Approximation of Differentiable and Analytic Functions by Splines on the Torus

We consider a continuous kernel K on the d-dimensional torus and we study the rate of convergence in Lq, of functions of the type f=K*ϕ where ϕ is a function in a Lp-space, by its interpolating sk-splines. The rate of convergence is obtained for functions in classes of Sobolev, of infinitely differentiable functions and of analytic functions, and it provides optimal error estimates of the same order as best trigonometric approximation, in several cases.