  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 $f=K*\varphi$ 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 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.
