Skip to main content
SpringerLink
Log in

Banach Gelfand Triples for Gabor Analysis

  • Chapter
Pseudo-Differential Operators

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1949))

Abstract

It is the purpose of this survey note to show the relevance of a Gelfand triple which is closely connected with time–frequency analysis and Gabor analysis. The Segal algebra S 0(ℝd) and its dual can be shown to be — for a large variety of concrete cases – a convenient substitute for the Schwartz space S(ℝd) and it’s dual, the space of tempered distributions S′(ℝd). This concrete pair of Banach spaces is actually a Gelfand triple, which allows to describe in a very intuitive way the properties of the classical Fourier transform and other unitary operators arising in the treatment of various mathematical questions, e.g., multipliers in harmonic analysis. We will demonstrate the usefulness of the Banach Gelfand triple (S 0(ℝd), L 2(ℝd), S 0(ℝd)) within time–frequency analysis, with a special emphasis on questions from time–frequency analysis and Gabor analysis.

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 PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.95
Price excludes VAT (USA)
  • Compact, lightweight 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. R. Balan, P. Casazza, C. Heil, and Z. Landau. Deficits and excesses of frames. Adv. Comput. Math., 18(2-4):93–116, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  2. R. Balan, P. Casazza, C. Heil, and Z. Landau. Excesses of Gabor frames. Appl. Comput. Harmon. Anal., 14(2):87–106, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  3. R. Balian. Un principe d’incertitude fort en théorie du signal on en mécanique quantique. C. R. Acad. Sci. Paris, 292:1357–1362, 1981.

    MathSciNet  Google Scholar 

  4. O. Christensen and Y. C. Eldar. Oblique dual frames and shift-invariant spaces. Appl. and Comp. Harm. Anal., 17(1):48–68, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  5. E. Cordero and K. Gröchenig. Time-frequency analysis of Localization operators. J. Funct. Anal., 205(1):107–131, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  6. E. Cordero and K. Gröchenig. Necessary conditions for Schatten class localization operators,Proc. Amer. Math. Soc., 133:3573–3579, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  7. E. Cordero and K. Gröchenig. Symbolic Calculus and Fredholm Property for Localization Operators. J. Fourier Anal. Appl., 2006.

    Google Scholar 

  8. S. Dahlke, M. Fornasier, and T. Raasch. Adaptive frame methods for elliptic operator equations. to appear in Adv. Comp. Math., 2005.

    Google Scholar 

  9. I. Daubechies, S. Jaffard, and J.L. Journé. A simple Wilson orthonormal basis with exponential decay. SIAM J. Math. Anal., 22(2):554–572, 1991.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. M. Dörfler, H. G. Feichtinger, and K. Gröchenig. Time–frequency partitions for the Gelfand triple (S 0,L 2,S 0′). Submitted, 2004.

    Google Scholar 

  12. R. Duffin and A. Schaeffer. A class of nonharmonic Fourier series. Trans. Amer. Math. Soc., 72:341–366, 1952.

    Article  MATH  MathSciNet  Google Scholar 

  13. H. G. Feichtinger, Modulation spaces on locally compact abelian groups, TechnicalReport, University Vienna, 1983. and also in Wavelets and Their Applications, M. Krishna, R. Radha, S. Thangavelu, editors, Allied Publishers, 99–140, 2003.

    Google Scholar 

  14. H. G. Feichtinger, K. Gröchenig, and D. Walnut. Wilson bases and modulation spaces. Math. Nachrichten, 155:7–17, 1992.

    Article  MATH  Google Scholar 

  15. H. G. Feichtinger and N. Kaiblinger. Varying the time–frequency lattice of Gabor frames. Trans. Am. Math. Soc., 356(5):2001–2023, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  16. H. G. Feichtinger and W. Kozek. Quantization of TF–lattice invariant operators on elementary LCA groups. In H.G. Feichtinger and T. Strohmer, editors, Gabor Analysis and Algorithms: Theory and Applications, chapter 7, pages 233–266. Birkhäuser, Boston, 1998.

    Google Scholar 

  17. H. G. Feichtinger and F. Luef. Wiener amalgam spaces for the fundamental identity of Gabor analysis. In Proc. Conf. El-Escorial, summer 2004, El-Escorial, 2005. To appear.

    Google Scholar 

  18. H. G. Feichtinger, F. Luef, and T. Werther. A Guided Tour from Linear Algebra to the Foundations of Gabor Analysis. 2005.

    Google Scholar 

  19. H. G. Feichtinger and K. Nowak. A first survey of Gabor multipliers. In H.G. Feichtinger and T. Strohmer, editors, Advances in Gabor Analysis, Appl. Numer. Harmon. Anal., pages 99–128. Birkhäuser, 2003.

    Google Scholar 

  20. H. G. Feichtinger and G. Zimmermann. A Banach space of test functions for Gabor analysis. In H. G. Feichtinger and T. Strohmer, editors, Gabor Analysis and Algorithms: Theory and Applications, pages 123–170. Birkhäuser, Boston, 1998.

    Google Scholar 

  21. G. B. Folland. A course in abstract harmonic analysis. Studies in Advanced Mathematics. Boca Raton, FL: CRC Press, 1995.

    Google Scholar 

  22. D. Gabor. Theory of communication. Proc. IEE (London), 93(III):429–457, November 1946.

    Google Scholar 

  23. I. M. Gel’fand and A. G. Kostyuchenko. Entwicklung nach Eigenfunktionen von Differentialoperatoren und anderen Operatoren. Dokl. Akad. Nauk SSSR, 103:349–352, 1955.

    MATH  MathSciNet  Google Scholar 

  24. I. M. Gelfand and N. Ya. Vilenkin. Applications of harmonic analysis. New York and London: Academic Press. XIV, 1964.

    Google Scholar 

  25. K. Gröchenig. An uncertainty principle related to the Poisson summation formula. Stud. Math., 121(1):87–104, 1996.

    MATH  Google Scholar 

  26. K. Gröchenig. Foundations of Time–Frequency Analysis. Birkhäuser, 2001.

    Google Scholar 

  27. K. Gröchenig and M. Leinert. Wiener’s Lemma for twisted convolution and Gabor frames. J. Amer. Math. Soc., 17(1):1–18, 2003.

    Article  Google Scholar 

  28. C. Heil. A basis theory primer. Unpublished Manuscript, July 1997.

    Google Scholar 

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

    Article  MATH  Google Scholar 

  30. A. J. E. M. Janssen. Classroom proof of the density theorem for Gabor systems. Unpublished Manuscript, 2005.

    Google Scholar 

  31. N. Kaiblinger. Approximation of the Fourier transform and the dual Gabor window. J. Fourier Anal. Appl., 11(1):25–42, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  32. J. G. Kirkwood. Quantum statistics of almost classical assemblies. Phys. Rev., II. Ser. 44:31–37, 1933.

    MATH  Google Scholar 

  33. S. Li and H. Ogawa. Pseudoframes for subspaces with applications. J. Four. Anal. and Appl., 10(4):409–431, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  34. F. Low. Complete sets of wave packets. In C. DeTar, editor, A Passion for Physics - Essay in Honor of Geoffrey Chew, pages 17–22. World Scientific, Singapore, 1985.

    Google Scholar 

  35. Y. I. Lyubarskii. Frames in the Bargmann space of entire functions. Adv. Soviet Math., 429:107–113, 1992.

    Google Scholar 

  36. A. Ron and Z. Shen. Weyl-Heisenberg frames and Riesz basis on L 2(ℝ d). Duke Math. J., 89(2):273–282, 1997.

    Article  MathSciNet  Google Scholar 

  37. W. Rudin. Functional Analysis. Mc Graw-Hill, 2nd edition, 1991.

    Google Scholar 

  38. K. Seip and R. Wallsten. Density theorems for sampling and interpolation in the Bargmann-Fock space II. J. reine angewandte Mathematik, 429:107–113, 1992.

    MATH  MathSciNet  Google Scholar 

  39. T. Strohmer. Numerical algorithms for discrete Gabor expansions. In H. G. Feichtinger and T. Strohmer, editors, Gabor Analysis and Algorithms: Theory and Applications, pages 267–294. Birkhäuser, Boston, 1998.

    Google Scholar 

  40. J. Wexler and S. Raz. Discrete Gabor expansions. Signal Processing, 21(3):207–220, 1990.

    Article  Google Scholar 

  41. E. P. Wigner. On the quantum correction for thermo-dynamic equilibrium. Phys. Rerv. Letters, 40:749–759, 1932.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universität Wien, Nordbergstrasse 15, Vienna, Austria

    H. Feichtinger & F. Luef

  2. University of Torino, Turin, Italy

    E. Cordero

Authors
  1. H. Feichtinger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. Luef
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. Cordero
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica, Universitá di Torino, Via Carlo Alberto 10, 10123, Torino, Italy

    Luigi Rodino

  2. Department of Mathematics and Statistics, York University, Keele Street 4700, M3J 1P3, Toronto, Ontario, Canada

    M. W. Wong

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Feichtinger, H., Luef, F., Cordero, E. (2008). Banach Gelfand Triples for Gabor Analysis. In: Rodino, L., Wong, M.W. (eds) Pseudo-Differential Operators. Lecture Notes in Mathematics, vol 1949. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68268-4_1

Download citation

Publish with us

Policies and ethics

Search

Navigation