Illinois Journal of Mathematics

Uncertainty principles for compact groups

Gorjan Alagic and Alexander Russell

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

First available in Project Euclid: 18 November 2009

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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


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

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