Peter G. Casazza
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Research Interests
Peter G. Casazza has a total of 62 co-authors. Here are the top co-authors by number of shared publications:
- Radu Balan
Siemens Corporate Research, 755 College Road East, Princeton, New Jersey 08540
7 shared publications - Gitta Kutyniok
Ludwig-Maximilians-Universität München
6 shared publications - Christopher Heil
School of Mathematics, Georgia Institute of Technology, Atlanta, USA
4 shared publications - Dan Edidin
Department of Mathematics , University of Missouri-Columbia , Columbia, Missouri, USA
4 shared publications - Janet C. Tremain
University of Missouri
4 shared publications - Zeph Landau
Department of Mathematics R8133, The City College of New York, Convent Ave at 138th Street, New York, NY 10031, USA
4 shared publications
Latest Publications
- Open Access
Real matrices whose columns have equal modulus coordinates
We study $m \times n$ matrices whose columns are of the form \[\{(a_{1j},\ldots, a_{nj}): \quad a_{1j} = λ_j,\ a_{ij} = \pmλ_j\ , \ λ_j >0 ,\ j=1,2,\ldots,n\}.\] We explicitly construct for all $a = (a_1,\ldots, a_{\frac{m(m- 1)}{2}}) \in \mathbb{R}^{\frac{m(m-1)}{2}}$ a matrix of the above form whose rows have pairwise dot product equal to $a$. Using Hardamard matrices constructed by Sylvester we classify all matrices of the above form whose rows have pairwise dot product equal to $a$. We also use our results to reformulate the Hadamard conjecture.Open AccessReal matrices whose columns have equal modulus coordinates
We study $m \times n$ matrices whose columns are of the form \[\{(a_{1j},\ldots, a_{nj}): \quad a_{1j} = λ_j,\ a_{ij} = \pmλ_j\ , \ λ_j >0 ,\ j=1,2,\ldots,n\}.\] We explicitly construct for all $a = (a_1,\ldots, a_{\frac{m(m- 1)}{2}}) \in \mathbb{R}^{\frac{m(m-1)}{2}}$ a matrix of the above form whose rows have pairwise dot product equal to $a$. Using Hardamard matrices constructed by Sylvester we classify all matrices of the above form whose rows have pairwise dot product equal to $a$. We also use our results to reformulate the Hadamard conjecture. - Open Access
A Survey of Fusion Frames in Hilbert Spaces
Fusion frames are a very active area of research today because of their myriad of applications in pure mathematics, applied mathematics, engineering, medicine, signal and image processing and much more. They provide a great flexibility for designing sets of vectors for applications and are therefore prominent in all these areas, including e.g. mitigating the effects of noise in a signal or giving robustness to erasures. In this chapter, we present the fundamentals of fusion frame theory with an emphasis on their delicate relation to frame theory. The goal here is to provide researchers and students with an easy entry into this topic. Proofs for fusion frames will be self-contained and differences between frames and fusion frames are analyzed. In particular, we focus on the subtleties of fusion frame duality. We also provide a reproducible research implementation.
Open AccessA Survey of Fusion Frames in Hilbert Spaces
Fusion frames are a very active area of research today because of their myriad of applications in pure mathematics, applied mathematics, engineering, medicine, signal and image processing and much more. They provide a great flexibility for designing sets of vectors for applications and are therefore prominent in all these areas, including e.g. mitigating the effects of noise in a signal or giving robustness to erasures. In this chapter, we present the fundamentals of fusion frame theory with an emphasis on their delicate relation to frame theory. The goal here is to provide researchers and students with an easy entry into this topic. Proofs for fusion frames will be self-contained and differences between frames and fusion frames are analyzed. In particular, we focus on the subtleties of fusion frame duality. We also provide a reproducible research implementation.
- 17 February 2023
- Published by Springer Nature in Journal of Fourier Analysis and Applications
- Vol. 29 (2) , 1-16
The Core of a Grassmannian Frame
- 17 February 2023
- Published by Springer Nature in Journal of Fourier Analysis and Applications
- Vol. 29 (2) , 1-16
The Core of a Grassmannian Frame
- Open Access
Classifying weak phase retrieval
We will give several surprising equivalences and consequences of weak phase retrieval. These results give a complete understanding of the difference between weak phase retrieval and phase retrieval. We also answer two longstanding open problems on weak phase retrieval: (1) We show that the families of weak phase retrievable frames $\{x_{i}\}_{i=1}^{m}$ in $\mathbb{R}^n$ are not dense in the family of $m$-element sets of vectors in $\mathbb{R}^n$ for all $m\ge 2n-2$; (2) We show that any frame $\{x_i\}_{i=1}^{2n-2}$ containing one or more canonical basis vectors in $\mathbb{R}^n$ cannot do weak phase retrieval. We provide numerous examples to show that the obtained results are best possible.Open AccessClassifying weak phase retrieval
We will give several surprising equivalences and consequences of weak phase retrieval. These results give a complete understanding of the difference between weak phase retrieval and phase retrieval. We also answer two longstanding open problems on weak phase retrieval: (1) We show that the families of weak phase retrievable frames $\{x_{i}\}_{i=1}^{m}$ in $\mathbb{R}^n$ are not dense in the family of $m$-element sets of vectors in $\mathbb{R}^n$ for all $m\ge 2n-2$; (2) We show that any frame $\{x_i\}_{i=1}^{2n-2}$ containing one or more canonical basis vectors in $\mathbb{R}^n$ cannot do weak phase retrieval. We provide numerous examples to show that the obtained results are best possible. - 5 December 2022
- Published by Springer Nature in Annals of Functional Analysis
- Vol. 14 (1) , 1-18
Fundamental results on weak phase retrieval
- 5 December 2022
- Published by Springer Nature in Annals of Functional Analysis
- Vol. 14 (1) , 1-18
Fundamental results on weak phase retrieval
- Open Access
The core of a Grassmannian frame
Let $X=\{x_i\}_{i=1}^m$ be a set of unit vectors in $\RR^n$. The coherence of $X$ is $\coh(X):=\max_{i\not=j}|\langle x_i, x_j\rangle|$. A vector $x\in X$ is said to be isolable if there are no unit vectors $x'$ arbitrarily close to $x$ such that $|\langle x', y\rangle|<\coh(X)$ for all other vectors $y$ in $X$. We define the {\bf core} of a Grassmannian frame $X=\{x_i\}_{i=1}^m$ in $\RR^n$ at angle $\alpha$ as a maximal subset of $X$ which has coherence $\alpha$ and has no isolable vectors. In other words, if $Y$ is a subset of $X$, $\coh(Y)=\alpha$, and $Y$ has no isolable vectors, then $Y$ is a subset of the core. We will show that every Grassmannian frame of $m>n$ vectors for $\RR^n$ has the property that each vector in the core makes angle $\alpha$ with a spanning family from the core. Consequently, the core consists of $\ge n+1$ vectors. We then develop other properties of Grassmannian frames and of the core.Open AccessThe core of a Grassmannian frame
Let $X=\{x_i\}_{i=1}^m$ be a set of unit vectors in $\RR^n$. The coherence of $X$ is $\coh(X):=\max_{i\not=j}|\langle x_i, x_j\rangle|$. A vector $x\in X$ is said to be isolable if there are no unit vectors $x'$ arbitrarily close to $x$ such that $|\langle x', y\rangle|<\coh(X)$ for all other vectors $y$ in $X$. We define the {\bf core} of a Grassmannian frame $X=\{x_i\}_{i=1}^m$ in $\RR^n$ at angle $\alpha$ as a maximal subset of $X$ which has coherence $\alpha$ and has no isolable vectors. In other words, if $Y$ is a subset of $X$, $\coh(Y)=\alpha$, and $Y$ has no isolable vectors, then $Y$ is a subset of the core. We will show that every Grassmannian frame of $m>n$ vectors for $\RR^n$ has the property that each vector in the core makes angle $\alpha$ with a spanning family from the core. Consequently, the core consists of $\ge n+1$ vectors. We then develop other properties of Grassmannian frames and of the core. - 1 September 2016
- Published by Elsevier in Journal of Mathematical Analysis and Applications
- Vol. 441 (1) , 475-498
Riesz outer product Hilbert space frames: Quantitative bounds, topological properties, and full geometric characterization
- 1 September 2016
- Published by Elsevier in Journal of Mathematical Analysis and Applications
- Vol. 441 (1) , 475-498
Riesz outer product Hilbert space frames: Quantitative bounds, topological properties, and full geometric characterization
- Open Access
The Unconditional Constants for Hilbert Space Frame Expansions
The most fundamental notion in frame theory is the frame expansion of a vector. Although it is well known that these expansions are unconditionally convergent series, no characterizations of the unconditional constant were known. This has made it impossible to get accurate quantitative estimates for problems which require using subsequences of a frame. We will prove some new results in frame theory by showing that the unconditional constants of the frame expansion of a vector in a Hilbert space are bounded by $\sqrt{\frac{B}{A}}$, where $A,B$ are the frame bounds of the frame. Tight frames thus have unconditional constant one, which we then generalize by showing that Bessel sequences have frame expansions with unconditional constant one if and only if the sequence is an orthogonal sum of tight frames. We give further results concerning frame expansions, in which we examine when $\sqrt{\frac BA}$ is actually attained or not. We end by discussing the connections of this work to {\it frame multipliers}. These results hold in both real and complex Hilbert spaces.Open AccessThe Unconditional Constants for Hilbert Space Frame Expansions
The most fundamental notion in frame theory is the frame expansion of a vector. Although it is well known that these expansions are unconditionally convergent series, no characterizations of the unconditional constant were known. This has made it impossible to get accurate quantitative estimates for problems which require using subsequences of a frame. We will prove some new results in frame theory by showing that the unconditional constants of the frame expansion of a vector in a Hilbert space are bounded by $\sqrt{\frac{B}{A}}$, where $A,B$ are the frame bounds of the frame. Tight frames thus have unconditional constant one, which we then generalize by showing that Bessel sequences have frame expansions with unconditional constant one if and only if the sequence is an orthogonal sum of tight frames. We give further results concerning frame expansions, in which we examine when $\sqrt{\frac BA}$ is actually attained or not. We end by discussing the connections of this work to {\it frame multipliers}. These results hold in both real and complex Hilbert spaces. - 1 May 2015
- Published by Institute of Electrical and Electronics Engineers (IEEE)
- p. 362-366
Unconditional convergence constants of Hilbert space frame expansions
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 to a classification of such frames by showing that for Bessel sequences whose frame operator can be diagonalized, the frame expansions have unconditional constant one if and only if the Bessel sequence is an orthogonal sum of tight frames. We then prove similar results for cross frame expansions but here the results are no longer a classification. We also give examples to show that our results are best possible. These results should have been done 20 years ago but somehow we overlooked this topic.
- 1 May 2015
- Published by Institute of Electrical and Electronics Engineers (IEEE)
- p. 362-366
Unconditional convergence constants of Hilbert space frame expansions
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 to a classification of such frames by showing that for Bessel sequences whose frame operator can be diagonalized, the frame expansions have unconditional constant one if and only if the Bessel sequence is an orthogonal sum of tight frames. We then prove similar results for cross frame expansions but here the results are no longer a classification. We also give examples to show that our results are best possible. These results should have been done 20 years ago but somehow we overlooked this topic.
- 1 January 2015
- Published by Cambridge University Press (CUP)
- Vol. 83, 253-254
Foreword to Why I Became a Mathematician
- 1 January 2015
- Published by Cambridge University Press (CUP)
- Vol. 83, 253-254
Foreword to Why I Became a Mathematician