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


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.


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

Mathematics Subject Classification



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© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, 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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