A note on finite dual frame pairs

Christensen, Ole; Powell, Alexander; Xiao, Xiang Chun

doi:10.1090/S0002-9939-2012-11256-0

A note on finite dual frame pairs
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by Ole Christensen, Alexander M. Powell and Xiang Chun Xiao PDF

Proc. Amer. Math. Soc. 140 (2012), 3921-3930 Request permission

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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