This article develops direct and inverse estimates for certain finitedimensional spaces arising in kernel approximation. Both the direct and inverseestimates are based on approximation spaces spanned by local Lagrange functionswhich are spatially highly localized. The construction of such functions iscomputationally efficient and generalizes the construction given by the authorsfor restricted surface splines on $\mathbb{R}^d$. The kernels for which thetheory applies includes the Sobolev-Mat\'ern kernels for closed, compact,connected, $C^\infty$ Riemannian manifolds.
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机译：本文针对由核近似产生的某些有限维空间开发了直接和逆估计。正和反估计均基于空间上高度局部化的局部拉格朗日函数所跨越的近似空间。这些函数的构造在计算上是有效的，并且推广了作者针对$ \ mathbb {R} ^ d $上的受限曲面样条给出的构造。适用于该理论的内核包括Sobolev-Mat'ern内核，用于封闭，紧凑，连通，$ C ^ \ infty $黎曼流形。
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