Abstract
The concept of R-duals of a frame was introduced by Casazza, Kutyniok and Lammers in 2004, with the motivation to obtain a general version of the duality principle in Gabor analysis. For tight Gabor frames and Gabor Riesz bases the three authors were actually able to show that the duality principle is a special case of general results for R-duals. In this paper we introduce various alternative R-duals, with focus on what we call R-duals of type II and III. We show how they are related and provide characterizations of the R-duals of type II and III. In particular, we prove that for tight frames these classes coincide with the R-duals by Casazza et al., which is desirable in the sense that the motivating case of tight Gabor frames already is well covered by these R-duals. On the other hand, all the introduced types of R-duals generalize the duality principle for larger classes of Gabor frames than just the tight frames and the Riesz bases; in particular, the R-duals of type III cover the duality principle for all Gabor frames.
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Acknowledgments
Diana Stoeva was supported by the Austrian Science Fund (FWF) START-project FLAME (‘Frames and Linear Operators for Acoustical Modeling and Parameter Estimation’; Y 551-N13). She is grateful for the hospitality of the Technical University of Denmark, where most of the work on the paper was done. Both authors thank the reviewer for useful suggestions that improved the presentation.
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Communicated by Peter G. Casazza.
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Stoeva, D.T., Christensen, O. On R-Duals and the Duality Principle in Gabor Analysis. J Fourier Anal Appl 21, 383–400 (2015). https://doi.org/10.1007/s00041-014-9376-8
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DOI: https://doi.org/10.1007/s00041-014-9376-8