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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 of d-dimensional torus is considered and convergence rates are proved when the kernel is a product of one-dimensional kernels, and the data are gridded. 1999Academic Press

For a fixed, continuous, periodic kernel K, an sk-spline is a function of the form sk(x)=c 0 + n i=1 c i K(x&x i ). In this paper we consider a generalization of the univariate sk-spline to the d-dimensional torus (d 2), and give almost optimal error estimates of the same order, in power scale, as best trigonometric approximation on Sobolev's classes in L q . An important component of our method is that the interpolation nodes are generated using number theoretic ideas. Academic Press