Skip to main content
SpringerLink
Log in

Gabor Frames: Characterizations and Coarse Structure

  • Chapter
  • First Online:
New Trends in Applied Harmonic Analysis, Volume 2

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

This chapter offers a systematic and streamlined exposition of the most important characterizations of Gabor frames over a lattice.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 69.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The terminology is often a bit different, e.g., [37] uses a “fiberization technique.”

  2. 2.

    It suffices to take the Fourier coefficients of the multivariate Fejer kernel \(\hat{F}_n(k) = \prod _{j=1}^d \big (1 - \frac{|k_j|}{n+1} \big )_{+}\) and set \(K_n (\nu ) = \hat{F}_n(A^T \nu ) = \hat{F}_n(k)\) for \(\nu = (A^T)^{-1}k \in \varLambda ^\perp \).

References

  1. G. Ascensi, H.G. Feichtinger, N. Kaiblinger, Dilation of the Weyl symbol and Balian-Low theorem. Trans. Amer. Math. Soc. 366(7), 3865–3880 (2014)

    Article  MathSciNet  Google Scholar 

  2. E.A. Azoff, Borel measurability in linear algebra. Proc. Amer. Math. Soc. 42, 346–350 (1974)

    Article  MathSciNet  Google Scholar 

  3. B. Bekka, Square integrable representations, von Neumann algebras and an application to Gabor analysis. J. Fourier Anal. Appl. 10(4), 325–349 (2004)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. O. Christensen, An introduction to frames and Riesz bases. Appl. Numer. Harmon. Anal. 2 edn. Birkhäuser/Springer, Cham (2016)

    Google Scholar 

  6. O. Christensen, H.O. Kim, R.Y. Kim, Gabor windows supported on \([-1,1]\) and compactly supported dual windows. Appl. Comput. Harmon. Anal. 28(1), 89–103 (2010)

    Article  MathSciNet  Google Scholar 

  7. W. Czaja, A.M. Powell, Recent developments in the Balian-Low theorem, in Harmonic Analysis and Applications, Applications Numerical Harmonic Analysis, pp. 79–100. Birkhäuser Boston, Boston, MA (2006)

    Google Scholar 

  8. I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992)

    Book  Google Scholar 

  9. I. Daubechies, A. Grossmann, Y. Meyer, Painless nonorthogonal expansions. J. Math. Phys. 27(5), 1271–1283 (1986)

    Article  MathSciNet  Google Scholar 

  10. I. Daubechies, H.J. Landau, Z. Landau, Gabor time-frequency lattices and the Wexler-Raz identity. J. Fourier Anal. Appl. 1(4), 437–478 (1995)

    Article  MathSciNet  Google Scholar 

  11. R.J. Duffin, A.C. Schaeffer, A class of nonharmonic Fourier series. Trans. Amer. Math. Soc. 72, 341–366 (1952)

    Article  MathSciNet  Google Scholar 

  12. H.G. Feichtinger, K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. I. J. Funct. Anal. 86(2), 307–340 (1989)

    Article  MathSciNet  Google Scholar 

  13. H.G. Feichtinger, K. Gröchenig, Gabor Wavelets and the Heisenberg group: gabor expansions and short time Fourier transform from the group theoretical point of view, in Wavelets: A Tutorial in Theory and Applications, ed. by C.K. Chui (Academic Press, Boston, MA, 1992), pp. 359–398

    Chapter  Google Scholar 

  14. H.G. Feichtinger, N. Kaiblinger, Varying the time-frequency lattice of Gabor frames. Trans. Amer. Math. Soc. 356(5), 2001–2023 (electronic) (2004)

    Article  MathSciNet  Google Scholar 

  15. H. G. Feichtinger, W. Kozek, Quantization of TF lattice-invariant operators on elementary LCA groups, in Gabor Analysis and Algorithms, Applications Numerical Harmonic Analysis, pp. 233–266. Birkhäuser Boston, Boston, MA (1998)

    Chapter  Google Scholar 

  16. H.G. Feichtinger, F. Luef, Wiener amalgam spaces for the fundamental identity of Gabor analysis. Collect. Math. (Vol. Extra), 233–253 (2006)

    Google Scholar 

  17. D. Gabor, Theory of communication. J. IEE 93(26), 429–457 (1946)

    Google Scholar 

  18. K. Gröchenig, Foundations of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis (Birkhäuser Boston Inc, Boston, MA, 2001)

    Book  Google Scholar 

  19. K. Gröchenig, Gabor frames without inequalities. Int. Math. Res. Not. IMRN (23), Article ID rnm111, 21 (2007)

    Google Scholar 

  20. K. Gröchenig, The mystery of Gabor frames. J. Fourier Anal. Appl. 20(4), 865–895 (2014)

    Article  MathSciNet  Google Scholar 

  21. K. Gröchenig, A.J.E.M. Janssen, Letter to the editor: a new criterion for Gabor frames. J. Fourier Anal. Appl. 8(5), 507–512 (2002)

    Article  MathSciNet  Google Scholar 

  22. K. Gröchenig, J. Ortega-Cerdà, J.L. Romero, Deformation of Gabor systems. Adv. Math. 277, 388–425 (2015)

    Article  MathSciNet  Google Scholar 

  23. K. Gröchenig, J.L. Romero, J. Stöckler, Sampling theorems for shift-invariant spaces, Gabor frames, and totally positive functions. Invent. Math. 211(3), 1119–1148 (2018)

    Article  MathSciNet  Google Scholar 

  24. K. Gröchenig, J. Stöckler, Gabor frames and totally positive functions. Duke Math. J. 162(6), 1003–1031 (2013)

    Article  MathSciNet  Google Scholar 

  25. C. Heil, History and evolution of the density theorem for Gabor frames. J. Fourier Anal. Appl. 13(2), 113–166 (2007)

    Article  MathSciNet  Google Scholar 

  26. M.S. Jakobsen, J. Lemvig, Density and duality theorems for regular Gabor frames. J. Funct. Anal. 270(1), 229–263 (2016)

    Article  MathSciNet  Google Scholar 

  27. A.J.E.M. Janssen, Signal analytic proofs of two basic results on lattice expansions. Appl. Comput. Harmon. Anal. 1(4), 350–354 (1994)

    Article  MathSciNet  Google Scholar 

  28. A.J.E.M. Janssen, Duality and biorthogonality for Weyl-Heisenberg frames. J. Fourier Anal. Appl. 1(4), 403–436 (1995)

    Article  MathSciNet  Google Scholar 

  29. A.J.E.M. Janssen, The duality condition for Weyl-Heisenberg frames, in Gabor Analysis and Algorithms, Applications Numerical Harmonic Analysis, pp. 33–84. Birkhäuser Boston, Boston, MA (1998)

    Chapter  Google Scholar 

  30. A.J.E.M. Janssen, On generating tight Gabor frames at critical density. J. Fourier Anal. Appl. 9(2), 175–214 (2003)

    Article  MathSciNet  Google Scholar 

  31. A.J.E.M. Janssen, Zak transforms with few zeros and the tie, in Advances in Gabor analysis, Applications Numerical Harmonic Analysis, pp. 31–70. Birkhäuser Boston, Boston, MA (2003)

    Chapter  Google Scholar 

  32. J. Lemvig, K. Haahr Nielsen, Counterexamples to the B-spline conjecture for Gabor frames. J. Fourier Anal. Appl. 22(6), 1440–1451 (2016)

    Article  MathSciNet  Google Scholar 

  33. Y. Lyubarskii, P.G. Nes, Gabor frames with rational density. Appl. Comput. Harmon. Anal. 34(3), 488–494 (2013)

    Article  MathSciNet  Google Scholar 

  34. Y.I. Lyubarskiĭ, Frames in the Bargmann space of entire functions, in Entire and subharmonic functions, vol. 11 of Adv. Soviet Math., pp. 167–180. Amer. Math. Soc., Providence, RI (1992)

    Google Scholar 

  35. J.V. Neumann, Mathematische Grundlagen der Quantenmechanik. Springer, Berlin, English translation: “Mathematical Foundations of Quantum Mechanics,” (Princeton Univ, Press, 1932), p. 1955

    Google Scholar 

  36. M.A. Rieffel, Projective modules over higher-dimensional noncommutative tori. Canad. J. Math. 40(2), 257–338 (1988)

    Article  MathSciNet  Google Scholar 

  37. A. Ron, Z. Shen, Weyl-Heisenberg frames and Riesz bases in \(L_2(\mathbf{R}^d)\). Duke Math. J. 89(2), 237–282 (1997)

    Article  MathSciNet  Google Scholar 

  38. K. Seip, Density theorems for sampling and interpolation in the Bargmann-Fock space. I. J. Reine Angew. Math. 429, 91–106 (1992)

    MathSciNet  MATH  Google Scholar 

  39. E.M. Stein, G. Weiss, Introduction to Fourier analysis on Euclidean spaces. Princeton University Press, Princeton, N.J. (1971). Princeton Mathematical Series, No. 32

    Google Scholar 

  40. D.F. Walnut, Lattice size estimates for Gabor decompositions. Monatsh. Math. 115(3), 245–256 (1993)

    Article  MathSciNet  Google Scholar 

  41. J. Wexler, S. Raz, Discrete Gabor expansions. Signal Process. 21(3), 207–220 (1990)

    Article  Google Scholar 

  42. Y.Y. Zeevi, M. Zibulski, M. Porat, Multi-window Gabor schemes in signal and image representations, in Gabor Analysis and Algorithms, Applications Numerical Harmonic Analysis, pp. 381–407. Birkhäuser Boston, Boston, MA (1998)

    Chapter  Google Scholar 

  43. M. Zibulski, Y.Y. Zeevi, Analysis of multiwindow Gabor-type schemes by frame methods. Appl. Comput. Harmon. Anal. 4(2), 188–221 (1997)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

    Karlheinz Gröchenig & Sarah Koppensteiner

Authors
  1. Karlheinz Gröchenig
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sarah Koppensteiner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Karlheinz Gröchenig .

Editor information

Editors and Affiliations

  1. Vanderbilt University, Nashville, TN, USA

    Akram Aldroubi

  2. Department of Mathematics (FCEyN), Universidad de Buenos Aires, Buenos Aires, Argentina

    Carlos Cabrelli

  3. Department of Mathematics (LAMA), Université Paris-Est Créteil, Creteil, Paris, France

    Stéphane Jaffard

  4. Department of Mathematics (FCEyN), Universidad de Buenos Aires, Buenos Aires, Argentina

    Ursula Molter

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Gröchenig, K., Koppensteiner, S. (2019). Gabor Frames: Characterizations and Coarse Structure. In: Aldroubi, A., Cabrelli, C., Jaffard, S., Molter, U. (eds) New Trends in Applied Harmonic Analysis, Volume 2. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-32353-0_4

Download citation

Publish with us

Policies and ethics

Search

Navigation