Illinois Journal of Mathematics

Uncertainty principles for compact groups

Gorjan Alagic and Alexander Russell

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Abstract

We establish an uncertainty principle over arbitrary compact groups, generalizing several previous results. Specifically, we show that if $\mathrm{P}$ and $\mathrm{R}$ are operators on $L^2(G)$ such that $\mathrm{P}$ commutes with projection onto every measurable subset of $G$ and $\mathrm{R}$ commutes with left-multiplication by elements of $G$, then $\|\operatorname{PR}\| \leq\|\mathrm{P} \cdot\chi_G \|_2 \|\mathrm {R}\|_2$, where $\chi_G : g \mapsto1$ is the characteristic function of $G$. As a consequence, we show that every nonzero function $f$ in $L^2(G)$ satisfies $\mu(\operatorname{\mathbf{supp}} f)\cdot \sum_{\rho\in\hat G} d_\rho\operatorname{\mathbf{rank}} \hat f(\rho) \geq1$.

Article information

Source
Illinois J. Math., Volume 52, Number 4 (2008), 1315-1324.

Dates
First available in Project Euclid: 18 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258554365

Digital Object Identifier
doi:10.1215/ijm/1258554365

Mathematical Reviews number (MathSciNet)
MR2595770

Zentralblatt MATH identifier
1184.43004

Subjects
Primary: 43A30: Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. 43A65: Representations of groups, semigroups, etc. [See also 22A10, 22A20, 22Dxx, 22E45] 43A77: Analysis on general compact groups

Citation

Alagic, Gorjan; Russell, Alexander. Uncertainty principles for compact groups. Illinois J. Math. 52 (2008), no. 4, 1315--1324. doi:10.1215/ijm/1258554365. https://projecteuclid.org/euclid.ijm/1258554365


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References

  • G. Alagic and A. Russell, Uncertainty principles over finite groups, available at ArXiv: math.RT/0608702, 2006.
  • J. B. Conway, A course in operator theory, Amer. Math. Soc., Providence, RI, 2000.
  • D. L. Donoho and P. B. Stark, Uncertainty principles and signal recovery, SIAM J. Appl. Math. 49 (1989), 906–931.
  • S. Echterhoff, E. Kaniuth and A. Kumar, A qualitative uncertainty principle for certain locally compact groups, Forum Math. 3 (1991), 355–369.
  • J. M. Fell and R. S. Doran, Representations of *-algebras, locally compact groups, and Banach *-algebraic bundles, vol. 1, Academic Press, San Diego, CA, 1988.
  • G. Folland and A. Sitaram, The uncertainty principle: A mathematical survey, J. Fourier Anal. Appl. 3 (1997), 207–228.
  • J. A. Hogan, A qualitative uncertainty principle for unimodular groups of type I, Trans. Amer. Math. Soc. 340 (1993), 587–594.
  • E. Kaniuth, Minimizing functions for an uncertainty principle on locally compact groups of bounded representation dimension, Proc. Amer. Math. Soc. 135 (2007), 217–227.
  • C. Kok Seng and N. Wee Seng, A simple proof of the uncertainty principle for compact groups, Expos. Math. 23 (2005), 147–150.
  • G. Kutyniok, A weak qualitative uncertainty principle for compact groups, Illinois J. Math. 47 (2003), 709–724.
  • R. Meshulam, An uncertainty inequality for groups of order pq, European J. Combin. 13 (1992), 401–407.
  • K. T. Smith, The uncertainty principle on groups, SIAM J. Appl. Math. 50 (1990), 876–882.
  • A. Terras, Fourier analysis on finite groups and applications, London Mathematical Society Student Texts, vol. 43, Cambridge University Press, Cambridge, UK, 1999.