On non-uniqueness of phase retrieval for continuous and discrete Fourier transforms
Abstract
We consider the problem of 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. We also consider the problem of phase retrieval with background information. We show that this problem is not uniquely solvable if the known and unknown function parts are not sufficiently disjoint.
Related
- Fourier level inequalities on the discrete cube via complex hypercontractivity and Riesz transforms
- Fourier-Laplace transforms in polynomial Ornstein-Uhlenbeck volatility models
- Fourier-Mukai transforms by a family of stable sheaves and preservation of stability
- Birational maps of moduli spaces of stable sheaves on a K3 surface induced by Fourier-Mukai transforms
- Fractional symplectic Fourier transforms and their actions on function and distribution spaces
BibTeX
@misc{novikov2026nonuniqueness,
title = {{On non-uniqueness of phase retrieval for continuous and discrete Fourier transforms}},
author = {Roman Novikov and Tianli Xu},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05604553v2},
note = {CC BY-NC 4.0},
}
RIS
TY - UNPB
TI - On non-uniqueness of phase retrieval for continuous and discrete Fourier transforms
AU - Roman Novikov
AU - Tianli Xu
PY - 2026
DA - 2026/09/23
PB - HAL
UR - https://hal.science/hal-05604553v2
AB - We consider the problem of 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. We also consider the problem of phase retrieval with background information. We show that this problem is not uniquely solvable if the known and unknown function parts are not sufficiently disjoint.
ER -