The Balian-Low theorem and regularity of Gabor systems

  • John J. Benedetto
  • Wojciech Czaja
  • Przemystaw Gadziński
  • Alexander M. Powell


For any positive real numbers A, B, and d satisfying the conditions\(\frac{1}{A} + \frac{1}{B} = 1\), d>2, we construct a Gabor orthonormal basis for L2(ℝ), such that the generating function g∈L2(ℝ) satisfies the condition:∫|g(x)|2(1+|x|A)/logd(2+|x|)dx < ∞ and\(\int_{\hat {\mathbb{R}}} {\left| {\hat g(\xi )} \right|^2 (1 + \left| \xi \right|^B )/\log ^d (2 + \left| \xi \right|)d\xi< \infty } \).

Math Subject Classifications


Key Words and Phrases

Gabor orthonormal bases Balian-Low theorem Zak transform regularity of functions 


  1. [1]
    Auslander, A. and Tolimieri, R. Abelian harmonic analysis, theta functions and function algebras on a nilmanifold,Lecture Notes in Mathematics,436, Springer-Verlag, Berlin, (1975).MATHGoogle Scholar
  2. [2]
    Bacry, H., Grossman, A., and Zak, J. Proof of completeness of lattice states in the kq representation,Phys. Rev. B.,12, 1118–1120, (1975).CrossRefGoogle Scholar
  3. [3]
    Balian, R. A strong uncertainty principle in signal theory or in quantum mechanics (in French),C.R. Acad. Sci. Paris,292(20), 1357–1362, (1981).MathSciNetGoogle Scholar
  4. [4]
    Benedetto, J.J. Gabor representations and wavelets,Contemporary Mathematics,91, 9–27, (1989).MathSciNetGoogle Scholar
  5. [5]
    Benedetto, J.J. Frame decompositions, sampling, and uncertainty principle inequalities, inWavelets: Mathematics and Applications, 247–304. CRC Press, Boca Raton, FL, (1994).Google Scholar
  6. [6]
    Benedetto, J.J., Heil, C., and Walnut, D. Differentiation and the Balian-Low theorem,J. Fourier Anal. Appl.,1(4), 355–402, (1995).MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Bonami, A., Demange, B., and Jaming, P. Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms, to appear inRev. Mat. Iberoamericana.Google Scholar
  8. [8]
    Brezin, J. Harmonic analysis on nilmanifolds,Trans. Am. Math. Soc.,150, 611–618, (1970).MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Folland, G. and Sitaram, A. The uncertainty principle: a mathematical survey,J. Fourier Anal. Appl.,3(3), 207–238, (1997).MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Feichtinger, H.G. and Strohmer, T., Eds., Gabor analysis and algorithms, Theory and applications,Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, MA, (1998).MATHGoogle Scholar
  11. [11]
    Feichtinger, H.G. and Strohmer, T., Eds.,Advances in Gabor Analysis, Birkhäuser, Boston, MA, (2003).MATHGoogle Scholar
  12. [12]
    Gelfand, I.M. Expansion in characteristic functions of an equation with periodic coefficients, (in Russian),Doklady Akad. Nauk SSSR (N.S.),73, 1117–1120, (1950).MathSciNetGoogle Scholar
  13. [13]
    Gröchenig, K. An uncertainty principle related to the Poisson summation formula,Studia Math.,121(4), 87–104, (1996).MATHMathSciNetGoogle Scholar
  14. [14]
    Gröchenig, K. Foundations of time-frequency analysis,Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, MA, (2001).MATHGoogle Scholar
  15. [15]
    Havin, V. and Jöricke, B.The Uncertainty Principle in Harmonic Analysis, Springer-Verlag, Berlin, (1994).MATHGoogle Scholar
  16. [16]
    Heil, C. and Walnut, D. Continuous and discrete wavelet transforms,SIAM Rev.,31, 628–666, (1989).MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    Hernández, E. and Weiss G.A First Course in Wavelets, CRC Press, Boca Raton, FL, (1996).Google Scholar
  18. [18]
    Hörmander, L. A uniqueness theorem of Beurling for Fourier transform pairs,Ark. Mat.,29, 237–240, (1991).MATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    Igusa, J.Theta Functions, Springer-Verlag, New York, Heidelberg, (1972).MATHGoogle Scholar
  20. [20]
    Janssen, A.J.E.M. Bargmann transform, Zak transform, and coherent states,J. Math. Phys.,23(5), 720–731, (1982).MATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    Janssen, A.J.E.M. The Zak transform: a signal transform for sampled time-continuous signalsPhilips J. Res.,43(1), 23–69, (1988).MATHMathSciNetGoogle Scholar
  22. [22]
    Low, F. Complete sets of wave packets, inA Passion for Physics-Essays in Honor of Geoffrey Chew, DeTar, C., et al., Eds., World Scientific, Singapore, 17–22, (1985).Google Scholar
  23. [23]
    Rado, T. and Reichelderfer, P.Continuous Transformations in Analysis, Springer-Verlag, Berlin, New York, (1955).MATHGoogle Scholar
  24. [24]
    Weil, A. On some groups of unitary operators, (in French),Acta Math.,111, 143–211, (1964).MATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    Wilczok, E. New uncertainty principles for the continuous Gabor transform and the continuous wavelet transform,Doc. Math.,5, 201–226, (2001).MathSciNetGoogle Scholar
  26. [26]
    Zak, J. Finite translations in solid state physics,Phys. Rev. Lett.,19, 1385–1387, (1967).CrossRefGoogle Scholar
  27. [27]
    Zak, J. The kq-representation in the dynamics of electrons in solids,Solid State Physics,27(1), 1–62, (1972).CrossRefGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2003

Authors and Affiliations

  • John J. Benedetto
    • 1
  • Wojciech Czaja
    • 1
    • 2
  • Przemystaw Gadziński
    • 2
  • Alexander M. Powell
    • 1
  1. 1.Department of MathematicsUniversity of MarylandCollege Park
  2. 2.Mathematical InstituteUniversity of WrocławWrocławPoland

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