# Norm Retrieval and Phase Retrieval by Projections

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**frame**if there are constants $0<A\le B<\infty $ so that for all $x\in {\mathbb{H}}^{N}$,

- The constants A and B are called the
**lower and upper frame bounds**of the frame, respectively. - If $A=B$, the frame is called an
**A-tight frame**(or a tight frame). In particular, if $A=B=1$, the frame is called a**Parseval frame**. - Φ is an
**equal norm frame**if $\parallel {\varphi}_{i}\parallel =\parallel {\varphi}_{j}\parallel $ for all $i,j$ and is called a**unit norm frame**if $\parallel {\varphi}_{i}\parallel =1$ for all $i=1,2,\xb7\xb7\xb7,n$. - If, only the right hand side inequality holds in (1), the frame is called a
**B-Bessel family with Bessel bound B**.

**analysis operator**associated with Φ is defined as the operator $T:{\mathbb{H}}^{N}\to {\ell}_{2}^{M}$ to be

**synthesis operator**of the frame Φ. It can be shown that ${T}^{*}\left({e}_{i}\right)={\varphi}_{i}.$

**frame operator**for the frame Φ is defined as $S:{T}^{*}T:{\mathbb{H}}^{N}\to {\mathbb{H}}^{N}.$ That is,

**Definition**

**2.**

**phase retrieval**if for all $x,y\in {\mathbb{H}}^{N}$ satisfying $\parallel {P}_{i}x\parallel =\parallel {P}_{i}y\parallel $ for all $i=1,2,\xb7\xb7\xb7,M$ then $x=cy$ for some scalar c such that $\left|c\right|=1$

**Definition**

**3.**

**phase retrieval**with respect to an orthonormal basis ${\left\{{e}_{i}\right\}}_{i=1}^{N}$ if there is a $\left|c\right|=1$ such that ${x}_{i}=c{y}_{i}$, for all $i=1,2,\xb7\xb7\xb7,N$, where ${x}_{i}=\langle x,{e}_{i}\rangle $.

**Definition**

**4.**

**complement property**if for all subsets $I\subset \{1,2,\xb7\xb7\xb7,M\}$, either ${\left\{{\varphi}_{i}\right\}}_{i\in I}$ or ${\left\{{\varphi}_{i}\right\}}_{i\in {I}^{c}}$ spans the whole space ${\mathbb{H}}^{N}$.

**Definition**

**5.**

**spark**of Φ is defined as the cardinality of the smallest linearly dependent subset of Φ. When spark$\left(\mathsf{\Phi}\right)=N+1$, every subset of size N is linearly independent, and in that case, Φ is said to be

**full spark**.

**Theorem 1**(Naimark’s Theorem).

## 3. Beginnings of Norm Retrieval

**Definition**

**6.**

**norm retrieval**if for all $x,y\in {\mathbb{H}}^{N}$ satisfying $\parallel {P}_{i}x\parallel =\parallel {P}_{i}y\parallel $ for all $i=1,2,\xb7\xb7\xb7,M$ then $\parallel x\parallel =\parallel y\parallel $.

**Remark**

**1.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Proposition**

**1.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

## 4. Phase Retrieval and Norm Retrieval

**Theorem**

**6**

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Example**

**1.**

**Lemma**

**2.**

**Proof.**

**Example**

**2.**

**Corollary**

**3.**

**Proof.**

**Proposition**

**5**

**Proof.**

**Theorem**

**7.**

**Example**

**3.**

**Proposition**

**6.**

**Proof.**

**Example**

**4.**

## 5. Classification of Norm Retrieval

**Theorem**

**8.**

**Theorem**

**9.**

- ${\left\{{P}_{i}\right\}}_{i=1}^{M}$ does norm retrieval,
- Given any orthonormal bases ${\left\{{\varphi}_{ij}\right\}}_{j=1}^{{I}_{i}}$ of ${W}_{i}$ and any subcollection $S\subseteq \{(i,j):1\le i\le M,1\le j\le {I}_{i}\}$ then$$span\phantom{\rule{4pt}{0ex}}{\left\{{\varphi}_{ij}\right\}}_{(i,j)\in S}^{\perp}\perp span\phantom{\rule{4pt}{0ex}}{\left\{{\varphi}_{ij}\right\}}_{(i,j)\in {S}^{c}}^{\perp},$$
- For any orthonormal basis ${\left\{{\varphi}_{ij}\right\}}_{j=1}^{{I}_{i}}$ of ${W}_{i}$, then the collection of vectors ${\left\{{\varphi}_{ij}\right\}}_{(i,j)}$ do norm retrieval.

**Proof.**

**Corollary**

**4.**

**Proof.**

**Proposition**

**7.**

**Proof.**

**Corollary**

**5.**

**Proof.**

**Corollary**

**6.**

- Φ does norm retrieval.
- For $i\in \left[M\right]$ if ${W}_{1}=span{\left\{{\varphi}_{i}\right\}}_{i\in I}$ and ${W}_{2}=span{\left\{{\varphi}_{i}\right\}}_{i\in {I}^{c}}$ then, ${{W}_{1}}^{\perp}\subseteq {W}_{2}$.

**Proof.**

**Corollary**

**7.**

**Proof.**

**Corollary**

**8.**

**Proof.**

**Corollary**

**9.**

**Corollary**

**10.**

**Proof.**

**Proposition**

**8.**

**Proof.**

**Theorem**

**10.**

- Φ does norm retrieval.
- By Proposition 8 if $T:{\mathbb{R}}^{N}\to {\ell}_{2}^{2M-1}$ is an isometry and ${\left\{{e}_{i}\right\}}_{i=1}^{2M-1}$ is the unit vector basis for ${\ell}_{2}^{2M-1}$ then for every $\varphi ,\psi \in {\mathbb{R}}^{N}$ with $|\langle \varphi ,{\varphi}_{i}\rangle |=|\langle \psi ,{\varphi}_{i}\rangle |$ for $i\in \left[M\right]$, we have$$\parallel \sum _{i=M+1}^{2M-1}\langle T\varphi ,{e}_{i}\rangle {e}_{i}{\parallel}^{2}=\sum _{i=M+1}^{2M-1}|\langle T\varphi ,{e}_{i}\rangle {|}^{2}=\parallel \sum _{i=M+1}^{2M-1}\langle T\psi ,{\varphi}_{i}\rangle {e}_{i}{\parallel}^{2}=\sum _{i=M+1}^{2M-1}{\left|\langle T\psi ,{\varphi}_{i}\rangle \right|}^{2}.$$

**Proof.**

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Casazza, P.G.; Ghoreishi, D.; Jose, S.; Tremain, J.C.
Norm Retrieval and Phase Retrieval by Projections. *Axioms* **2017**, *6*, 6.
https://doi.org/10.3390/axioms6010006

**AMA Style**

Casazza PG, Ghoreishi D, Jose S, Tremain JC.
Norm Retrieval and Phase Retrieval by Projections. *Axioms*. 2017; 6(1):6.
https://doi.org/10.3390/axioms6010006

**Chicago/Turabian Style**

Casazza, Peter G., Dorsa Ghoreishi, Shani Jose, and Janet C. Tremain.
2017. "Norm Retrieval and Phase Retrieval by Projections" *Axioms* 6, no. 1: 6.
https://doi.org/10.3390/axioms6010006