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While inverse estimates in the context of radial basis function approximation on boundary-free domains have been known for at least ten years, such theorems for the more important and difficult setting of bounded domains have been notably absent. This article develops inverse estimates for finite dimensional spaces arising in radial basis function approximation and meshless methods. The inverse estimates we consider control Sobolev norms of linear combinations of a localized basis by the $L_p$ norm over a bounded domain. The localized basis is generated by forming local Lagrange functions for certain types of RBFs (namely Matern and surface spline RBFs). In this way it extends the boundary-free construction of Fuselier, Hangelbroek, Narcowich, Ward and Wright.

An inverse theorem for compact Lipschitz regions in ℝ^{𝕕} using localized kernel bases