Interpolation on the Torus using sk-Splines with number theoretic knots

Abstract

For a fixed, continuous, periodic kernelK, an sk-spline is a function of the form sk(x)=c0+∑ni=1ciK(x-xi). In this paper we consider a generalization of the univariate sk-spline to thed-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 inLq. An important component of our method is that the interpolation nodes are generated using number theoretic ideas. © 1999 Academic Press.