Interpolation on the torus using \(sk\)-splines with number theoretic knots (Q1288878)

For a fixed, continuous, periodic kernel \(K\), an \(sk\)-spline is a function of the form \(sk(x)=c_0+\sum_{i=1}^nc_iK(x-x_i)\). This paper considers a generalization of the univariate \(sk\)-spline to the \(d\)-dimensional torus (\(d\geq 2\)), and gives almost optimal error estimates of the same order, in power scale, as best trigonometric approximation of Sobolev's classes in \(L_q\). An important component of this method is that the interpolation nodes are generated using number theoretic ideas.