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Fixed a continuous kernel K on the $d$-dimensional torus, we consider ageneralization of the univariate $sk$-spline to the torus, associated with thekernel K. It is proved an estimate which provides the rate of convergence of agiven function by its interpolating $sk$-splines, in the norm of $L^q$ forfunctions of the type $f=K*\varphi$ where $\varphi \in L^p$ and $1\leq p \leq 2\leq q \leq \infty,\ 1/p - 1/q \geq 1/2$. The rate of convergence is obtainedfor functions f in Sobolev classes and this rate gives optimal error estimateof the same order as best trigonometric approximation, in a special case.

Optimal Approximation by $sk$-Splines on the Torus