We provide a new method for constructing equiangular tight frames (ETFs). The construction is valid in both the real and complex settings, and shows that many of the few previously-known examples of ETFs are but the first representatives of infinite families of such frames. It provides great freedom in terms of the frame's size and redundancy. This method also explicitly constructs the frame vectors in their native domain, as opposed to implicitly defining them via their Gram matrix. Moreover, in this domain, the frame vectors are very sparse. The construction is extremely simple: a tensor-like combination of a Steiner system and a regular simplex. This simplicity permits us to resolve an open question regarding ETFs and the restricted isometry property (RIP): we show that the RIP behavior of some ETFs is unfortunately no better than their coherence indicates.
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. This work gives all these areas common ground on which to interact as well as explaining why each area has volumes of literature on their respective problems without a satisfactory resolution.
Summary. We find finite tight frames when the lengths of the frame elements are predetermined. In particular, we derive a "fundamental inequality" which completely characterizes those sequences which arise as the lengths of a tight frame's elements. Furthermore, using concepts from classical physics, we show that this characterization has an intuitive physical interpretation.
We will show that the famous, intractible 1959 Kadison-Singer problem in C * -algebras is equivalent to fundamental unsolved problems in a dozen areas of research in pure mathematics, applied mathematics and Engineering. This gives all these areas common ground on which to interact as well as explaining why each of these areas has volumes of literature on their respective problems without a satisfactory resolution. In each of these areas we will reduce the problem to the minimum which needs to be proved to solve their version of Kadison-Singer. In some areas we will prove what we believe will be the strongest results ever available in the case that Kadison-Singer fails. Finally, we will give some directions for constructing a counter-example to Kadison-Singer.
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.
Abstract-Real equiangular tight frames can be especially useful in practice because of their structure. The problem is that very few of them are known. We will look at recent advances on the problem of classifying the equiangular tight frames and as a consequence give a classification of this family of frames for all real Hilbert spaces of dimension less than or equal to 50.
We give simple necessary and sufficient conditions on Bessel sequences { f i } and {g i } and operators L 1 , L 2 on a Hilbert space H so that {L 1 f i + L 2 g i } is a frame for H. This allows us to construct a large number of new Hilbert space frames from existing frames.
Abstract. We will give some new techniques for working with problems surrounding the BourgainTzafriri Restricted Invertibility Theorem. First we show that the parameters which work in the theorem for all T 2 √ 2 closely approximate the parameters which work for all operators. This yields a generalization of the theorem which simultaneously does restricted invertibility on a small partition of the vectors and yields a direct proof that the Bourgain-Tzafriri Conjecture is equivalent to the Feichtinger Conjecture. We also fill in two gaps in the theory involving the relationship between paving results for norm one operators with zero diagonal and restricted invertibility results. Mathematics subject classification (2000): Primary: 46B03, 46B07, 47A05.
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