Continuous Approach to Phase (Norm) Retrieval Frames

TL;DR

This paper explores the properties of continuous frames in Hilbert spaces, focusing on phase and norm retrieval, introducing continuous near-Riesz bases, and analyzing stability and tensor product frames.

Contribution

It introduces continuous near-Riesz bases, proves their invariance under invertible operators, and establishes equivalences for phase and norm retrieval in tensor product frames.

Findings

Continuous near-Riesz bases are invariant under invertible operators.

Conditions for phase and norm retrieval in continuous frames are characterized.

Phase and norm retrieval properties are equivalent between components and their tensor products.

Abstract

This paper investigates the properties of continuous frames, with a particular focus on phase retrieval and norm retrieval in the context of Hilbert spaces. We introduce the concept of continuous near-Riesz bases and prove their invariance under invertible operators. Some equivalent conditions for phase and norm retrieval property of continuous frames are presented. We study the stability of phase retrieval under perturbations. Furthermore, tensor product frames for separable Hilbert spaces are studied, and we establish the equivalence of phase retrieval and norm retrieval properties between components and their tensor products.