Optimal Approximation by $sk$-Splines on the Torus

Abstract

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. It is proved an estimate which provides the rate of convergence of a given function by its interpolating sk-splines, in the norm of L q for functions of the type f = K * ϕ where ϕ ∈ L p and 1 ≤ p ≤ 2 ≤ q ≤ ∞, 1/p − 1/q ≥ 1/2. 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.