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Erb, W.

Uncertainty principles on compact Riemannian manifolds.

Appl. Comput. Harmon. Anal. 29, 182-197 (2010)

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Open Access Green as soon as Postprint is submitted to ZB.

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Based on a result of Rosier and Voit for ultraspherical polynomials, we derive an uncertainty principle for compact Riemannian manifolds M. The frequency variance of a function in L-2(M) is therein defined by means of the radial part of the Laplace-Beltrami operator. The proof of the uncertainty rests upon Dunkl theory. In particular, a special differential-difference operator is constructed which plays the role of a generalized root of the radial Laplacian. Subsequently, we prove with a family of Gaussian-like functions that the deduced uncertainty is asymptotically sharp. Finally, we specify in more detail the uncertainty principles for well-known manifolds like the d-dimensional unit sphere and the real projective space.

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