The Density of Translates of Zonal Kernels on Compact Homogeneous Spaces

Abstract

Compact manifolds embedded in Euclidean space which have a transitive group G of linear isometries, such as the spheres with the rotation group or the “flat” tori with the group of rotations in each coordinate direction, admit a natural notion of a continuous G-invariant kernel function k(x, y), which generalizes the idea of a radial or distance-dependent function on the spheres and tori. In connection with a study of quasi-interpolation on these spaces, we have reproved and extended results of Sun for the spheres to characterize those kernels for which the span of the translates, ∑ank(x, yn), is dense in the continuous functions. The essence of the characterization is that the integral operator with G-invariant kernel k(x, y) must be non-singular when restricted to the space of nth degree polynomial functions. This requires that the polynomials be invariant under all such linear operators, which is true for many compact homogeneous M including the spheres, tori, and others. In fact the non-singularity must hold only on any finite-dimensional space of zonal polynomials, those which are pointwise fixed by the subgroup of all isometries fixing a single point on M. In practical terms this later condition is verified by choosing one point on the manifold (the north pole on the spheres or the identity element on the flat tori), picking some basis for the polynomials of given degree which are fixed under the isometries leaving the pole invariant, and testing whether the integral operator (which leaves this space invariant) has a non-singular matrix. In all the cases considered, where the family of G-invariant kernels lead to commuting operator families, there are diagonalizing bases for this restricted operator, and the characterization becomes the non-vanishing of the appropriate Fourier-like coefficients.