Generalised sk-Spline Interpolation on Compact Abelian Groups

The notion of sk-spline is generalised to arbitrary compact Abelian groups. A class of conditionally positive definite kernels on the group is identified, and a subclass corresponding to the generalised sk-spline is used for constructing interpolants, on scattered data, to continuous functions on the group. The special case ofd-dimensional torus is considered and convergence rates are proved when the kernel is a product of one-dimensional kernels, and the data are gridded.