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In this paper we provide a priori error estimates with explicit constants forboth the $L^2$-projection and the Ritz projection onto spline spaces ofarbitrary smoothness defined on arbitrary grids. This extends the resultsrecently obtained for spline spaces of maximal smoothness. The presented errorestimates are in agreement with the numerical evidence found in the literaturethat smoother spline spaces exhibit a better approximation behavior per degreeof freedom, even for low smoothness of the functions to be approximated. Firstwe introduce results for univariate spline spaces, and then we addressmultivariate tensor-product spline spaces and isogeometric spline spacesgenerated by means of a mapped geometry, both in the single-patch and in themulti-patch case.

Explicit error estimates for spline approximation of arbitrary smoothness in isogeometric analysis