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

TL;DR

This paper investigates phase and norm retrieval in finite and infinite-dimensional real Hilbert spaces, establishing conditions for hyperplanes, frames, and Riesz bases to perform norm retrieval, with results on their density and structure.

Contribution

It provides new results on when hyperplanes and frames in real Hilbert spaces can perform norm retrieval, including density properties and the characterization of Riesz bases that do so.

Findings

Hyperplanes do norm retrieval under certain conditions.

Families of norm retrievable frames are not dense in finite dimensions.

Riesz bases that do norm retrieval in  are orthogonal sequences.

Abstract

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.