Journal of Fourier Analysis and Applications

, Volume 17, Issue 4, pp 640–655

On the Duality Principle by Casazza, Kutyniok, and Lammers

Article

Abstract

The R-dual sequences of a frame {fi}iI, 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 {ωj}jI to be an R-dual of a given frame {fi}iI. In particular we show that the R-duals {ωj}jI can be characterized in terms of frame properties of an associated sequence {ni}iI. 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.

Keywords

Duality principle Frame Riesz basis Gabor system Wexler-Raz theorem 

Mathematics Subject Classification (2000)

42C15 42C40 42A38 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsTechnical University of DenmarkLyngbyDenmark
  2. 2.Department of Mathematical SciencesKAISTDaejeonRepublic of Korea
  3. 3.Department of MathematicsYeungnam UniversityGyeongsangbuk-doRepublic of Korea

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