Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Discrete sets of coherent states and their use in signal analysis

  • Conference paper
  • First Online:
Differential Equations and Mathematical Physics

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1285))

Abstract

We discuss expansions of L 2-functions into {φmn; m,n∈Z}, where the φmn are generated from one function φ, either by translations in phase space, i.e. \(\phi _{mn} (x) = e^{imp_0 x} \phi (x - nq_0 )\), (p 0, q 0 fixed), or by translations and dilations, i.e. φmn(x)=a −m/20 φ(a −m0 x−nb 0). These expansions can be used for phase space localization.

This paper is partially supported by NSF grant MCS 8301662.

"Bevoegdverklaard Navorser" at the Belgian National Science Foundation; on leave from Vrije Universiteit Brussel, Belgium.

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 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.00
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. J.R.Klauder and B.-S. Skagerstam, "Coherent States. Applications in Physics and Mathematical Physics". World Sci. Pub. (Singapore, 1985).

    Google Scholar 

  2. I. Daubechies, A. Grossmann and Y. Meyer, "Painless non-orthogonal expansions." J. Math. Phys. 27 (1986) 1271–1283.

    Article  MathSciNet  MATH  Google Scholar 

  3. J. Morlet, G. Arens, I. Fourgeau and D. Giard, "Wave propagation and sampling theory." Geophysics 47 (1982) 203–236.

    Article  Google Scholar 

  4. E.Stein, "Singular integrals and differentiability prperties of functions." Princeton University Press (1970).

    Google Scholar 

  5. M.Frazier and B.Jawerth, "Decomposition of Besov spaces." To be published.

    Google Scholar 

  6. Y.Meyer. "La transformation en ondelettes et les nouveaux paraproduits." To be published in Actes du Colloque d'Analyse non lineaire du Ceremade, Univ. de Paris-Dauphine.

    Google Scholar 

  7. P.G.Lemarie and Y.Meyer, "Ondelettes et bases hilbertiennes." To be published in Revista Ibero-Americana.

    Google Scholar 

  8. R.R. Coifman and Y.Meyer, "The discrete wavelet transform." To be published. Y.Meyer, "Principe d'incertitude, bases hilbertiennes et algebres d'operateurs." Seminaire Bourbaki, 1985–1986, nr. 662.

    Google Scholar 

  9. R.J. Duffin and A.C. Schaeffer, "A class of nonharmonic Fourier series." Trans. Am. Math. Soc. 72 (1952) 341–366.

    Article  MathSciNet  MATH  Google Scholar 

  10. I.Daubechies and A.Grossmann, "Frames in the Bargmann space of entire functions." Submitted for publication.

    Google Scholar 

  11. I.Daubechies, "Frames of coherent states." In preparation.

    Google Scholar 

  12. R. Balian, "Un principe d'incertitude fort en theorie du signal ou en mecanique quantique." C. R. Acad. Sci. Paris 292 (serie 2) (1981) 1357–1362.

    MathSciNet  Google Scholar 

  13. R.R.Coifman and S.Semmes, private communication.

    Google Scholar 

  14. H. Bacry, A. Grossmann and J. Zak, "Proof of completeness of lattice states." Phys. Rev. B12 (1975) 1118–1120.

    Article  Google Scholar 

  15. A.J.E.M. Janssen, "Bargmann transform, Zak transform, and coherent states." J. Math. Phys. 23 (1982) 720–731.

    Article  MathSciNet  MATH  Google Scholar 

  16. D. Gabor, "Theory of communication." J. Inst. Elec. Engrs. (London) 93 (1946) 429–457.

    Google Scholar 

  17. A.J.E.M.Janssen, "Gabor representation and Wigner distribution of signals." Proc. IEEE Acoust., Speech and Signal Processing, April 1983, 41 B.2.1.-41 B.2.4.

    Google Scholar 

  18. V. Bargmann, P. Butero, L. Girardello and J.R. Klauder, "On the completeness of coherent states." Rep. Mod. Phys. 2 (1971) 221–228.

    MathSciNet  Google Scholar 

  19. A.M. Perelomov, "On the completeness of a system of coherent states." Theor. Math. Phys. 6 (1971) 156–164.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, 10012, New York, NY

    Ingrid Daubechies

Authors
  1. Ingrid Daubechies
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Ian W. Knowles Yoshimi Saitō

Rights and permissions

Reprints and permissions

Copyright information

© 1987 Springer-Verlag

About this paper

Cite this paper

Daubechies, I. (1987). Discrete sets of coherent states and their use in signal analysis. In: Knowles, I.W., Saitō, Y. (eds) Differential Equations and Mathematical Physics. Lecture Notes in Mathematics, vol 1285. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0080582

Download citation

  • DOI: https://doi.org/10.1007/BFb0080582

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-18479-9

  • Online ISBN: 978-3-540-47983-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation