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A Decomposition Theorem for frames and the Feichtinger Conjecture

Peter G. Casazza·Gitta Kutyniok·Darrin Speegle·Janet C. Tremain
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Abstract

In this paper we study the Feichtinger Conjecture in frame theory, which was recently shown to be equivalent to the 1959 Kadison-Singer Problem in $C^{*}$-Algebras. We will show that every bounded Bessel sequence can be decomposed into two subsets each of which is an arbitrarily small perturbation of a sequence with a finite orthogonal decomposition. This construction is then used to answer two open problems concerning the Feichtinger Conjecture: 1. The Feichtinger Conjecture is equivalent to the conjecture that every unit norm Bessel sequence is a finite union of frame sequences. 2. Every unit norm Bessel sequence is a finite union of sets each of which is $ω$-independent for $\ell_2$-sequences.

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Peter G. Casazza, Gitta Kutyniok, Darrin Speegle, Janet C. Tremain. 2007-02-08. A Decomposition Theorem for frames and the Feichtinger Conjecture. https://arxiv.org/abs/math/0702216

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