Computational Analysis Seminar

Wolfgang Erb

(TU Munich)

"Uncertainty principles on Riemannian manifolds"

Date: Tue, March 2, 2010
Time: 15:45
Place: Research I Seminar Room

Abstract: Based on a result of Rösler and Voit for ultraspherical polynomials, we derive an uncertainty principle for Riemannian manifolds M. Similar as for the classical Heisenberg principle or the Breitenberger principle on the unit circle, the proof of the uncertainty rests upon a commutator relation of two Hilbert space operators. As a frequency operator, we will construct a special differential-difference operator, a so called Dunkl operator, which plays the role of a generalized root of the radial part of the Laplace-Beltrami operator on M. Subsequently, we will show with a family of Gaussian-like functions that the deduced uncertainty inequalities are in fact asymptotically sharp.

Particularly interesting in these uncertainty relations is the formula for the space variance. For two-point homogeneous compact manifolds, we are able to find those harmonic polynomials of a fixed degree n that minimize the term for the space variance, i.e. those polynomials that are best localised in space with respect to the given uncertainty principle. Finally, we investigate the space-frequency behavior of some well-known approximation kernels on the compact two-point homogeneous spaces.