Factor posets of frames and dual frames in finite dimensions

TL;DR

This paper explores the structure of factor posets of frames in finite-dimensional Hilbert spaces, characterizing when certain posets arise and examining the relationship between frames and their duals, including duals optimized for a9^p minimization.

Contribution

It provides a characterization of factor posets of frames and investigates the connections between frames and their duals, including duals with a9^p minimization.

Findings

Characterization of when a poset is a factor poset of a frame

Analysis of the relationship between frames and their duals

Discussion of duals with respect to a9^p minimization

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 $\{f_i\}_{i \in J}$ 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.