Continuous Approach to Phase (Norm) Retrieval Frames
Ramin Farshchian, Rajab Ali Kamyabi-Gol, Fahimeh Arabyani-Neyshaburi, Fatemeh Esmaeelzadeh
TL;DR
This work develops a comprehensive theory of continuous frames in Hilbert spaces with a focus on phase retrieval and norm retrieval. It establishes foundational results, including norm-boundedness of Bessel mappings under a positive infimum condition, and introduces continuous near-Riesz bases with invariance under invertible operators. A key contribution is the mu-complement characterization of phase retrieval and the orthogonality criterion for norm retrieval, along with stability analyses under perturbations in finite and infinite measure spaces. The tensor-product analysis demonstrates that phase retrieval for $F=F_1\otimes F_2$ holds exactly when each factor is phase retrievable, and similarly links norm retrieval properties across components and their tensor products, enabling principled reconstruction in product spaces. Collectively, the results advance the understanding of magnitude-only reconstruction in continuous-frame settings with potential impact on signal processing and quantum mechanics.
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.