On phase and norm retrieval by subspaces

TL;DR

This paper explores the conditions under which subspaces and hyperplanes enable phase and norm retrieval in real vector spaces, providing new classifications and extending known results about the density of such frames.

Contribution

It introduces new properties of subspaces for phase and norm retrieval, classifies two subspaces that perform norm retrieval, and extends density results to subspaces.

Findings

Linearly independent vectors with hyperplanes must form an orthonormal basis for norm retrieval.

Complete classification of two subspaces enabling norm retrieval.

Extended non-density results of norm-retrievable frames to subspaces.

Abstract

This paper studies phase and norm retrieval by subspaces. We first investigate norm retrieval by hyperplanes. We show that if $N$ hyperplanes $\{\varphi_i^\perp\}_{i=1}^N\subset \mathbb{R}^N$ allow norm retrieval and the vectors $\{\varphi_i\}_{i=1}^N$ are linearly independent, then these vectors must be an orthonormal basis for $\mathbb{R}^N$. We then present several new properties of subspaces that allow phase and norm retrieval. In particular, we provide a complete classification of two proper subspaces that perform norm retrieval. It is known that the collection of norm-retrievable frames $\{\varphi_i\}_{i=1}^M$ in $\mathbb{R}^N$ is not dense in the set of all $M$-element frames when $M < 2N-1$. We extend this result to subspaces. Several alternative proofs of fundamental results in phase and norm retrieval are also provided.