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On the Duality Principle by Casazza, Kutyniok, and Lammers

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Abstract

The R-dual sequences of a frame {f i } i∈I , introduced by Casazza, Kutyniok and Lammers in (J. Fourier Anal. Appl. 10(4):383–408, 2004), provide a powerful tool in the analysis of duality relations in general frame theory. In this paper we derive conditions for a sequence {ω j } j∈I to be an R-dual of a given frame {f i } i∈I . In particular we show that the R-duals {ω j } j∈I can be characterized in terms of frame properties of an associated sequence {n i } i∈I . We also derive the duality results obtained for tight Gabor frames in (Casazza et al. in J. Fourier Anal. Appl. 10(4):383–408, 2004) as a special case of a general statement for R-duals of frames in Hilbert spaces. Finally we consider a relaxation of the R-dual setup of independent interest. Several examples illustrate the results.

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Authors and Affiliations

  1. Department of Mathematics, Technical University of Denmark, Building 303, 2800, Lyngby, Denmark

    Ole Christensen

  2. Department of Mathematical Sciences, KAIST, 373-1, Guseong-dong, Yuseong-gu, Daejeon, 305-701, Republic of Korea

    Hong Oh Kim

  3. Department of Mathematics, Yeungnam University, 214-1, Dae-dong, Gyeongsan-si, Gyeongsangbuk-do, 712-749, Republic of Korea

    Rae Young Kim

Authors
  1. Ole Christensen
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  2. Hong Oh Kim
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  3. Rae Young Kim
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Corresponding author

Correspondence to Rae Young Kim.

Additional information

Communicated by Peter G. Casazza.

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology(2010-0007614).

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Christensen, O., Kim, H.O. & Kim, R.Y. On the Duality Principle by Casazza, Kutyniok, and Lammers. J Fourier Anal Appl 17, 640–655 (2011). https://doi.org/10.1007/s00041-010-9151-4

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  • DOI: https://doi.org/10.1007/s00041-010-9151-4

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