Direct and Inverse Results on Bounded Domains for Meshless Methods via   Localized Bases on Manifolds
  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. 
Thomas Hangelbroek, Francis J. Narcowich, Christian Rieger, Joseph D.   Ward
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