On phase retrieval for continuous and discrete Fourier transforms

TL;DR

This paper explores non-uniqueness in phase retrieval for Fourier transforms, providing new examples and solutions to longstanding open questions using finite difference operators.

Contribution

It introduces a broad class of non-uniqueness examples in phase retrieval, including with sparsity, and addresses an open problem in the field.

Findings

Constructed numerous non-uniqueness examples with finite difference operators.

Identified large classes of non-trivial Pauli partners with identical intensities.

Provided solutions to an old open question in phase retrieval with background information.

Abstract

We continue studies on phase retrieval for continuous and discrete Fourier transforms in multidimensions. Using finite difference operators, we give a large class of unexpected examples of non-uniqueness for this problem, including examples with the sparsity condition. A prototype of this construction in the continuous case is given in the work Novikov, Xu (JFAA, 2026), using linear differential operators. The construction of the present work also yields a large class of non-trivial Pauli partners, i.e., different functions with the same intensities in both configuration and Fourier domains. Besides, our construction yields examples that solve an old open question in phase retrieval with background information arising in many areas including Fourier holography.