The Balian-Low theorem and regularity of Gabor systems

Journal of Geometric Analysis

Volumn 13, Issue 2, 2003, Pages 239-254

  Benedetto, John J ^a   Czaja, Wojciech ^a,b   Gadziński, Przemystaw ^b   Powell, Alexander M ^a  

^a Bldg 142   (United States)

^b UNIVERSITY OF WROC AW   (Poland)

Author keywords

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

Indexed keywords

EID: 84867923645     PISSN: 10506926     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02930696     Document Type: Article

References (27)

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).
2. H. Bacry A. Grossman J. Zak 1975 Proof of completeness of lattice states in the kq representation Phys. Rev. B. 12 1118 1120 10.1103/PhysRevB.12.1118
3. R. Balian 1981 A strong uncertainty principle in signal theory or in quantum mechanics (in French) C.R. Acad. Sci. Paris 292 20 1357 1362 644367
4. J.J. Benedetto 1989 Gabor representations and wavelets Contemporary Mathematics 91 9 27 1002584
5. Benedetto, J.J. Frame decompositions, sampling, and uncertainty principle inequalities, inWavelets: Mathematics and Applications, 247-304. CRC Press, Boca Raton, FL, (1994).
6. J.J. Benedetto C. Heil D. Walnut 1995 Differentiation and the Balian-Low theorem J. Fourier Anal. Appl. 1 4 355 402 0887.42026 10.1007/s00041-001-4016-5 1350699
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. J. Brezin 1970 Harmonic analysis on nilmanifolds Trans. Am. Math. Soc. 150 611 618 0205.13203 10.2307/1995541 279244
9. G. Folland A. Sitaram 1997 The uncertainty principle: a mathematical survey J. Fourier Anal. Appl. 3 3 207 238 0885.42006 10.1007/BF02649110 1448337
10. Feichtinger, H.G. and Strohmer, T.; Eds.; Gabor analysis and algorithms, Theory and applications,Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, MA, (1998).
11. Feichtinger, H.G. and Strohmer, T.; Eds.;Advances in Gabor Analysis, Birkhäuser, Boston, MA, (2003).
12. I.M. Gelfand 1950 Expansion in characteristic functions of an equation with periodic coefficients, (in Russian) Doklady Akad. Nauk SSSR (N.S.) 73 1117 1120 39154
13. K. Gröchenig 1996 An uncertainty principle related to the Poisson summation formula Studia Math. 121 4 87 104 0866.42005 1414896
14. Gröchenig, K. Foundations of time-frequency analysis,Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, MA, (2001).
15. Havin, V. and Jöricke, B.The Uncertainty Principle in Harmonic Analysis, Springer-Verlag, Berlin, (1994).
16. C. Heil D. Walnut 1989 Continuous and discrete wavelet transforms SIAM Rev. 31 628 666 0683.42031 10.1137/1031129 1025485
17. Hernández, E. and Weiss G.A First Course in Wavelets, CRC Press, Boca Raton, FL, (1996).
18. L. Hörmander 1991 A uniqueness theorem of Beurling for Fourier transform pairs Ark. Mat. 29 237 240 0755.42009 10.1007/BF02384339 1150375
19. Igusa, J.Theta Functions, Springer-Verlag, New York, Heidelberg, (1972).
20. A.J.E.M. Janssen 1982 Bargmann transform, Zak transform, and coherent states J. Math. Phys. 23 5 720 731 0486.46027 10.1063/1.525426 655886
21. A.J.E.M. Janssen 1988 The Zak transform: a signal transform for sampled time-continuous signals Philips J. Res. 43 1 23 69 0653.94002 947891
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).
23. Rado, T. and Reichelderfer, P.Continuous Transformations in Analysis, Springer-Verlag, Berlin, New York, (1955).
24. A. Weil 1964 On some groups of unitary operators, (in French) Acta Math. 111 143 211 0203.03305 10.1007/BF02391012 165033
25. E. Wilczok 2001 New uncertainty principles for the continuous Gabor transform and the continuous wavelet transform Doc. Math. 5 201 226 1758876
26. J. Zak 1967 Finite translations in solid state physics Phys. Rev. Lett. 19 1385 1387 10.1103/PhysRevLett.19.1385
27. J. Zak 1972 The kq-representation in the dynamics of electrons in solids Solid State Physics 27 1 1 62 10.1016/S0081-1947(08)60236-2