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Janet C. Tremain

Publications and source records attributed to Janet C. Tremain.

17 recordsLinked to original sources
arXiv2021

An elementary proof of a fundamental result in phase retrieval

Edidin [3] proved a fundamental result in phase retrieval: Theorem: A family of orthogonal projections $\{P_i\}_{i=1}^m$ does phase retrieval in $\mathbb{R}^n$ if and only if for every $0\not= x\in \mathbb{R}^n$, the family $\{P_ix\}_{i=1}^m$ spans $\mathbb{R}^n$. The proof of this result relies on Algebraic Geometry and so is inaccessible to many people in the field. We will give an elementary proof of this result without Algebraic Geometry. We will also solve the complex version of this result by showing that the "if" part fails and the "only if" part holds in $\mathbb{C}^n$. Finally, we will show that these techniques can be used to verify two classifications of norm retrieval.

math.FA↗
arXiv2019

Regular two distance sets

This paper makes a deep study of regular two-distance sets. A set of unit vectors $X$ in Euclidean space $\RR^n$ is said to be regular two-distance set if the inner product of any pair of its vectors is either $α$ or $β$, and the number of $α$ (and hence $β$) on each row of the Gram matrix of $X$ are the same. We present various properties of these sets as well as focus on the case where they form tight frames for the underling space. We then give some constructions of regular two-distance sets, in particular, two-distance frames, both tight and non-tight cases. Connections among two-distance sets, equiangular lines and quasi-symmetric designs are also given.

math.FA↗
arXiv2017

Phase retrieval by hyperplanes

We show that a scalable frame does phase retrieval if and only if the hyperplanes of its orthogonal complements do phase retrieval. We then show this result fails in general by giving an example of a frame for $\mathbb R^3$ which does phase retrieval but its induced hyperplanes fail phase retrieval. Moreover, we show that such frames always exist in $\mathbb R^d$ for any dimension $d$. We also give an example of a frame in $\mathbb R^3$ which fails phase retrieval but its perps do phase retrieval. We will also see that a family of hyperplanes doing phase retrieval in $\mathbb R^d$ must contain at least $2d-2$ hyperplanes. Finally, we provide an example of six hyperplanes in $\mathbb R^4$ which do phase retrieval.

math.FA↗
arXiv2017

Norm retrieval and phase retrieval by projections

We make a detailed study of norm retrieval. We give several classification theorems for norm retrieval and give a large number of examples to go with the theory. One consequence is a new result about Parseval frames: If a Parseval frame is divided into two subsets with spans $W_1,W_2$ and $W_1 \cap W_2=\{0\}$, then $W_1 \perp W_2$.

math.FA↗
arXiv2016

Weak phase retrieval and phaseless reconstruction

Phase retrieval and phaseless reconstruction for Hilbert space frames is a very active area of research. Recently, it was shown that these concepts are equivalent. In this paper, we make a detailed study of a weakening of these concepts to weak phase retrieval and weak phaseless reconstruction. We will give several necessary and/or sufficient conditions for frames to have these weak properties. We will prove three surprising results: (1) Weak phaseless reconstruction is equivalent to phaseless reconstruction. I.e. It never was "weak"; (2) Weak phase retrieval is not equivalent to weak phaseless reconstruction; (3) Weak phase retrieval requires at least $2m-2$ vectors in an m-dimensional Hilbert space. We also gives several examples illustrating the relationship between these concepts.

math.FA↗
arXiv2016

Partitioning Hadamard vectors into Hadamard matrices

We will show that in a space of dimension $m$, any family of $2^{m-1}$ distinct Hadamard vectors (where you can choose x or -x but not both) can be partitioned into Hadamard matrices if and only if $m=2^n$ for some n. We will solve this problem with a simple algorithm for assigning the vectors to the Hadamard matrices.

math.CO↗
arXiv2015

Integer Frames

Finite frame theory has become a powerful tool for many applications of mathematics. In this paper we introduce a new area of research in frame theory: Integer frames. These are frames having all integer coordinates with respect to a fixed orthonormal basis for a Hilbert space. Integer frames have potential to mitigate quantization errors and transmission losses as well as speeding up computation times. This paper gives the first systematic study of this important class of finite Hilbert space frames.

math.FA↗
arXiv2015

Phase Retrieval Versus Phaseless Reconstruction

In 2006, Balan/Casazza/Edidin \cite{BCE} introduced the frame theoretic study of phaseless reconstruction. Since then, this has turned into a very active area of research. Over the years, many people have replaced the term {\it phaseless reconstruction} with {\it phase retrieval}. Casazza then asked: {\it Are these really the same?} In this paper, we will show that phase retrieval is equivalent to phaseless reconstruction. We then show, more generally, that phase retrieval by projections is equivalent to phaseless reconstruction by projections. Finally, we study {\it weak phase retrieval} and discover that it is very different from phaseless reconstruction.

math.FA↗
arXiv2011

A Simple Algorithm for Constructing all Real Hessenberg Unitary Matrices

Unitary matrices which are zero below the secondary diagonal (Hessenberg unitary matrices) have many uses in analysis. Given a set of needed conditions on a unitary matrix, this algorithm will give the sparsest unitary matrix. We give an algorithm for constructing all real Hessenberg unitary matrices. The $n\times n$ unitary matrices given by the algorithm have $n-1$ variables which can be chosen to give additional properties needed for a particular application.

math.FA↗
arXiv2011

Algorithmic constructions of unitary matrices and tight frames

We give a number of algorithms for constructing unitary matrices and tight frames with specialized properties. These were produced at the request of researchers at the Frame Research Center (www.framerc.org) to help with their research on fusion frames, the Kadison-Singer Problem and equiangular tight frames.

math.FA↗
arXiv2010

Steiner equiangular tight frames

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.

math.FA↗
arXiv2010

The Bourgain-Tzafriri conjecture and concrete constructions of non-pavable projections

It is known that the Kadison-Singer Problem (KS) and the Paving Conjecture (PC) are equivalent to the Bourgain-Tzafriri Conjecture (BT). Also, it is known that (PC) fails for $2$-paving projections with constant diagonal $1/2$. But the proofs of this fact are existence proofs. We will use variations of the discrete Fourier Transform matrices to construct concrete examples of these projections and projections with constant diagonal $1/r$ which are not $r$-pavable in a very strong sense. In 1989, Bourgain and Tzafriri showed that the class of zero diagonal matrices with small entries (on the order of $\le 1/log^{1+ε}n$, for an $n$-dimensional Hilbert space) are {\em pavable}. It has always been assumed that this result also holds for the BT-Conjecture - although no one formally checked it. We will show that this is not the case. We will show that if the BT-Conjecture is true for vectors with small coefficients (on the order of $\le C/\sqrt{n}$) then the BT-Conjecture is true and hence KS and PC are true.

math.FA↗
arXiv2010

Concrete constructions of non-pavable projections

It is known that the paving conjecture fails for 2-paving projections with constant diagonal 1/2. But the proofs of this fact are existence proofs. We will give concrete examples of these projections and projections with constant diagonal $1/r$ which are not $r$-pavable in a very strong sense.

math.FA↗
arXiv2008

Concrete Constructions of Real Equiangular Line Sets

We give some concrete constructions of real equiangular line sets. The emphasis here is on {\em building blocks} for certain angles which are then used to build up larger equiangular line sets. We concentrate on angles greater than or equal to 1/7.

math.MG↗
arXiv2007

A Decomposition Theorem for frames and the Feichtinger Conjecture

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 $\ell_2$-sequences.

math.FA↗
arXiv2006

The Kadison-Singer Problem in Mathematics and Engineering

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.

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