A decomposition theorem for frames and the Feichtinger Conjecture

@inproceedings{Casazza2007ADT,
  title={A decomposition theorem for frames and the Feichtinger Conjecture},
  author={Peter G. Casazza and Gitta Kutyniok and Darrin Speegle and Janet C. Tremain},
  year={2007}
}
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… 
Spanning and independence properties of frame partitions
TLDR
It is proved that in finite dimensional Hilbert spaces, 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.
The Kadison–Singer and Paulsen Problems in Finite Frame Theory
We now know that some of the basic open problems in frame theory are equivalent to fundamental open problems in a dozen areas of research in both pure and applied mathematics, engineering, and
The Feichtinger Conjecture and Reproducing Kernel Hilbert Spaces
We prove two new equivalences of the Feichtinger conjecture that involve reproducing kernel Hilbert spaces. We prove that if for every Hilbert space, contractively contained in the Hardy space, each
A Discussion of the Kadison-Singer Conjecture
The Kadison-Singer problem is a major result in operator theory, recently proven by Marcus, Spielman and Srivastava. Since its formulation in 1959, it has generated a significant amount of interest,
Spanning and Independence Properties of Finite Frames
The fundamental notion of frame theory is redundancy. It is this property which makes frames invaluable in so many diverse areas of research in mathematics, computer science, and engineering, because
The Kadison–Singer Problem in mathematics and engineering
  • P. Casazza, J. Tremain
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 2006
We will see that the famous intractible 1959 Kadison–Singer Problem in C*-algebras is equivalent to fundamental open problems in a dozen different areas of research in mathematics and engineering.
Consequences of the Marcus/Spielman/Srivastava Solution of the Kadison-Singer Problem
It is known that the famous, intractable 1959 Kadison-Singer problem in C∗-algebras is equivalent to fundamental unsolved problems in a dozen areas of research in pure mathematics, applied
Feichtinger Conjectures, $R_\varepsilon$-Conjectures and Weaver's Conjectures for Banach spaces
Abstract: Motivated from two decades old famous Feichtinger conjectures for frames, Rε-conjecture and Weaver’s conjecture for Hilbert spaces (and their solution by Marcus, Spielman, and Srivastava),
Discrete Hilbert transforms on sparse sequences
Weighted discrete Hilbert transforms (an)n ↦ ∑n an vn/(z−γn) from ℓ2ν to a weighted L2‐space are studied, with Γ=(γn) a sequence of distinct points in the complex plane and v=(vn) a corresponding
...
...

References

SHOWING 1-10 OF 29 REFERENCES
Frames and the Feichtinger conjecture
We show that the conjectured generalization of the Bourgain-Tzafriri restricted-invertibility theorem is equivalent to the conjecture of Feichtinger, stating that every bounded frame can be written
The Feichtinger Conjecture for Wavelet Frames, Gabor Frames and Frames of Translates
Abstract The Feichtinger conjecture is considered for three special families of frames. It is shown that if a wavelet frame satisfies a certain weak regularity condition, then it can be written as
On a problem of Kadison and Singer.
In 1959, R. Kadison and I. Singer [10] raised the question whether every pure state (i.e. an extremal element in the space of states) on the C*-algebra D of the diagonal operators on (2 has a unique
KADISON-SINGER MEETS BOURGAIN-TZAFRIRI
Abstract. We show that the Kadison-Singer problem is equivalent to the (strong) restricted invertibility conjecture of Bourgain-Tzafriri. We also show that these two problems are equivalent to two
The Kadison–Singer Problem in mathematics and engineering
  • P. Casazza, J. Tremain
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 2006
We will see that the famous intractible 1959 Kadison–Singer Problem in C*-algebras is equivalent to fundamental open problems in a dozen different areas of research in mathematics and engineering.
Extensions, restrictions, and representations of states on *-algebras
In the first three sections the question of when a pure state g on a C*-subalgebra B of a C*-algebra A has a unique state extension is studied. It is shown that an extension/is unique if and only if
The Kadison-Singer problem in discrepancy theory
Invertibility of ‘large’ submatrices with applications to the geometry of Banach spaces and harmonic analysis
AbstractThe main problem investigated in this paper is that of restricted invertibility of linear operators acting on finite dimensionallp-spaces. Our initial motivation to study such questions lies
Frames of subspaces
One approach to ease the construction of frames is to first construct local components and then build a global frame from these. In this paper we will show that the study of the relation between a
Density, overcompleteness, and localization of frames
This work presents a quantitative framework for describing the overcompleteness of a large class of frames. It introduces notions of localization and approximation between two frames F = {fi}i∈I and
...
...