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The Balian-Low theorem and regularity of Gabor systems

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Abstract

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:∫ℝ|g(x)|2(1+|x|A)/logd(2+|x|)dx < ∞ and\(\int_{\hat {\mathbb{R}}} {\left| {\hat g(\xi )} \right|^2 (1 + \left| \xi \right|^B )/\log ^d (2 + \left| \xi \right|)d\xi< \infty } \).

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References

  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).

    MATH  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  4. Benedetto, J.J. Gabor representations and wavelets,Contemporary Mathematics,91, 9–27, (1989).

    MathSciNet  Google Scholar 

  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. Benedetto, J.J., Heil, C., and Walnut, D. Differentiation and the Balian-Low theorem,J. Fourier Anal. Appl.,1(4), 355–402, (1995).

    Article  MATH  MathSciNet  Google Scholar 

  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.

  8. Brezin, J. Harmonic analysis on nilmanifolds,Trans. Am. Math. Soc.,150, 611–618, (1970).

    Article  MATH  MathSciNet  Google Scholar 

  9. Folland, G. and Sitaram, A. The uncertainty principle: a mathematical survey,J. Fourier Anal. Appl.,3(3), 207–238, (1997).

    Article  MATH  MathSciNet  Google Scholar 

  10. Feichtinger, H.G. and Strohmer, T., Eds., Gabor analysis and algorithms, Theory and applications,Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, MA, (1998).

    MATH  Google Scholar 

  11. Feichtinger, H.G. and Strohmer, T., Eds.,Advances in Gabor Analysis, Birkhäuser, Boston, MA, (2003).

    MATH  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  13. Gröchenig, K. An uncertainty principle related to the Poisson summation formula,Studia Math.,121(4), 87–104, (1996).

    MATH  MathSciNet  Google Scholar 

  14. Gröchenig, K. Foundations of time-frequency analysis,Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, MA, (2001).

    MATH  Google Scholar 

  15. Havin, V. and Jöricke, B.The Uncertainty Principle in Harmonic Analysis, Springer-Verlag, Berlin, (1994).

    MATH  Google Scholar 

  16. Heil, C. and Walnut, D. Continuous and discrete wavelet transforms,SIAM Rev.,31, 628–666, (1989).

    Article  MATH  MathSciNet  Google Scholar 

  17. Hernández, E. and Weiss G.A First Course in Wavelets, CRC Press, Boca Raton, FL, (1996).

    Google Scholar 

  18. Hörmander, L. A uniqueness theorem of Beurling for Fourier transform pairs,Ark. Mat.,29, 237–240, (1991).

    Article  MATH  MathSciNet  Google Scholar 

  19. Igusa, J.Theta Functions, Springer-Verlag, New York, Heidelberg, (1972).

    MATH  Google Scholar 

  20. Janssen, A.J.E.M. Bargmann transform, Zak transform, and coherent states,J. Math. Phys.,23(5), 720–731, (1982).

    Article  MATH  MathSciNet  Google Scholar 

  21. Janssen, A.J.E.M. The Zak transform: a signal transform for sampled time-continuous signalsPhilips J. Res.,43(1), 23–69, (1988).

    MATH  MathSciNet  Google Scholar 

  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. Rado, T. and Reichelderfer, P.Continuous Transformations in Analysis, Springer-Verlag, Berlin, New York, (1955).

    MATH  Google Scholar 

  24. Weil, A. On some groups of unitary operators, (in French),Acta Math.,111, 143–211, (1964).

    Article  MATH  MathSciNet  Google Scholar 

  25. Wilczok, E. New uncertainty principles for the continuous Gabor transform and the continuous wavelet transform,Doc. Math.,5, 201–226, (2001).

    MathSciNet  Google Scholar 

  26. Zak, J. Finite translations in solid state physics,Phys. Rev. Lett.,19, 1385–1387, (1967).

    Article  Google Scholar 

  27. Zak, J. The kq-representation in the dynamics of electrons in solids,Solid State Physics,27(1), 1–62, (1972).

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Maryland, 20742, College Park, MD

    John J. Benedetto, Wojciech Czaja & Alexander M. Powell

  2. Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384, Wrocław, Poland

    Wojciech Czaja & Przemystaw Gadziński

Authors
  1. John J. Benedetto
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  2. Wojciech Czaja
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  3. Przemystaw Gadziński
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  4. Alexander M. Powell
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Correspondence to John J. Benedetto.

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Communicated by Guido Weiss

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Benedetto, J.J., Czaja, W., Gadziński, P. et al. The Balian-Low theorem and regularity of Gabor systems. J Geom Anal 13, 239–254 (2003). https://doi.org/10.1007/BF02930696

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