Spanning and independence properties of frame partitions

Bodmann, Bernhard; Casazza, Peter; Paulsen, Vern; Speegle, Darrin

doi:10.1090/S0002-9939-2011-11072-4

Spanning and independence properties of frame partitions
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by Bernhard G. Bodmann, Peter G. Casazza, Vern I. Paulsen and Darrin Speegle PDF

Proc. Amer. Math. Soc. 140 (2012), 2193-2207 Request permission

Abstract:

We answer a number of open problems in frame theory concerning the decomposition of frames into linearly independent and/or spanning sets. We prove that Parseval frames with norms bounded away from $1$ can be decomposed into a number of sets whose complements are spanning, where the number of these sets only depends on the norm bound. Further, we prove a stronger result for Parseval frames whose norms are uniformly small, which shows that in addition to the spanning property, the sets can be chosen to be independent and the complement of each set can contain a number of disjoint, spanning sets.

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