---
OA_place: repository
OA_type: green
_id: '22175'
abstract:
- lang: eng
  text: We show how a number of well-known uncertainty principles for the Fourier
    transform, such as the Heisenberg uncertainty principle, the Donoho–Stark uncertainty
    principle, and Meshulam’s nonabelian uncertainty principle, have little to do
    with the structure of the Fourier transform itself. Rather, all of these results
    follow from very weak properties of the Fourier transform (shared by numerous
    linear operators), namely that it is bounded as an operator  L1 → L∞, and that
    it is unitary. Using a single, simple proof template, and only these (or weaker)
    properties, we obtain some new proofs and many generalizations of these basic
    uncertainty principles, to new operators and to new settings, in a completely
    unified way. Together with our general overview, this paper can also serve as
    a survey of the many facets of the phenomena known as uncertainty principles.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Avi
  full_name: Wigderson, Avi
  last_name: Wigderson
- first_name: Yuval
  full_name: Wigderson, Yuval
  id: 2d0023a0-1567-11f0-833d-d5c1e476d4b5
  last_name: Wigderson
citation:
  ama: 'Wigderson A, Wigderson Y. The uncertainty principle: Variations on a theme.
    <i>Bulletin of the American Mathematical Society</i>. 2021;58(2):225-261. doi:<a
    href="https://doi.org/10.1090/bull/1715">10.1090/bull/1715</a>'
  apa: 'Wigderson, A., &#38; Wigderson, Y. (2021). The uncertainty principle: Variations
    on a theme. <i>Bulletin of the American Mathematical Society</i>. American Mathematical
    Society. <a href="https://doi.org/10.1090/bull/1715">https://doi.org/10.1090/bull/1715</a>'
  chicago: 'Wigderson, Avi, and Yuval Wigderson. “The Uncertainty Principle: Variations
    on a Theme.” <i>Bulletin of the American Mathematical Society</i>. American Mathematical
    Society, 2021. <a href="https://doi.org/10.1090/bull/1715">https://doi.org/10.1090/bull/1715</a>.'
  ieee: 'A. Wigderson and Y. Wigderson, “The uncertainty principle: Variations on
    a theme,” <i>Bulletin of the American Mathematical Society</i>, vol. 58, no. 2.
    American Mathematical Society, pp. 225–261, 2021.'
  ista: 'Wigderson A, Wigderson Y. 2021. The uncertainty principle: Variations on
    a theme. Bulletin of the American Mathematical Society. 58(2), 225–261.'
  mla: 'Wigderson, Avi, and Yuval Wigderson. “The Uncertainty Principle: Variations
    on a Theme.” <i>Bulletin of the American Mathematical Society</i>, vol. 58, no.
    2, American Mathematical Society, 2021, pp. 225–61, doi:<a href="https://doi.org/10.1090/bull/1715">10.1090/bull/1715</a>.'
  short: A. Wigderson, Y. Wigderson, Bulletin of the American Mathematical Society
    58 (2021) 225–261.
date_created: 2026-06-29T10:57:49Z
date_published: 2021-01-04T00:00:00Z
date_updated: 2026-07-14T09:02:39Z
day: '04'
doi: 10.1090/bull/1715
extern: '1'
external_id:
  arxiv:
  - '2006.11206'
intvolume: '        58'
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2006.11206
mathsc:
- 81S07
- 43A25
- 20C15
- 94A12
month: '01'
oa: 1
oa_version: Preprint
page: 225-261
publication: Bulletin of the American Mathematical Society
publication_identifier:
  eissn:
  - 1088-9485
  issn:
  - 0273-0979
publication_status: published
publisher: American Mathematical Society
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'The uncertainty principle: Variations on a theme'
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 58
year: '2021'
...