In this paper we provide a priori error estimates in standard Sobolev
(semi-)norms for approximation in spline spaces of maximal smoothness on
arbitrary grids. The error estimates are expressed in terms of a power of the
maximal grid spacing, an appropriate derivative of the function to be
approximated, and an explicit constant which is, in many cases, sharp. Some of
these error estimates also hold in proper spline subspaces, which additionally
enjoy inverse inequalities. Furthermore, we address spline approximation of
eigenfunctions of a large class of differential operators, with a particular
focus on the special case of periodic splines. The results of this paper can be
used to theoretically explain the benefits of spline approximation under
k-refinement by isogeometric discretization methods. They also form a
theoretical foundation for the outperformance of smooth spline discretizations
of eigenvalue problems that has been numerically observed in the literature,
and for optimality of geometric multigrid solvers in the isogeometric analysis
context.Comment: 31 pages, 2 figures. Fixed a typo. Article published in M3A
