The density of translates of zonal kernels on compact homogeneous spaces
Classes of functions were studied which relate to approximation or to analysis of the approximating properties of other classes of functions. For quasi-interpolation the real symmetric (or self-adjoint) kernel \(k(x,y)\) and the approximating properties of linear combinations \(\sum a_i k(x,y)\) were analyzed. To this aim the density properties of such linear combinations are needed. The paper investigates the density properties of such linear combinations and shows how to use the density of polynomials to determine the kernel \({\mathcal K}\) for specific \(k\) forms. The kernel function generalizes the idea of a radial or distance-dependent function on spheres or tori. The \(G\)-invariant polynomial decomposition of \(L^2\) and the reduction of zonal kernel operators was discussed.
scientific article; zbMATH DE number 1443204