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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Irregular Gabor frames and their stability


Authors: Wenchang Sun and Xingwei Zhou
Journal: Proc. Amer. Math. Soc. 131 (2003), 2883-2893
MSC (2000): Primary 41A58, 42C15, 42C40
Published electronically: December 30, 2002
MathSciNet review: 1974346
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we give sufficient conditions for irregular Gabor systems to be frames. We show that for a large class of window functions, every relatively uniformly discrete sequence in $\mathbb R^2$ with sufficiently high density will generate a Gabor frame. Explicit frame bounds are given. We also study the stability of irregular Gabor frames and show that every Gabor frame with arbitrary time-frequency parameters is stable if the window function is nice enough. Explicit stability bounds are given.


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  • 1. H.Bacry, A.Grossmann, and J.Zak, Proofs of the completeness of lattice states in the $kq$-representation, Phys. Rev., B12(1975), 1118-1120.
  • 2. V.Bargmann, P.Butera, L.Girardello and J.R.Klauder, On the completeness of coherent states, Rep. Math. Phys., 2(1971), 221-228. MR 44:7860
  • 3. J.J.Benedetto, C.Heil, and D.Walnut, Differentiation and the Balian-Low theorem, J. Fourier Anal. Appl., 1(1995), 355-402. MR 96f:42002
  • 4. P.G.Casazza and O.Christensen, Weyl-Heisenberg frames for subspaces of $L^2({\mathbb{R}})$, Proc. Amer. Math. Soc., 129(2001), 145-154. MR 2001h:42046
  • 5. O.Christensen, Atomic decomposition via projective group representations, Rocky Mountain J. Math., 26(1996), 1289-1312. MR 98h:43004
  • 6. O.Christensen, Moment problems and stability results for frames with applications to irregular sampling and Gabor frames, Appl. Comp. Harmonic Anal., 3(1996), 82-86. MR 97f:44007
  • 7. O.Christensen, Perturbation of operators and applications to frame theory, J. Four. Anal. Appl., 3(1997), 543 - 557. MR 98j:47028
  • 8. O.Christensen, Frames, Riesz bases, and discrete Gabor/wavelet expansions, Bulletin (New series) of Amer. Math. Soc., 38(2001), 273-291. MR 2002c:42040
  • 9. O.Christensen, B.Deng, and C.Heil, Density of Gabor frames, Appl. Comput. Harmon. Anal., 7(1999), 292-304. MR 2000j:42043
  • 10. C.K.Chui and X.L.Shi, Inequalities of Littlewood-Paley type for frames and wavelets, SIAM J. Math. Anal., 24(1993), 263-277. MR 94d:42039
  • 11. I.Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory, 36(1990), 961-1005. MR 91e:42038
  • 12. I.Daubechies, Ten Lectures on Wavelets, SIAM Philadelphia, 1992. MR 93e:42045
  • 13. I.Daubechies and A.Grossmann, Frames of entire functions in the Bargmann space, Comm. Pure Appl. Math., 41(1988), 151-164. MR 89e:46028
  • 14. I.Daubechies, H.Landau and Z.Landau, Gabor time-frequency lattices and the Wexler-Raz identity, J. Fourier Anal. Appl., 4(1995), 437-478. MR 96i:42021
  • 15. S.Favier and R.Zalik, On the stability of frames and Riesz bases, Appl. and Comp. Harm. Anal., 2(1995), 160-173. MR 96e:42030
  • 16. H.G.Feichtinger and K.Gröchenig, Banach spaces related to integrable group representations and their atomic decomposition, J. funct. Anal., 86(1989), 307-340. MR 91g:43011
  • 17. H.G.Feichtinger and A.J.E.M.Janssen, Validity of $WH$-frame bound conditions depends on lattice parameters, Appl. Comput. Harmon. Anal., 8(2000), 104-112. MR 2000j:42044
  • 18. H.G.Feichtinger and T.Strohmer, Eds., Gabor Analysis and Algorithms: Theory and Applications, Birkhäuser, Boston, 1997. MR 98h:42001
  • 19. D.Gabor, Theory of communications, J. Inst. Elect. Eng.(London), 93(1943), 429-457.
  • 20. K. Gröchenig, Describing functions: atomic decompositions versus frames, Mh. Math., 112(1991), 1-41. MR 92m:42035
  • 21. K. Gröchenig, Irregular sampling of wavelet and short-time Fourier transforms, Constr. Approx., 9(1993), 283-297. MR 94m:42077
  • 22. G.Hardy, J.E.Littlewood, and G.Pólya, Inequalities, 2nd Ed.,Cambridge: Cambridge Univ. Press, 1952. MR 13:727e
  • 23. C.Heil and D.Walnut, Continuous and discrete wavelet transforms, SIAM Review, 31(1989), 628-666. MR 91c:42032
  • 24. Y.Lyubarskii, Frames in the Bargmann space of entire functions, in Entire and subharmonic functions, vol 11 of the series Advanced in Soviet Mathematics, B.Ya. Levin, ed., Springer-verlag, Berlin, 1992. MR 93k:30036
  • 25. A.M.Perelomov, On the completeness of a system of coherent states, Teor. Mat. Fiz., 6(1971), 213-224. MR 57:15050
  • 26. J.Ramanathan and T.Steger, Incompleteness of sparse coherent states, Appl. Comput. Harmon. Anal., 2(1995), 148-153. MR 96b:81049
  • 27. A.Ron and Z.Shen, Weyl-Heisenberg frames and Riesz bases in $L^2(\mathbb{R}^d)$, Duke Math. J., 89(1997), 148-153. MR 98i:42013
  • 28. K.Seip and R.Wallstén, Density theorems for sampling and interpolation in the Bargmann-Fock space II, J. Reine Angew. Math., 429(1992), 107-113. MR 93g:46026b
  • 29. W.Sun and X.Zhou, On the stability of Gabor frames, Advances in Applied Mathematics, 26(2001), 181-191. MR 2002c:42043
  • 30. W.Sun and X.Zhou, Irregular wavelet/Gabor frames, Appl. Comput. Harmon. Anal., 13 (2002), 63-76.
  • 31. R.Young, An Introduction to Non-Harmonic Fourier Series, Academic Press, New York, 1980. MR 81m:42027

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Additional Information

Wenchang Sun
Affiliation: Department of Mathematics, Nankai University, Tianjin 300071, People’s Republic of China
Email: sunwch@nankai.edu.cn

Xingwei Zhou
Affiliation: Department of Mathematics, Nankai University, Tianjin 300071, People’s Republic of China
Email: xwzhou@nankai.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06931-9
PII: S 0002-9939(02)06931-9
Keywords: Gabor frames, Weyl-Heisenberg frames, stability
Received by editor(s): August 29, 2001
Received by editor(s) in revised form: March 2, 2002, and April 11, 2002
Published electronically: December 30, 2002
Additional Notes: This work was supported by the National Natural Science Foundation of China (Grant Nos. 10171050 and 10201014), the Mathematical Tianyuan Foundation (Grant No. TY10126007), the Research Fund for the Doctoral Program of Higher Education, and the Liuhui Center for Applied Mathematics.
Communicated by: David R. Larson
Article copyright: © Copyright 2002 American Mathematical Society