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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A note on finite dual frame pairs


Authors: Ole Christensen, Alexander M. Powell and Xiang Chun Xiao
Journal: Proc. Amer. Math. Soc. 140 (2012), 3921-3930
MSC (2010): Primary 42C15
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.


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Additional Information

Ole Christensen
Affiliation: Department of Mathematics, Technical University of Denmark, Building 303, 2800 Lyngby, Denmark
Email: Ole.Christensen@mat.dtu.dk

Alexander M. Powell
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
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

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11256-0
PII: S 0002-9939(2012)11256-0
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
Article copyright: © Copyright 2012 American Mathematical Society