Related papers: A Decomposition Theorem for frames and the Feichti…
We prove the Feichtinger Conjecture for a class of Bessel sequences of unit norm vectors in a Hilbert space. Also, we prove that every Bessel sequence of unit vectors in a Hilbert space can be partitioned into finitely many uniformly…
The Kadison-Singer problem asks: does every pure state on the diagonal sublgebra of the C*-algebra of bounded operators on a separable infinite dimensional Hilbert space admit a unique extension? A yes answer is equivalent to several open…
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 Bessel sequence of normalized kernel…
We answer a number of open problems in frame theory concerning the decomposition of frames into linearly independent and/or spanning sets. We prove that in finite dimensional Hilbert spaces, Parseval frames with norms bounded away from 1…
The Feichtinger Conjecture, if true, would have as a corollary that for each set $E\subset \T$ and $\Lambda \subset \Z$, there is a partition $\Lambda_1,...,\Lambda_N$ of $\Z$ such that for each $1\le i \le N$, $\{\exp(2\pi i x\lambda):…
We prove that if a Bessel sequence in a Hilbert space, that is indexed by a countably infinite group in an invariant manner, can be partitioned into finitely many Riesz basic sequences, then each of the sets in the partition can be chosen…
We show an extension of a probabilistic result of Marcus, Spielman, and Srivastava, which resolved the Kadison-Singer problem, for block diagonal positive semidefinite random matrices. We use this result to show several selector results,…
It is known that the tensor product of two sequences, in the tensor product of two separable Hilbert spaces, is a frame if and only if each component of that product is a frame. This paper proposes a sort of generalization of the…
In this paper we ask when it is possible to transform a given sequence into a frame or a lower semi frame by multiplying the elements by numbers. In other words, we ask when a given sequence is a weighted frame or a weighted lower semi…
An equivalent definition of the Fibonacci numbers is that they are the unique sequence such that every integer can be written uniquely as a sum of non-adjacent terms. We can view this as we have bins of length 1, we can take at most one…
For any unitary matrix there exists a ZXZ decomposition, according to a theorem by Idel and Wolf. For any even-dimensional unitary matrix there exists a block-ZXZ decomposition, according to a theorem by F\"uhr and Rzeszotnik. We conjecture…
Zeckendorf's theorem states that every positive integer can be uniquely decomposed as a sum of nonconsecutive Fibonacci numbers. The distribution of the number of summands converges to a Gaussian, and the individual measures on gaps between…
Frame theory provides a robust method for recovering vectors in a Hilbert space from inner product data, though the associated decomposition formula can be computationally demanding. We relax the frame condition by studying sequences that…
Finite frames, or spanning sets for finite-dimensional Hilbert spaces, are a ubiquitous tool in signal processing. There has been much recent work on understanding the global structure of collections of finite frames with prescribed…
The Union Closed Sets Conjecture states that in every finite, nontrivial set family closed under taking unions there is an element contained in at least half of all the sets of the family. We investigate two new directions with respect to…
We prove a general decomposition theorem for the modal $\mu$-calculus $L_\mu$ in the spirit of Feferman and Vaught's theorem for disjoint unions. In particular, we show that if a structure (i.e., transition system) is composed of two…
We formulate a quantitative finite-dimensional conjecture about frame multipliers and prove that it is equivalent to Conjecture 1 in [SB2]. We then present solutions to the conjecture for certain classes of frame multipliers. In particular,…
With the aid of utilising tensor products, we give a simplified proof to the fundamental theorem of Benedetto and Fickus about the existence and characterisation of finite, normalised tight frames. We also establish unit-norm tensor…
Zeckendorf proved that every integer can be written uniquely as a sum of non-consecutive Fibonacci numbers $\{F_n\}$, and later researchers showed that the distribution of the number of summands needed for such decompositions of integers in…
Two Bessel sequences are orthogonal if the composition of the synthesis operator of one sequence with the analysis operator of the other sequence is the 0 operator. We characterize when two Bessel sequences are orthogonal when the Bessel…