  This article develops direct and inverse estimates for certain finite
dimensional spaces arising in kernel approximation. Both the direct and inverse
estimates are based on approximation spaces spanned by local Lagrange functions
which are spatially highly localized. The construction of such functions is
computationally efficient and generalizes the construction given by the authors
for restricted surface splines on $\mathbb{R}^d$. The kernels for which the
theory applies includes the Sobolev-Mat\'ern kernels for closed, compact,
connected, $C^\infty$ Riemannian manifolds.
