In this paper we provide a priori error estimates with explicit constants for both the $$L^2$$ L 2 -projection and the Ritz projection onto spline spaces of arbitrary smoothness defined on arbitrary grids. This extends the results recently obtained for spline spaces of maximal smoothness. The presented error estimates are in agreement with the numerical evidence found in the literature that smoother spline spaces exhibit a better approximation behavior per degree of freedom, even for low smoothness of the functions to be approximated. First we introduce results for univariate spline spaces, and then we address multivariate tensor-product spline spaces and isogeometric spline spaces generated by means of a mapped geometry, both in the single-patch and in the multi-patch case.

中文翻译：

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等几何分析中任意平滑度样条近似的显式误差估计

在本文中，我们为 $$L^2$$L 2 投影和 Ritz 投影到任意网格上定义的任意平滑度的样条空间提供了带有显式常量的先验误差估计。这扩展了最近获得的最大平滑样条空间的结果。所提供的误差估计与文献中发现的数值证据一致，即更平滑的样条空间在每个自由度上表现出更好的逼近行为，即使要逼近的函数的平滑度较低。首先，我们介绍单变量样条空间的结果，然后我们解决了通过映射几何生成的多元张量积样条空间和等几何样条空间，无论是在单面片情况下还是在多面片情况下。