arXiv subjects

F. Akrami

Publications and source records attributed to F. Akrami.

3 recordsLinked to original sources

arXiv2023

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.

math.FA↗

arXiv2021

A note on (weak) phase and norm retrievable Real Hilbert space frames and projections

\begin{abstract} In this manuscript, we answer a list of longstanding open problems on weak phase retrieval including: (1) A complete classification of the vectors $\{x_i\}_{i=1}^2$ in $\RR^3$ that do weak phase retrieval; (2) We show that frames doing weak phase retrieval in $\RR^n$ must span $\RR^n$; (3) We give an example of a set of vectors doing phase retrieval but their orthogonal complement hyperplanes fail weak phase retrieval; (4) We give a classification of weak phase retrievable frames - which makes clear the difference between phase retrieval and weak phase retrieval; (5) We classify when weak phase retrievable frames also do norm retrieval. We then introduce the notion of weak phase retrieval by projections and develop their basic properties. We then look at phase (norm) retrieval by projections. We end with some open problems. We provide numerous examples to show that our results are best possible. \end{abstract}

math.FA↗

arXiv2021

A note on phase (norm) retrievable Real Hilbert space (fusion) frames

In this manuscript, we present several new results in finite and countable dimensional real Hilbert space phase retrieval and norm retrieval by vectors and projections. We make a detailed study of when hyperplanes do norm retrieval. Also, we show that the families of norm retrievable frames $\{f_{i}\}_{i=1}^{m}$ in $\mathbb{R}^n$ are not dense in the family of $m\leq (2n-2)$-element sets of vectors in $\mathbb{R}^n$ for every finite $n$ and the families of vectors which do norm retrieval in $\ell^2$ are not dense in the infinite families of vectors in $\ell^2$. We also show that if a Riesz basis does norm retrieval in $\ell^2$, then it is an orthogonal sequence. We provide numerous examples to show that our results are best possible.

math.FA↗