Approximation by $sk$-spline interpolants in $L_q$

Rates of convergence of $sk$-spline interpolants are established on a wide range of sets of smooth functions $\RR \oplus K \ast U_{p}$ in $L_q$ (including sets of differentiable, infinitely differentiable, analytic and entire functions) for any $1 < p, q < \infty$. We show that on such function classes $sk$-spline interpolants with equidistant knots and points of interpolation give the same order of convergence as the subspace of trigonometric polynomials of the same dimension.