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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(More)
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(More)
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(More)
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(More)
For finite and infinite dimensional Hilbert spaces H we classify the sequences of positive real numbers {a n } ∞ n=1 so that there is a tight frame {ϕ n } ∞ n=1 for H satisfying: ϕ n = a n , for all n = 1, 2, 3, · · ·. In the finite dimensional case we will identify the frames which are closest to being tight (in the sense of minimizing potential enerty)(More)
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(More)
We will give some new techniques for working with problems surrounding the Bourgain-Tzafriri 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(More)
We provide a new method for constructing equiangular tight frames (ETFs). This method 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. The construction is extremely simple: a tensor-like combination of a Steiner system and a(More)
We will prove some new, fundamental results in frame theory by computing the unconditional constant (for all definitions of unconditional) for the frame expansion of a vector in a Hilbert space and see that it is √B/A, where A, B are the frame bounds of the frame. It follows that tight frames have unconditional constant one. We then generalize this(More)