Vol. 9, No. 2, 2016

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Factor posets of frames and dual frames in finite dimensions

Kileen Berry, Martin S. Copenhaver, Eric Evert, Yeon Hyang Kim, Troy Klingler, Sivaram K. Narayan and Son T. Nghiem

Vol. 9 (2016), No. 2, 237–248
Abstract

We consider frames in a finite-dimensional Hilbert space, where frames are exactly the spanning sets of the vector space. A factor poset of a frame is defined to be a collection of subsets of $I$, the index set of our vectors, ordered by inclusion so that nonempty $J\subseteq I$ is in the factor poset if and only if ${\left\{{f}_{i}\right\}}_{i\in J}$ is a tight frame. We first study when a poset $P\subseteq {2}^{I}$ is a factor poset of a frame and then relate the two topics by discussing the connections between the factor posets of frames and their duals. Additionally we discuss duals with regard to ${\ell }^{p}$-minimization.

Keywords
frames, tight frames, factor poset, $\ell_p$-norm
Mathematical Subject Classification 2010
Primary: 42C15, 05B20, 15A03