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Integral Equations and Operator Theory

, Volume 84, Issue 4, pp 577–590 | Cite as

On Various R-duals and the Duality Principle

  • Diana T. StoevaEmail author
  • Ole Christensen
Open Access
Article
  • 428 Downloads

Abstract

The duality principle states that a Gabor system is a frame if and only if the corresponding adjoint Gabor system is a Riesz sequence. In general Hilbert spaces and without the assumption of any particular structure, Casazza, Kutyniok and Lammers have introduced the so-called R-duals that also lead to a characterization of frames in terms of associated Riesz sequences; however, it is still an open question whether this abstract theory is a generalization of the duality principle. In this paper we prove that a modified version of the R-duals leads to a generalization of the duality principle that keeps all the attractive properties of the R-duals. In order to provide extra insight into the relations between a given sequence and its R-duals, we characterize all the types of R-duals that are available in the literature for the special case where the underlying sequence is a Riesz basis.

Keywords

Frames Riesz bases R-duals R-duals of type II R-duals of type III Riesz sequences Duality principle 

Mathematics Subject Classification

42C15 

References

  1. 1.
    Casazza P.G., Kutyniok G., Lammers M.C.: Duality principles in frame theory. J. Fourier Anal. Appl. 10(4), 383–408 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Christensen, O.: An Introduction to Frames and Riesz Bases. Second expanded edition. Birkhäuser, Boston (2015)Google Scholar
  3. 3.
    Christensen O., Xiao X.C., Zhu Y.C.: Characterizing R-duality in Banach spaces. Acta Math. Sin. Engl. Ser. 29(1), 75–84 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chuang, Z., Zhao, J.: On equivalent conditions of two sequences to be R-dual. J. Inequal. Appl. 2015:10 (2015)Google Scholar
  5. 5.
    Daubechies I., Landau H.J., Landau Z.: Gabor time-frequency lattices and the Wexler-Raz identity. J. Fourier Anal. Appl. 1(4), 437–478 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dutkay D., Han D., Larson D.: A duality principle for groups. J. Funct. Anal. 257(4), 1133–1143 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fan, Z., Ji, H., Shen, Z.: Dual Gramian analysis: duality principle and unitary extension principle. Math. Comput. (2015). doi: 10.1090/mcom/2987
  8. 8.
    Fan, Z., Heinecke, A., Shen, Z.: Duality for frames. J. Fourier Anal. Appl. (2015). doi: 10.1007/s00041-015-9415-0
  9. 9.
    Gröchenig K.: Foundations of Time-Frequency Analysis. Birkhäuser, Boston (2001)CrossRefzbMATHGoogle Scholar
  10. 10.
    Heil, C.: A Basis Theory Primer. Expanded edition. Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, New York (2011)Google Scholar
  11. 11.
    Janssen A.J.E.M.: Duality and biorthogonality for Weyl–Heisenberg frames. J. Fourier Anal. Appl. 1(4), 403–436 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ron A., Shen Z.: Weyl–Heisenberg frames and Riesz bases in \({L^2({\mathbb{R}}^d)}\). Duke Math. J. 89(2), 237–282 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Stoeva D.T., Christensen O.: On R-duals and the duality principle in Gabor analysis. J. Fourier Anal. Appl. 21(2), 383–400 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Xiao X., Zhu Y.: Duality principles of frames in Banach spaces. Acta. Math. Sci. Ser. A Chin. Ed. 29(1), 94–102 (2009)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Wigner E.P.: Normal form of antiunitary operators. J. Math. Phys. 1, 409–413 (1960)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Acoustics Research InstituteViennaAustria
  2. 2.Technical University of Denmark, DTU ComputeLyngbyDenmark

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