Optimally space localised polynomial approximation kernels on compact two-point homogeneous spaces.

Based on uncertainty principles for orthogonal expansions in terms of Gegenbauer and Jacobi polynomials and, in particular, the Dunkl operators used therein, we derive an uncertainty relation for compact two-point homogeneous spaces. Particularly interesting in these uncertainty relations is the formula for the space variance. Investigating this formula in more detail, we were 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, using these optimally space localised polynomials as approximation kernels on the compact two-point homogeneous spaces, we investigate some of their properties