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In this paper we provide a priori error estimates in standard Sobolev(semi-)norms for approximation in spline spaces of maximal smoothness onarbitrary grids. The error estimates are expressed in terms of a power of themaximal grid spacing, an appropriate derivative of the function to beapproximated, and an explicit constant which is, in many cases, sharp. Some ofthese error estimates also hold in proper spline subspaces, which additionallyenjoy inverse inequalities. Furthermore, we address spline approximation ofeigenfunctions of a large class of differential operators, with a particularfocus on the special case of periodic splines. The results of this paper can beused to theoretically explain the benefits of spline approximation under$k$-refinement by isogeometric discretization methods. They also form atheoretical foundation for the outperformance of smooth spline discretizationsof eigenvalue problems that has been numerically observed in the literature,and for optimality of geometric multigrid solvers in the isogeometric analysiscontext.

Sharp error estimates for spline approximation: explicit constants, $n$-widths, and eigenfunction convergence