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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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