MATH  MathSciNet  Google Scholar 

  118. Weyl, H. (1928).Gruppentheorie und Quantenmechanik. S. Hirzel, Leipzig. Revised English edition:The Theory of Groups and Quantum Mechanics Methuen, London, 1931; reprinted by Dover, New York, 1950.

    MATH  Google Scholar 

  119. Wiener, N. (1956).I Am a Mathematician. MIT Press, Cambridge.

    MATH  Google Scholar 

  120. Williams, D. N. (1979). New mathematical proof of the uncertainty relation.Amer. J. Phys. 47, 606–607. MR 80d:81059

    Article  MathSciNet  Google Scholar 

  121. Wolf, J. A. (1992). The uncertainty principle for Gelfand pairs.Nova J. Algebra Geom. 1, 383–396. MR 94k:43006

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Washington, 98195-4350, Seattle, WA

    Gerald B. Folland

  2. Statistics and Mathematics Division, Indian Statistical Institute, R. V. College Post Office, 560 059, Bangalore, India

    Alladi Sitaram

Authors
  1. Gerald B. Folland
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alladi Sitaram
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Folland, G.B., Sitaram, A. The uncertainty principle: A mathematical survey. The Journal of Fourier Analysis and Applications 3, 207–238 (1997). https://doi.org/10.1007/BF02649110

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02649110

Math Subject Classifications

Keywords and Phrases

Search

Navigation