Optimal Approximation by sk-Splines on the Torus

Fixed a continuous kernel K on the d-dimensional torus, we consider a generalization of the univariate sk-spline to the torus, associated with the kernel K. We prove an estimate which provides the rate of convergence of a given function by its interpolating sk-splines, in the norm of Lq for functions of convolution type f=K*ϕ where ϕ is a function in a Lp-space. The rate of convergence is obtained for functions f in Sobolev classes and this rate gives optimal error estimate of the same order as best trigonometric approximation, in a special case.