Uncertainty principles on compact Riemannian manifolds

Abstract

Based on a result of Rösler 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.