The uncertainty principle: Variations on a theme
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- by Avi Wigderson and Yuval Wigderson HTML | PDF
- Bull. Amer. Math. Soc. 58 (2021), 225-261 Request permission
Abstract:
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 $L^1 \to L^\infty$, 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.References
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Additional Information
- Avi Wigderson
- Affiliation: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540
- MR Author ID: 182810
- Email: avi@ias.edu
- Yuval Wigderson
- Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
- MR Author ID: 1156799
- ORCID: 0000-0001-5909-9250
- Email: yuvalwig@stanford.edu
- Received by editor(s): August 26, 2020
- Published electronically: January 4, 2021
- Additional Notes: The first author’s research was supported by NSF grant CCF-1900460
The second author’s research was supported by NSF GRFP Grant DGE-1656518. - © Copyright 2021 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 58 (2021), 225-261
- MSC (2020): Primary 81S07, 43A25; Secondary 20C15, 94A12
- DOI: https://doi.org/10.1090/bull/1715
- MathSciNet review: 4229152
Dedicated: Dedicated to the memory and work of Jean Bourgain