A note on finite dual frame pairs
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- by Ole Christensen, Alexander M. Powell and Xiang Chun Xiao
- Proc. Amer. Math. Soc. 140 (2012), 3921-3930
- DOI: https://doi.org/10.1090/S0002-9939-2012-11256-0
- Published electronically: April 25, 2012
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Abstract:
The purpose of this note is to extend certain key results in frame theory from the setting of tight frames to dual pairs of frames. We provide various characterizations of dual frame pairs $\{e_n\}_{n=1}^N, \{f_n\}_{n=1}^N$ for a $d$-dimensional Hilbert space $\mathbb {K}^d.$ Based on this we characterize those scalar sequences $\{\alpha _n\}_{n=1}^N$ for which there exist dual pairs of frames $\{e_n\}_{n=1}^N, \{f_n\}_{n=1}^N$ for $\mathbb {K}^d$ such that $\alpha _n = \langle e_n, f_n \rangle .$ This generalizes the well-known fundamental inequality of tight frames.References
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Bibliographic Information
- Ole Christensen
- Affiliation: Department of Mathematics, Technical University of Denmark, Building 303, 2800 Lyngby, Denmark
- MR Author ID: 339614
- Email: Ole.Christensen@mat.dtu.dk
- Alexander M. Powell
- Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
- MR Author ID: 712100
- Email: alexander.m.powell@vanderbilt.edu
- Xiang Chun Xiao
- Affiliation: Department of Mathematics, Xiamen University, Xiamen 361005, People’s Republic of China
- Email: xxc570@163.com
- Received by editor(s): January 7, 2011
- Received by editor(s) in revised form: May 10, 2011
- Published electronically: April 25, 2012
- Communicated by: Michael T. Lacey
- © Copyright 2012 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 140 (2012), 3921-3930
- MSC (2010): Primary 42C15
- DOI: https://doi.org/10.1090/S0002-9939-2012-11256-0
- MathSciNet review: 2944732