Advertisement

Journal of Fourier Analysis and Applications

, Volume 25, Issue 1, pp 101–107 | Cite as

Gabor Frames in \({\ell }^2({\mathbb {Z}})\) and Linear Dependence

  • Ole ChristensenEmail author
  • Marzieh Hasannasab
Article
  • 148 Downloads

Abstract

We prove that an overcomplete Gabor frame in \({\ell }^2({\mathbb {Z}})\) generated by a finitely supported sequence is always linearly dependent. This is a particular case of a general result about linear dependence versus independence for Gabor systems in \({\ell }^2({\mathbb {Z}})\) with modulation parameter 1 / M and translation parameter N for some \(M,N\in {\mathbb {N}},\) and generated by a finite sequence g in \({\ell }^2({\mathbb {Z}})\) with K nonzero entries.

Keywords

Frames Gabor system in \({\ell }^2({\mathbb {Z}})\) Linear dependency of Gabor systems 

Mathematics Subject Classification

42C15 

Notes

Acknowledgements

The authors would like to thank Guido Janssen, Chris Heil and Shahaf Nitzan for useful comments and references.

References

  1. 1.
    Christensen, O.: An Introduction to Frames and Riesz Bases, 2nd edn. Birkhäuser, Boston (2016)zbMATHGoogle Scholar
  2. 2.
    Cvetković, Z., Vetterli, M.: Oversampled filter banks. IEEE Trans. Signal. Proc. 46(5), 1245–1255 (1998)CrossRefGoogle Scholar
  3. 3.
    Demeter, C., Gautam, S.Z.: On the finite linear independence of lattice Gabor systems. Proc. Am. Math. Soc. 141(5), 1735–1747 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Heil, C., Ramanathan, J., Topiwala, P.: Linear independence of time-frequency translates. Proc. Am. Math. Soc. 124, 2787–2795 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Heil, C.: Linear independence of finite Gabor systems. In: Harmonic Analysis and Applications, Appl. Numer. Harmon. Anal., pp. 171–206. Birkhäuser, Boston (2006)Google Scholar
  6. 6.
    Janssen, A.J.E.M.: From continuous to discrete Weyl-Heisenberg frames through sampling. J. Fourier Anal. Appl. 3(5), 583–596 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Jakobsen, M.S., Lemvig, J.: Co-compact Gabor systems on locally compact groups. J. Fourier Anal. Appl. 22, 36–70 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Jitomirskaya, S.Y.: Metal-insulator transition for the almost Mathieu operator. Ann. Math. 150, 1159–1175 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kutyniok, G.: Linear independence of time-frequency shifts under a generalized Schrödinger representation. Arch. Math. 78(2), 135–144 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lawrence, J., Pfander, G.E., Walnut, D.: Linear independence of Gabor systems in finite dimensional vector spaces. J. Fourier Anal. Appl. 11(6), 715–726 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Linnell, P.: Von Neumann algebras and linear independence of translates. Proc. Am. Math. Soc. 127(11), 3269–3277 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lopez, J., Han, D.: Discrete Gabor frames in \(\ell ^2(\mathbb{Z}^d)\). Proc. Amer. Math. Soc. 141(11), 3839–3851 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.DTU ComputeTechnical University of DenmarkLyngbyDenmark

Personalised recommendations