The Kadison–Singer Problem in mathematics and engineering

Casazza, Peter G.; Tremain, Janet C.

doi:10.1073/pnas.0507888103

Cited by 101 publications

(117 citation statements)

References 30 publications

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“…Recently, it has been shown by Casazza and Tremain that the Feichtinger Conjecture is equivalent to the Kadison-Singer Conjecture, which is the deepest open problem in operator theory today [39].…”

Section: Redundancymentioning

confidence: 99%

History and Evolution of the Density Theorem for Gabor Frames

Heil

2007

J Fourier Anal Appl

176

137

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ABSTRACT. The Density Theorem for Gabor Frames is one of the fundamental results of time-frequency analysis. This expository survey attempts to reconstruct the long and very involved history of this theorem and to present its context and evolution, from the onedimensional rectangular lattice setting, to arbitrary lattices in higher dimensions, to irregular Gabor frames, and most recently beyond the setting of Gabor frames to abstract localized frames. Related fundamental principles in Gabor analysis are also surveyed, including the Wexler-Raz biorthogonality relations, the Duality Principle, the Balian-Low Theorem, the Walnut and Janssen representations, and the Homogeneous Approximation Property. An extended bibliography is included.

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Section: Redundancymentioning

confidence: 99%

History and Evolution of the Density Theorem for Gabor Frames

Heil

2007

J Fourier Anal Appl

176

137

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“…Much work has been done on the Feichtinger Conjecture in just the last few years [15,6,12,11,8,4]. In particular, by employing the equivalence of the Paving Conjecture to the Kadison-Singer Problem shown by Anderson in 1979 [1] and by using the Bourgain-Tzafriri Conjecture [3] which arose from the "restricted invertibility principle" by Bourgain and Tzafriri from 1987 [2], the series of papers [6,12,11,8] proves the equivalence between the Kadison-Singer Problem and the Feichtinger Conjecture.…”

Section: Conjecture 11 (Feichtinger Conjecture) Every Bounded Framementioning

confidence: 99%

“…Recently, there has been a flurry of activity around this problem due to a fundamental paper by the first and fourth authors [11] (cf. also the longer version joint with M. Fickus and E. Weber [8]), which connects the Kadison-Singer Conjecture with many longstanding open conjectures in a variety of different research areas -in Hilbert space theory, Banach space theory, frame theory, harmonic analysis, time-frequency analysis, and even in engineering -by proving that these conjectures are in fact equivalent to the Kadison-Singer Problem.…”

Section: Introductionmentioning

confidence: 99%

A decomposition theorem for frames and the Feichtinger Conjecture

Casazza¹,

Kutyniok²,

Speegle³

et al. 2008

Proc. Amer. Math. Soc.

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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 2 -sequences.

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“…For instance, in wireless communications frames ensure robustness of transmission against erasures [21], in image processing they serve as building blocks for novel directional representation systems [3], and just recently it has been discovered that the theory of frames may provide a means to attack the Kadison-Singer Conjecture in operator theory from 1959 [4].…”

Section: Introductionmentioning

confidence: 99%

Beurling Dimension of Gabor Pseudoframes for Affine Subspaces

Czaja

Kutyniok

Speegle

2008

J Fourier Anal Appl

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Abstract. Pseudoframes for subspaces have been recently introduced by S. Li and H. Ogawa as a tool to analyze lower dimensional data with arbitrary flexibility of both the analyzing and the dual sequence.In this paper we study Gabor pseudoframes for affine subspaces by focusing on geometrical properties of their associated sets of parameters. We first introduce a new notion of Beurling dimension for discrete subsets of R d by employing a certain generalized Beurling density. We present several properties of Beurling dimension including a comparison with other notions of dimension showing, for instance, that our notion includes the mass dimension as a special case. Then we prove that Gabor pseudoframes for affine subspaces satisfy a certain Homogeneous Approximation Property, which implies invariance under time-frequency shifts of an approximation by elements from the pseudoframe.The main result of this paper is a classification of Gabor pseudoframes for affine subspaces by means of the Beurling dimension of their sets of parameters. This provides us, in particular, with a Nyquist dimension which separates sets of parameters of pseudoframes from those of non-pseudoframes and which links a fixed value to sets of parameters of pseudo-Riesz sequences. These results are even new for the special case of Gabor frames for an affine subspace.

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The Kadison–Singer Problem in mathematics and engineering

Cited by 101 publications

References 30 publications

History and Evolution of the Density Theorem for Gabor Frames

History and Evolution of the Density Theorem for Gabor Frames

A decomposition theorem for frames and the Feichtinger Conjecture

Beurling Dimension of Gabor Pseudoframes for Affine Subspaces

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