Open Access
Winter 2008 Uncertainty principles for compact groups
Gorjan Alagic, Alexander Russell
Illinois J. Math. 52(4): 1315-1324 (Winter 2008). DOI: 10.1215/ijm/1258554365


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 $\mu(\operatorname{\mathbf{supp}} f)\cdot \sum_{\rho\in\hat G} d_\rho\operatorname{\mathbf{rank}} \hat f(\rho) \geq1$.


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Gorjan Alagic. Alexander Russell. "Uncertainty principles for compact groups." Illinois J. Math. 52 (4) 1315 - 1324, Winter 2008.


Published: Winter 2008
First available in Project Euclid: 18 November 2009

zbMATH: 1184.43004
MathSciNet: MR2595770
Digital Object Identifier: 10.1215/ijm/1258554365

Primary: 43A30 , ‎43A65 , 43A77

Rights: Copyright © 2008 University of Illinois at Urbana-Champaign

Vol.52 • No. 4 • Winter 2008
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