Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 23 min ago
PreprintarXiv licencearXiv

Fourier sampling on a lattice

  • Oleg Asipchuk
  • Sinai Robins
Open at sourcePDFDOICite

Abstract

We characterize the ambiguity in reconstructing bounded measurable sets from their Fourier transforms sampled on a lattice. For a full-rank lattice ℒ⊂Rd, we prove that two bounded measurable sets have identical indicator Fourier transforms on ℒ* if and only if they are equidecomposable, modulo null sets, into finitely many measurable pieces by translations in ℒ. If the sets are connected finite unions of convex bodies and no two distinct points of either set are congruent modulo ℒ, we show that this equidecomposition reduces to a single translation by a vector of ℒ. In particular, two such sets that are symmetric about the origin must coincide. We also construct counterexamples in dimensions two and three to a conjecture of the second author, which had a similar hypothesis, but lacked the anti-aliasing hypothesis. Together with the first author's counterexamples in dimensions at least four, these show that the conjecture fails in every dimension d≥ 2. The positive results here are an outgrowth of refinements of this conjecture.

Related

  1. Smallest Singular Value Estimates for Nonuniform Fourier Matrices via Periodic Nonuniform Sampling
    Liang Chen, Rongrong Lin, Haizhang Zhang
    arXivPreprintCC BY 4.0Full textNumerical analysis & scientific computingarXiv
  2. Alias Geometry and Density Compensation in Perturbed Fourier Sampling
    Erli Wind-Andersen
    arXivPreprintCC BY 4.0Full textNumerical analysis & scientific computingarXiv
  3. Sector-Resolved Flow Sampling for Topologically Frozen Lattice Gauge Theories
    Ankur Singha, Jacob Kauffmann, Karl Jansen +3
    arXivPreprintCC BY 4.0Full textMathematical physicsHigh-energy & particle physicsarXiv
  4. Scalable generative sampling and multilevel estimation for lattice field theories near criticality
    A. Singha, J. Kauffmann, E. Cellini +2
    Physical review. D/Physical review. D.Peer-reviewedCC BY10 citedStatisticsOpenAlex
  5. Multilevel Plaquette-Space Sampling for Lattice Gauge Theories with Local Constraint Solves
    Ankur Singha, Jacob Finkenrath, Karl Jansen +2
    arXivPreprintCC BY 4.0Full textMathematical physicsHigh-energy & particle physicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{asipchuk2026fourier,
  title = {{Fourier sampling on a lattice}},
  author = {Oleg Asipchuk and Sinai Robins},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.29217},
  eprint = {2608.29217},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.29217},
}
RIS
TY  - UNPB
TI  - Fourier sampling on a lattice
AU  - Oleg Asipchuk
AU  - Sinai Robins
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2608.29217
UR  - https://arxiv.org/abs/2608.29217
AB  - We characterize the ambiguity in reconstructing bounded measurable sets from their Fourier transforms sampled on a lattice. For a full-rank lattice ℒ⊂R^d, we prove that two bounded measurable sets have identical indicator Fourier transforms on ℒ^* if and only if they are equidecomposable, modulo null sets, into finitely many measurable pieces by translations in ℒ. If the sets are connected finite unions of convex bodies and no two distinct points of either set are congruent modulo ℒ, we show that this equidecomposition reduces to a single translation by a vector of ℒ. In particular, two such sets that are symmetric about the origin must coincide. We also construct counterexamples in dimensions two and three to a conjecture of the second author, which had a similar hypothesis, but lacked the anti-aliasing hypothesis. Together with the first author's counterexamples in dimensions at least four, these show that the conjecture fails in every dimension d≥ 2. The positive results here are an outgrowth of refinements of this conjecture.
ER  -