---
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.
Bulletin of the American Mathematical Society. 2021;58(2):225-261. doi:10.1090/bull/1715'
apa: 'Wigderson, A., & Wigderson, Y. (2021). The uncertainty principle: Variations
on a theme. Bulletin of the American Mathematical Society. American Mathematical
Society. https://doi.org/10.1090/bull/1715'
chicago: 'Wigderson, Avi, and Yuval Wigderson. “The Uncertainty Principle: Variations
on a Theme.” Bulletin of the American Mathematical Society. American Mathematical
Society, 2021. https://doi.org/10.1090/bull/1715.'
ieee: 'A. Wigderson and Y. Wigderson, “The uncertainty principle: Variations on
a theme,” Bulletin of the American Mathematical Society, 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.” Bulletin of the American Mathematical Society, vol. 58, no.
2, American Mathematical Society, 2021, pp. 225–61, doi:10.1090/bull/1715.'
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'
...
---
_id: '8510'
abstract:
- lang: eng
text: In this paper, using the ideas of Bessi and Mather, we present a simple mechanical
system exhibiting Arnold diffusion. This system of a particle in a small periodic
potential can be also interpreted as ray propagation in a periodic optical medium
with a near-constant index of refraction. Arnold diffusion in this context manifests
itself as an arbitrary finite change of direction for nearly constant index of
refraction.
article_processing_charge: No
article_type: original
author:
- first_name: Vadim
full_name: Kaloshin, Vadim
id: FE553552-CDE8-11E9-B324-C0EBE5697425
last_name: Kaloshin
orcid: 0000-0002-6051-2628
- first_name: Mark
full_name: Levi, Mark
last_name: Levi
citation:
ama: Kaloshin V, Levi M. An example of Arnold diffusion for near-integrable Hamiltonians.
Bulletin of the American Mathematical Society. 2008;45(3):409-427. doi:10.1090/s0273-0979-08-01211-1
apa: Kaloshin, V., & Levi, M. (2008). An example of Arnold diffusion for near-integrable
Hamiltonians. Bulletin of the American Mathematical Society. American Mathematical
Society. https://doi.org/10.1090/s0273-0979-08-01211-1
chicago: Kaloshin, Vadim, and Mark Levi. “An Example of Arnold Diffusion for Near-Integrable
Hamiltonians.” Bulletin of the American Mathematical Society. American
Mathematical Society, 2008. https://doi.org/10.1090/s0273-0979-08-01211-1.
ieee: V. Kaloshin and M. Levi, “An example of Arnold diffusion for near-integrable
Hamiltonians,” Bulletin of the American Mathematical Society, vol. 45,
no. 3. American Mathematical Society, pp. 409–427, 2008.
ista: Kaloshin V, Levi M. 2008. An example of Arnold diffusion for near-integrable
Hamiltonians. Bulletin of the American Mathematical Society. 45(3), 409–427.
mla: Kaloshin, Vadim, and Mark Levi. “An Example of Arnold Diffusion for Near-Integrable
Hamiltonians.” Bulletin of the American Mathematical Society, vol. 45,
no. 3, American Mathematical Society, 2008, pp. 409–27, doi:10.1090/s0273-0979-08-01211-1.
short: V. Kaloshin, M. Levi, Bulletin of the American Mathematical Society 45 (2008)
409–427.
date_created: 2020-09-18T10:48:20Z
date_published: 2008-07-01T00:00:00Z
date_updated: 2021-01-12T08:19:47Z
day: '01'
doi: 10.1090/s0273-0979-08-01211-1
extern: '1'
intvolume: ' 45'
issue: '3'
keyword:
- Applied Mathematics
- General Mathematics
language:
- iso: eng
month: '07'
oa_version: None
page: 409-427
publication: Bulletin of the American Mathematical Society
publication_identifier:
issn:
- 0273-0979
publication_status: published
publisher: American Mathematical Society
quality_controlled: '1'
status: public
title: An example of Arnold diffusion for near-integrable Hamiltonians
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 45
year: '2008'
...