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Rational time-frequency multi-window subspace Gabor frames and their Gabor duals

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Abstract

This paper addresses the theory of multi-window subspace Gabor frame with rational time-frequency parameter products. With the help of a suitable Zak transform matrix, we characterize multi-window subspace Gabor frames, Riesz bases, orthonormal bases and the uniqueness of Gabor duals of type I and type II. Using these characterizations we obtain a class of multi-window subspace Gabor frames, Riesz bases, orthonormal bases, and at the same time we derive an explicit expression of their Gabor duals of type I and type II. As an application of the above results, we obtain characterizations of multi-window Gabor frames, Riesz bases and orthonormal bases for L 2(ℝ), and derive a parametric expression of Gabor duals for multi-window Gabor frames in L 2(ℝ).

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References

  1. Akinlar M A, Gabardo J-P. Oblique duals associated with rational subspace Gabor frames. J Integral Equations Appl, 2008, 20: 283–309

    Article  MATH  MathSciNet  Google Scholar 

  2. Casazza P G, Christensen O. Weyl-Heisenberg frames for subspaces of L 2(ℝ). Proc Amer Math Soc, 2001, 129: 145–154

    Article  MATH  MathSciNet  Google Scholar 

  3. Christensen O. An Introduction to Frames and Riesz Bases. Boston: Birkhäuser, 2003

    Book  MATH  Google Scholar 

  4. Daubechies I. The wavelet transform, time-frequency localization and signal analysis. IEEE Trans Inform Theory, 1990, 36: 961–1005

    Article  MATH  MathSciNet  Google Scholar 

  5. Feichtinger H G, Onchis D M. Constructive reconstruction from irregular sampling in multi-window spline-type spaces. In: Progress in Analysis and its Applications. Hackensack, NJ: World Sci Publ, 2010, 257–265

    Google Scholar 

  6. Feichtinger H G, Onchis D M. Constructive realization of dual systems for generators of multi-window spline-type spaces. J Comput Appl Math, 2010, 234: 3467–3479

    Article  MATH  MathSciNet  Google Scholar 

  7. Feichtinger H G, Strohmer T. Gabor Analysis and Algorithms, Theory and Applications. Boston: Birkhäuser, 1998

    Book  MATH  Google Scholar 

  8. Feichtinger H G, Strohmer T. Advances in Gabor Analysis. Boston: Birkhäuser, 2003

    Book  MATH  Google Scholar 

  9. Gabardo J-P, Han D. Subspace Weyl-Heisenberg frames. J Fourier Anal Appl, 2001, 7: 419–433

    Article  MATH  MathSciNet  Google Scholar 

  10. Gabardo J-P, Han D. Balian-Low phenomenon for subspace Gabor frames. J Math Phys, 2004, 45: 3362–3378

    Article  MATH  MathSciNet  Google Scholar 

  11. Gabardo J-P, Han D. The uniqueness of the dual of Weyl-Heisenberg subspace frames. Appl Comput Harmon Anal, 2004, 17: 226–240

    Article  MATH  MathSciNet  Google Scholar 

  12. Gabardo J-P, Li Y Z. Density results for Gabor systems associated with periodic subsets of the real line. J Approx Theory, 2009, 157: 172–192

    Article  MATH  MathSciNet  Google Scholar 

  13. Gröchenig K. Foundations of Time-Frequency Analysis. Boston: Birkhäuser, 2001

    Book  MATH  Google Scholar 

  14. Heil C. History and evolution of the density theorem for Gabor frames. J Fourier Anal Appl, 2007, 13: 113–166

    Article  MATH  MathSciNet  Google Scholar 

  15. Jaillet F, Torrésani B. Time-frequency jigsaw puzzle: adaptive multiwindow and multilayered Gabor expansions. Int J Wavelets Multiresolut Inf Process, 2007, 5: 293–315

    Article  MATH  MathSciNet  Google Scholar 

  16. Li S. Discrete multi-Gabor expansions. IEEE Trans Inform Theory, 1999, 45: 1954–1967

    Article  MATH  MathSciNet  Google Scholar 

  17. Li Y Z, Lian Q F. Gabor systems on discrete periodic sets. Sci China Ser A, 2009, 52: 1639–1660

    Article  MATH  MathSciNet  Google Scholar 

  18. Lian Q F, Li Y Z. Gabor frame sets for subspaces. Adv Comput Math, 2011, 34: 391–411

    Article  MATH  MathSciNet  Google Scholar 

  19. Lian Q F, Li Y Z. The duals of Gabor frames on discrete periodic sets. J Math Phys, 2009, 50: 013534, 22pp

    Article  MathSciNet  Google Scholar 

  20. Young R M. An Introduction to Nonharmonic Fourier Series. New York: Academic Press, 1980

    MATH  Google Scholar 

  21. Zeevi Y Y. Multiwindow Gabor-type representations and signal representation by partial information. In: Twentieth century Harmonic Analysis-a Celebration (Il Ciocco, 2000), NATO Sci Ser II Math Phys Chem, 33. Dordrecht: Kluwer Acad Publ, 2001, 173–199

    Chapter  Google Scholar 

  22. Zibulski M, Zeevi Y Y. Analysis of multiwindow Gabor-type schemes by frame methods. Appl Comput Harmon Anal, 1997, 4: 188–221

    Article  MATH  MathSciNet  Google Scholar 

  23. Zeevi Y Y, Zibulski M, Porat M. Multi-window Gabor schemes in signal and image representations. In: Gabor Analysis and Algorithms, Appl Numer Harmon Anal. Boston, MA: Birkhäuser Boston, 1998, 381–407

    Google Scholar 

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Authors and Affiliations

  1. College of Applied Sciences, Beijing University of Technology, Beijing, 100124, China

    Yan Zhang & YunZhang Li

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  1. Yan Zhang
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  2. YunZhang Li
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Correspondence to YunZhang Li.

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Zhang, Y., Li, Y. Rational time-frequency multi-window subspace Gabor frames and their Gabor duals. Sci. China Math. 57, 145–160 (2014). https://doi.org/10.1007/s11425-013-4610-4

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